Studies in Applied Welfare Analysis : Papers from the Third ECINEQ Meeting [1 ed.] 9780857241467, 9780857241450

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STUDIES IN APPLIED WELFARE ANALYSIS: PAPERS FROM THE THIRD ECINEQ MEETING

RESEARCH ON ECONOMIC INEQUALITY Series Editor: John A. Bishop

RESEARCH ON ECONOMIC INEQUALITY VOLUME 18

STUDIES IN APPLIED WELFARE ANALYSIS: PAPERS FROM THE THIRD ECINEQ MEETING EDITED BY

JOHN A. BISHOP East Carolina University, Greenville, NC, USA

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

Emerald Group Publishing Limited Howard House, Wagon Lane, Bingley BD16 1WA, UK First edition 2010 Copyright r 2010 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: 978-0-85724-145-0 ISSN: 1049-2585 (Series)

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CONTENTS LIST OF CONTRIBUTORS

vii

INTRODUCTION

xi

CHAPTER 1 REFINING THE BASIC NEEDS APPROACH: A MULTIDIMENSIONAL ANALYSIS OF POVERTY IN LATIN AMERICA Marı´a Emma Santos, Marı´a Ana Lugo, Luis Felipe Lo´pez-Calva, Guillermo Cruces and Diego Battisto´n

1

CHAPTER 2 MULTIDIMENSIONAL POVERTY AMONG CHILDREN IN URUGUAY Vero´nica Amarante, Rodrigo Arim and Andrea Vigorito

31

CHAPTER 3 EXPLORING INTERGENERATIONAL EDUCATIONAL MOBILITY IN ARGENTINA Ana Ine´s Navarro

55

CHAPTER 4 ARE INFORMALITY AND POVERTY DYNAMICALLY INTERRELATED? EVIDENCE FROM ARGENTINA Francesco Devicienti, Fernando Groisman and Ambra Poggi CHAPTER 5 INEQUALITY EVOLUTION IN BRAZIL: THE ROLE OF CASH TRANSFER PROGRAMS AND OTHER INCOME SOURCES Luiz Guilherme Scorzafave and E´rica Marina Carvalho de Lima CHAPTER 6 INEQUALITY REDUCING TAXATION RECONSIDERED Udo Ebert v

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107

131

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CONTENTS

CHAPTER 7 COUNTING POVERTY ORDERINGS AND DEPRIVATION CURVES Ma Casilda Lasso de la Vega

153

CHAPTER 8 TESTING FOR MOBILITY DOMINANCE Ye´le´ Maweki Batana and Jean-Yves Duclos

173

CHAPTER 9 DISTRIBUTIONAL CHANGE, REFERENCE GROUPS, AND THE MEASUREMENT OF RELATIVE DEPRIVATION Jacques Silber and Paolo Verme

197

CHAPTER 10 ECONOMETRIC IDENTIFICATION OF THE COST OF MAINTAINING A CHILD Martina Menon and Federico Perali

219

CHAPTER 11 RISING INCOMES AND NUTRITIONAL INEQUALITY IN CHINA John A. Bishop, Haiyong Liu and Buhong Zheng

257

LIST OF CONTRIBUTORS Vero´nica Amarante

Instituto de Economı´ a, Universidad de la Repu´blica, Uruguay

Rodrigo Arim

Instituto de Economı´ a, Universidad de la Repu´blica, Uruguay

Ye´le´ Maweki Batana

PEP, De´partement d’e´conomique, Universite´ Laval, Laval, Que., Canada

Diego Battisto´n

Centro de Estudios Distributivos, Laborales y Sociales (CEDLAS), FCE, Universidad Nacional de La Plata (UNLP); Consejo Nacional de Investigaciones Cientificas y Te´cnicas (CONICET), Argentina

John A. Bishop

Department of Economics, East Carolina University, Greenville, NC, USA

E´rica Marina Carvalho de Lima

Department of Economics, University of Sa˜o Paulo, Ribeira˜o Preto, SP, Brazil

Guillermo Cruces

Centro de Estudios Distributivos, Laborales y Sociales (CEDLAS), FCE, Universidad Nacional de La Plata (UNLP); Consejo Nacional de Investigaciones Cientificas y Te´cnicas (CONICET), Argentina

Francesco Devicienti

Dipartimento di Scienze Economiche e Finanziarie ‘G. Prato’, Universita` di Torino, Torino, Italy; LABORatorio R. Revelli, Collegio Carlo Alberto, Torino, Italy De´partement d’e´conomique and CIRPE´E, Universite´ Laval, Laval, Que., Canada; Institut d’Ana`lisi Econo`mica (CSIC), Barcelona, Spain

Jean-Yves Duclos

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viii

LIST OF CONTRIBUTORS

Udo Ebert

Department of Economics, University of Oldenburg, Oldenburg, Germany

Fernando Groisman

University of Buenos Aires (UBA) and National Council of Scientific and Technical Research (CONICET), Repu´blica Argentina

Ma Casilda Lasso de la Vega

Department of Applied Economics IV, University of the Basque Country, Bilbao, Spain

Haiyong Liu

Department of Economics, East Carolina University, Greenville, NC, USA

Luis Felipe Lo´pez-Calva

United Nations Development Programme, Regional Bureau for Latin America and the Caribbean, New York, NY, USA

Marı´a Ana Lugo

Department of Economics, University of Oxford, UK

Martina Menon

Department of Economics, University of Verona, Verona, Italy

Ana Ine´s Navarro

Departamento de Economı´ a, Facultad de Ciencias Empresariales, Universidad Austral, Rosario, Santa Fe´, Argentina

Federico Perali

Department of Economics, University of Verona, Verona, Italy

Ambra Poggi

Department of Economics, University of Milan Bicocca, Milan, Italy; LABORatorio R. Revelli, Collegio Carlo Alberto, Torino, Italy

Marı´a Emma Santos

Oxford Poverty and Human Development Initiative (OPHI), University of Oxford, UK; Consejo Nacional de Investigaciones Cientı´ ficas y Te´cnicas (CONICET) – Universidad Nacional del Sur, Argentina

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List of Contributors

Luiz Guilherme Scorzafave

Department of Economics, University of Sa˜o Paulo, Ribeira˜o Preto, SP, Brazil

Jacques Silber

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

Paolo Verme

Department of Economics, ‘‘S. Cognetti de Martiis’’, University of Torino, Torino, Italy

Andrea Vigorito

Instituto de Economı´ a, Universidad de la Repu´blica, Uruguay

Buhong Zheng

Department of Economics, University of Colorado, Denver, CO, USA

INTRODUCTION Volume 18 of Research on Economic Inequality contains selected papers from the Society for the Study of Economic Inequality’s Third Annual Meeting (July, 2009) in Buenos Aires, Argentina. The volume collects eleven papers, five of which focus on inequality and poverty in Latin America. The Latin American papers address basic needs and poverty, multidimensional poverty, educational mobility, poverty dynamics, and the role of cash transfer programs in addressing inequality. The second half of Volume 18 collects other papers by ECINEQ members. The topics covered include taxation and inequality, evaluating poverty orderings, testing for mobility dominance, measuring relative deprivation, estimation of the costs of maintaining a child, and evaluating nutritional inequality. In Chapter 1, Marı´ a Emma Santos, Marı´ a Ana Lugo, Luis Felipe Lo´pezCalva, Guillermo Cruces, and Diego Battisto´n introduce a hybrid approach to multidimensional poverty measurement by including income into the well-established ‘‘unsatisfied basic needs approach.’’ They argue that adding income has several advantages: it can act as a surrogate for unmeasured dimensions of poverty, it has a low correlation with currently included factors, and represents a household’s freedom to choose among bundles of goods. The authors implement their procedure by providing ‘‘a set of comparable poverty estimates for six Latin American countries between 1992 and 2006.’’ They find enormous improvement in poverty reduction in all six countries, yet there are still troublingly high levels of rural poverty. Chapter 2 also studies poverty in a multidimensional framework. Vero´nica Amarante, Rodrigo Arim, and Andrea Vigorito consider four attributes (income, nutrition, household crowding, and education) and three possible aggregation procedures (two based on generalized FGT indices, and one based on fuzzy sets). Using a new dataset (ESN06) especially designed to study multidimensional poverty the authors estimate the degree of child poverty in Uruguay. They find very low correlations among the four attributes and highlight the finding that an income-only approach can neglect children with important social deficits. They caution that the ‘‘cardinalizations of the three families of indexes y yield very different results.’’ xi

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INTRODUCTION

Ana Ine´s Navarro (Chapter 3) also considers the role education plays in the determination of individual well-being. Her chapter investigates the degree of educational mobility in Argentina including young adults as well as teenage children. Exploiting new retrospective data the chapter contributes to the literature on how to measure mobility when no long panel datasets exist. Navarro finds that educational mobility is lower than previous studies without including young adults have implied. Chapter 4 also uses data from Argentina. Francesco Devicienti, Fernando Groisman, and Ambra Poggi propose a random effects probit model to study the causality between informal employment and poverty. The authors find that past poverty has a positive impact on the likelihood of current employment being in the informal sector. At the same time an individual’s past informal employment also has a positive impact on the risk of being poor. Additionally, their findings suggest that for Argentina, poverty and informal employment are highly persistent processes. The authors provide some policy recommendations to overcome the ‘‘poverty and informality traps.’’ Luiz Guilherme Scorzafave and E´rica Marina Carvalho de Lima (Chapter 5) offer the most detailed income source decomposition to date for Brazilian incomes. Their 1993 to 2007 study includes the effects of an expanded cash transfer program and efforts to reduce child labor in Brazil. Among the interesting findings of this chapter is that informal sector labor income reduces inequality while public sector earnings increase inequality. The role of pensions on Brazilian inequality is growing along with an aging population. Social transfers appear to have a limited but positive effect on overall income inequality. In Chapter 6, Udo Ebert revisits the notion of inequality reducing taxation. He clearly points out if one accepts the relative definition of inequality that this is a largely settled issue of Lorenz dominance of after-tax incomes over pre-tax incomes. However, empirical studies question whether people evaluate income distributions according to the accepted criterion. Furthermore, writers such as Kolm have long considered alternative views of inequality. This chapter shows how ordinary Lorenz dominance can be modified to test for tax progressivity under assumptions other than relative inequality. As highlighted in the first two chapters, most researchers now recognize that poverty and deprivation are best studied in a multidimensional framework. This necessitates the need for widely accepted unambiguous multidimensional poverty orderings. In Chapter 7, Casilda Lasso de la Vega addresses this problem and examines dominance conditions for

Introduction

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multidimensional poverty in a counting framework. In particular, she shows that the multidimensional headcount ratio and adjusted headcount ratios correspond to the first and second degree stochastic dominance conditions. The author introduces ‘‘dimension deprivation curves’’ to test for multidimensional poverty dominance. Ye´le´ Maweki Batana and Jean-Yves Duclos add to the literature on tests for mobility dominance in Chapter 8. Their chapter proposes a test for making partial ordering of mobility based on the null of nondominance. Two types of mobility are considered, absolute mobility and transition matrix mobility. An empirical example is provided that suggests that it ‘‘may be difficult to obtain rankings of distributions that are robust over wide classes of mobility indices.’’ Formal statistics tests are advocated when using sample rankings to infer population rankings. In Chapter 9, Jacques Silber and Paolo Verme suggest that economists have not paid enough attention to the concept of ‘‘reference group’’ in their treatment of relative deprivation. Their chapter explicitly integrates the concept of reference group in the measurement of relative deprivation. The authors propose a measure of relative deprivation that compares the actual income of an individual to his expected income, which based on a set of observable characteristics of a similar set of people. A unique aspect of their approach is that it takes account not only the gap between actual and expected income but also between an individual’s actual and expected ranks. The method is applied to Moldova, the poorest country in Europe. Finally, the authors explore the use of the method to study wage deprivation across genders. Martina Menon and Federico Perali (Chapter 10) investigate the cost of maintaining a child within a demand system modified to include demographic characteristics. They explicitly differentiate between the cost of raising a child, which varies with income, and the cost of maintaining a child, which does not. The authors contribute to the literature in two important ways. First, their work clarifies the important issues related to the econometric identification of equivalence scales. Second, it separates differences in needs from differences in lifestyle (or household technologies). In the final chapter, John A. Bishop, Haiyong Liu, and Buhong Zheng find that the doubling of Chinese incomes between 1991 and 2004 did not improve the nutritional outcomes. Using Kakwani’s method to estimate nutrient elasticity’s they find that the income elasticities for calorie and protein intakes are generally zero, while fat is a normal good. The authors question the conventional wisdom that improving the status of women

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INTRODUCTION

improves household nutrition with a finding that the nutrient elasticities with respect to women’s schooling or women’s wages are also zero. I would like to thank the many anonymous referees for their timely and thoughtful comments. Ms Cindy Mills of East Carolina University’s Economics Department contributed valuable editorial assistance. Mr. Chris Hart, Commissioning Editor at Emerald provided me with patient encouragement. John A. Bishop Editor

CHAPTER 1 REFINING THE BASIC NEEDS APPROACH: A MULTIDIMENSIONAL ANALYSIS OF POVERTY IN LATIN AMERICA Marı´ a Emma Santos, Marı´ a Ana Lugo, Luis Felipe Lo´pez-Calva, Guillermo Cruces and Diego Battisto´n ABSTRACT Latin America has a longstanding tradition in multidimensional poverty measurement through the unsatisfied basic needs (UBN) approach. However, the method has been criticized on several grounds, including the selection of indicators, the implicit weighting scheme and the aggregation methodology, among others. The estimates by the UBN approach have traditionally been complemented (or replaced) with income poverty estimates. Under the premise that poverty is inherently multidimensional, in this chapter we propose three methodological refinements to the UBN approach. Using the proposed methodology we provide a set of comparable poverty estimates for six Latin American countries between 1992 and 2006.

Studies in Applied Welfare Analysis: Papers from the Third ECINEQ Meeting Research on Economic Inequality, Volume 18, 1–29 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2010)0000018004

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1. INTRODUCTION Poverty is being increasingly recognized as an inherently multidimensional phenomenon. Welfarists stress both the existence of market imperfections and incompleteness and the lack of perfect correlation between relevant dimensions of well-being (Atkinson, 2003; Bourguignon & Chakravarty, 2003; Duclos & Araar, 2006). Nonwelfarists point to the need to move away from the space of utilities to a different and usually wider space, where multiple dimensions are both instrumentally and intrinsically important. Among the nonwelfarists, there are two main strands: the basic needs approach and the capability approach (Duclos & Araar, 2006). The first approach focuses on a set of primary goods that are constituent elements of well-being and considered necessary to live a good life. ‘‘Basic needs may be interpreted in terms of minimum specified quantities of such things as food, clothing, shelter, water and sanitation that are necessary to prevent ill health, undernourishment, and the like’’ (Streeten, Burki, Haq, Hicks, & Stewart, 1981, p. 25). ‘‘A basic needs approach to development (y) tries to ensure access to particular resources (such as caloric adequacy) for particular groups (defined by age, sex, or activity) that are deficient in these resources’’ (Streeten et al., 1981, pp. 33–34). The second approach, introduced by Sen (1992, 2009), argues that attention should be shifted from the means of living to the actual opportunities a person has, and therefore argues that the relevant space of well-being should be the set of capabilities. These are defined as ‘‘the various combinations of functionings (beings and doings) that the person can achieve. Capability is, thus, a set of vectors of functionings, reflecting the person’s freedom to lead one type of life or another y to choose from possible livings’’ (Sen, 1992, p. 40). The actual implementation of the capability approach poses some difficulties since one would need to observe the whole set of potential functionings. A ‘‘second best’’ alternative to implement this approach is to consider the actual functionings people have and evaluate poverty according to that. This presents a strong difference with the basic needs approach in that it requires evaluating the functioning and not the access to the resource that could eventually permit such functioning. For example, it would require looking into the actual nutritional status of each household member, and not the food consumption level, evaluating their cognitive skills and not whether they have had access to schooling. We understand that this is indeed the desirable approach to follow and we claim that surveys should be redesigned in order to capture actual functionings rather than access to resources. Even more, they should eventually aim at measuring capability sets. However, so

Multidimensional Analysis of Poverty in Latin America

3

far, the availability of data poses serious constraints to what can be done, and that has probably been the main reason for the widespread use of the basic needs approach over the capability approach. Moreover, very frequently, and as it is the case of this chapter, the basic needs approach can be seen as a (very) imperfect proxy of the capability approach. In this context, and until we can count with better sources of information that allow identifying actual functionings, we propose three specific refinements to the basic needs approach and we empirically implement them in six Latin American countries, Argentina, Brazil, Chile, El Salvador, Mexico, and Uruguay, over a period of 14 years (1992–2006).

2. THE BASIC NEEDS AND THE INCOME APPROACH TO POVERTY MEASUREMENT IN THE REGION Latin America has a longstanding tradition in multidimensional poverty measurement making use of the basic needs approach. Promoted in the region by the United Nation’s Economic Commission for Latin America and the Caribbean (ECLAC), the approach was employed extensively since the beginning of the 1980s (Feres & Mancero, 2001). In a context where household surveys were not as widespread as nowadays and income and consumption were difficult variables to measure, the census-based unsatisfied basic needs (UBN) measures became the poverty analysis tool of widespread use in the region, while income poverty studies were restricted to specific surveys and individual studies.1 Most commonly, the UBN approach combines population census information on the condition of households (construction material and number of people per room), access to sanitary services, children attending school and education, and economic capacity of household members (generally the household head). The UBN indicators are often reported by administrative areas in terms of the proportion of households unable to satisfy one, two, three, or more basic needs, and are often presented using poverty maps (Feres & Mancero, 2001). Thus, in practice, the approach does not offer a unique index but rather the percentage of population with different number of unmet basic needs. As household surveys started to be regularly administered and progressively available to the public, distributional studies using income became widespread as well as official income poverty estimates started to be reported periodically. Since then both methods have coexisted.

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The UBN method has also been called the ‘‘direct method’’ to measure poverty since it looks directly at whether certain needs are met or not, as opposed to the ‘‘indirect (or poverty line) method,’’ which looks at the income level and compares it to the income level necessary to achieve these needs (Feres & Mancero, 2001). As argued in the Introduction, in practice, the basic needs approach considers access to resources (means rather than actual ends), so the reference as a direct method is arguable. It is however true that it looks at a wider range of resources than the income approach, and in particular it incorporates access to services such as education, sanitation, and water. Given the existence of imperfect markets, capturing access to basic services represents a value-added over the income approach. Consistent with a multidimensional understanding of poverty, it has been long argued that both methods capture partial aspects of poverty, that both income as well as the UBN indicators are relevant for assessing well-being, and that there are significant errors in targeting the poor (either of inclusion or exclusion) when only one of them is used.2 Therefore, one of the contributions of this chapter is to expand the UBN approach into a hybrid one, incorporating income. The reasons for doing so are that income (a) can act as a surrogate for all the other nonconsidered dimensions due to data restrictions,3 (b) it has been found to have relatively low correlations with the other considered indicators,4 (c) even when merely a means, having purchasing power provides the household with some freedom to choose the bundle of goods. We think that having a set of relevant indicators combined into one single measure can prove helpful for monitoring poverty and for policy design. In terms of the UBN indicators themselves, there have been critiques arguing the selection to be arbitrary. Clearly in any multidimensional poverty measure (and in any composite indicator in general), the selection of indicators will be problematic, and some selection criteria are preferable to others from a methodological point of view (see Alkire, 2008 on such different methods). While the selection of indicators seems to have been originally highly influenced by data availability, it gained some form of public consensus over the years as estimates were periodically released. Drawing on such gained consensus, we use similar indicators to those of the UBN (with some minor adjustments explained in Section 3.2). However, we understand that this set of indicators is indeed very limited. We therefore advocate for an improvement in the collection of survey data that allows capturing other key variables today completely absent (such as health), as well as progressing toward the measurement of people’s functionings. We also consider, as suggested by the capability approach and especially by

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Sen (2009), that it would be important to reexamine and evaluate through public reasoning the dimensions to be considered in a multidimensional poverty index for the region. This could guide the subsequent improvement in data collection. The two other methodological contributions of the chapter derive from the measure used. On the one hand, by using the first measure of the family of multidimensional poverty measures proposed by Alkire and Foster (2007), one can account not only for the poverty incidence (the percentage of people that are multidimensionally poor) but also for the breadth of poverty, that is, on average, the proportion of the considered indicators in which the poor are deprived. This represents a strong advantage over the usual multidimensional headcounts reported by the UBN approach. In addition, the measure used allows for alternative weighting systems which gives flexibility to the index, providing another strong advantage over the traditional UBN headcounts which had an implicit weighting system sometimes criticized. In summary, the chapter proposes three specific refinements to the UBN methodology: (1) incorporating income as a proxy indicator for other nonincluded dimensions, (2) incorporating the breadth of poverty, (3) allowing for a flexible weighting system. Last but not least, the chapter contributes with empirical evidence on multidimensional poverty in the region. Six countries are covered with five observations over a period of 14 years (1992–2006). The countries under analysis are Argentina, Brazil, Chile, El Salvador, Mexico, and Uruguay, which altogether cover about 64 percent of the total population in Latin America in 2006. The existing studies in this specific area are limited in the region. Amarante, Arim, and Vigorito (2008) and Arim and Vigorito (2007) study the case of Uruguay. The first compares three alternative methodologies; namely, Bourguignon and Chakravarty (2003) indices, the fuzzy sets approach, and the stochastic dominance approach developed by Duclos, Sahn, and Younger (2006). The second one only uses Bourguignon and Chakravarty (2003) family of indices. They find that multidimensional poverty has decreased and that its evolution over time is smoother than that of the income poverty, as the first one includes less volatile indicators. Conconi and Ham (2007) also employ Bourguignon and Chakravarty indices (but using a relative approach to measurement) in a study on Argentina for the period around the last financial crisis (1998–2002). The authors find that the increased deprivation in employment and income is behind the rising trend in poverty in the study period. A number of other studies propose alternative measures of multidimensional poverty to study Latin American countries. Paes de Barros, De

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Carvalho, and Franco (2006) suggest using a weighted average of dichotomous indicators of deprivations as a multidimensional poverty measure for Brazil. Ballon and Krishnakumar (2008) develop a multidimensional capability deprivation index based on structural equation modeling and apply the method to a household survey dataset for Bolivia in 2002, focusing on two children capability domains: knowledge and living conditions. The authors find a strong interdependence between the two studied dimensions. Lo´pez-Calva and Rodrı´ guez-Chamussy (2005) and Lo´pez-Calva and Ortiz-Jua´rez (2009) have also adopted a multidimensional approach to studying poverty in Mexico. They estimate the magnitude of the ‘‘exclusion error’’ in targeting programes when a monetary measure is adopted instead of a multidimensional one. They find a large variability in the exclusion error depending on the selected criterion to identify the multidimensionally poor (union vs. intersection, explained in the next section). Finally, since 2004, the Programa Observatorio de la Deuda Social Argentina (Pontificia Universidad Cato´lica Argentina) implements a survey that collects information on housing conditions, health and subsistence, and computes a composite indicator of deprivation constructed using principal components analysis. The results of the present chapter are quite encouraging: a decreasing trend in multidimensional poverty is observed, both in incidence and breadth of poverty. However, strong disparities between countries as well as between urban and rural areas within countries remain, which demand renewed efforts to reduce poverty in the region.

3. METHODOLOGY 3.1. Multidimensional Poverty Measurement: From H to M0 In this section, we describe the measures used by the UBN approach and present a simple way in which the UBN index could be improved by using one of the members of the family of multidimensional poverty measures proposed by Alkire and Foster (2007). In the multidimensional context, distributional data are presented in the form of a matrix of size n  d, Xn;d , in which the typical element xij corresponds to the achievement of individual i in dimension j, with i ¼ 1; :::: ; n and j ¼ 1; :::: ; d. Vector xi contains the achievements of individual i in the d dimensions. Analogous to the unidimensional approach, the measurement of poverty in the multidimensional approach involves two

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steps (Sen, 1976): first the identification of the poor, second, the aggregation of the poor. The most common approach for identifying the poor in the multidimensional context is to first define a threshold level for each dimension j, below which a person is considered to be deprived. The collection of these thresholds can be expressed in a vector of poverty lines z ¼ ðz1 ; :::: ; zd Þ. In this way, whether a person is deprived or not in each dimension can be defined. However, unlike unidimensional measurement, a second decision needs to be made: among those who fall short in some dimension, who is to be considered multidimensionally poor? A natural starting point is to consider all those deprived in at least one dimension, the so-called union approach. Other more demanding criteria can be used, even to the extreme of requiring deprivation in all considered dimensions, the so-called intersection approach. In terms of Alkire and Foster (2007), the number of dimensions in which someone is required to be deprived so as to be identified as multidimensionally poor constitutes a second cut-off (the first cut-off were the dimension-specific ones contained in vector z). The authors name this second cut-off k and define ci to be the number of deprivations suffered by individual i. Then, an identification function rk ðxi ; zÞ is defined, such that: ( 1 if ci  k (1) rk ðxi ; zÞ ¼ 0 if ci ok In other words, if ci  k, the individual is identified as multidimensionally poor, and if ci ok, she is not, despite she may be experiencing some deprivation. For the aggregation step, one natural first measure is the headcount ratio, also frequently known as the poverty incidence, which is the fraction of the population identified as being multidimensionally poor. It is simply given by: Pn r ðxi ; zÞ q ¼ (2) H ¼ i¼1 k n n where q is the number of people identified as multidimensionally poor. Clearly, the value of H varies with the selected k cut-off, decreasing as k increases. The H measure is what the UBN approach has used, most frequently (but not always) using a k cut-off of one, that is, the union approach. Since the pioneer papers of Watts (1969) and Sen (1976), the limitations of the headcount ratio in the unidimensional approach have been repeatedly remarked, namely, that it is insensitive to the depth and distribution of poverty. In formal terms, it violates the monotonicity and transfer axioms.5

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Those critiques also apply to the multidimensional headcount, and therefore, to the UBN approach. Moreover, as noted by Alkire and Foster (2007), given a k value, if an individual identified as poor becomes deprived in an additional dimension, the multidimensional headcount does not change, that is, it violates what the authors call dimensional monotonicity. In simpler words, it is insensitive to the breadth of poverty: the number of deprivations suffered by the poor. Related to the latter point, Alkire and Foster (2007) argue that another informative measure is the average deprivation share across the poor, that is, the average fraction of dimensions in which the poor are deprived. This can be expressed as: Pn ci (3) A ¼ i¼1 qd H and A can be easily combined into one single measure, called by the authors as M0, which is just the headcount ratio ‘‘adjusted’’ (i.e., multiplied) by the breadth of poverty: M 0 ¼ HA

(4)

M0, also called the adjusted headcount ratio, is the first member of the Ma family of measures proposed by the authors. This family constitutes an extension of the unidimensional Foster, Greer, and Thorbecke (1984) (FGT or Pa) measures to the multidimensional context.6 Clearly, M0 is sensitive to the breadth of poverty, that is, it satisfies dimensional monotonicity. If someone becomes poor in one additional dimension, A will increase and therefore M0 will increase. Another example of the importance of the property is the following. Suppose two regions A and B, both with 50 percent of their population experiencing two or more deprivations. If in A that 50 percent experiences, on average, two deprivations out of six, while in B that 50 percent experiences on average four out of six, M0 will be higher in B than in A. It must be noted though that, as H, M0 is not sensitive to the depth of poverty, which has been a usual critique of the UBN approach. If someone becomes more deprived in one dimension, M0 will not change. The measure is also insensitive to the distribution of achievements among the poor. If one wants to account for the depth of poverty and for the distribution, other members of the Ma family need to be used (see Footnote 7). Alternatively, members of other proposed families of multidimensional poverty measures could be used, such as those introduced by Bourguignon and Chakravarty (2003), Tsui (2002), and Maasoumi and Lugo (2008). However, to incorporate considerations of depth and distribution with any of these

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measures one would need cardinality in all the considered indicators. Unfortunately, this is not the case of many of the indicators usually considered under a multidimensional approach.7 In summary, the failure of incorporating poverty depth in multidimensional measurement is not a consequence of the unavailability of appropriate aggregation methodologies, but of the nature of the indicators used. Given these constraints, the possibility that the M0 measures offers of at least accounting for the breadth of poverty constitutes an important advantage over H and, therefore, over the UBN aggregation methodology. Akin to their unidimensional counterpart, all the Ma measures can be decomposed by population subgroups, so that one can identify, for example, which are the regions that are contributing more to aggregate poverty.8 Moreover, once the poor have been identified, the measures can be brokendown by dimension, so it is possible to determine to which extent the deprivation in each dimension contributes to overall multidimensional poverty.9 This is a second advantage of M0 over H. There is a third advantage of the M0 measure over H (which is also present in the other Ma measures): it allows for alternative weighting systems of the dimensions, which affect both the identification and the aggregation steps. So far, we have implicitly assumed equal weights for all the considered dimensions (wj ¼ 1 for all j ¼ 1; . . . ; d). In that case, the identification cut-off ranges from k ¼ 1, corresponding to the union approach, to k ¼ d, corresponding to the intersection approach, and someone is multidimensionally poor when her number of deprivations is equal or greater than k: ci  k. However, one could use a ‘‘ranking weighting system’’ so that some dimensions receive higher weights than others. The sum of the weights needs to equal the total number of dimensions considered d. In that case, ci becomes the weighted number of deprivations in which the individual is deprived. For example, if an individual is deprived in income and health, and income has a weight of 2, while health has a weight of 0.5, then ci ¼ 2:5 and not 2, as it would be with equal weights. In this case, the minimum possible k value, which corresponds to the union approach, is given by the minimum weight: k ¼ min(wj), while the maximum possible k cut-off value remains to be d. With the mentioned change in the definition of ci, the definition of H, A, and consequently M0 are automatically adjusted to incorporate these weights.10 So far, we have referred to considering d dimensions, implicitly assuming that there is one indicator per dimension. However, in the practice of measurement, very often one considers more than one indicator referred to the same dimension. This is related to one critique sometimes done to the UBN approach: if there is more than one indicator corresponding to the

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MARI´A EMMA SANTOS ET AL.

same dimension, this means that some dimensions are weighted disproportionately more than others (Feres & Mancero, 2001). In those cases, the possibility of alternative weighting systems is an important feature of the M0 measure, which permits to overcome such critique. Indeed, one can use ‘‘nested weights.’’ For example, if there are five indicators, three of which correspond to the same dimension, say for example, housing, while the other two correspond to different dimensions, say income and health, then one can assign a weight of 5/3 to income and health each, and a value of 5/9 to each of the three housing indicators, assuring that each of the three considered dimensions are equally weighted. Other schemes can also be used, as considered appropriate by the nature of the measurement exercise.

3.2. Data, Indicators, and Poverty Lines The dataset used in the chapter corresponds to the Socioeconomic Database for Latin America and the Caribbean (SEDLAC), constructed by the Centro de Estudios Distributivos Laborales y Sociales (CEDLAS) and the World Bank. The dataset comprises household surveys of different Latin American countries that have been homogenized to make variables comparable across countries – the details of this process are covered in CEDLAS (2009). This study concentrates on a subset of the available database to maximize the possibilities for comparison across time and between countries.11 The study covers Argentina, Brazil, Chile and Uruguay, El Salvador and Mexico. Altogether, they account for about 64 percent of the total population in Latin America in 2006. The chapter performs estimates at five points in time between 1992 and 2006 for each country. Full details of survey names and sample sizes can be found in Table A1. In the case of Argentina and Uruguay, the data are representative only of urban areas and correspond to the years 1992, 1995, 2000, 2003, and 2006 in Argentina, and to the years 1992, 1995, 2000, 2003, and 2005 in Uruguay. In the other four countries data are nationally representative, including information from both urban and rural areas. In Brazil, data corresponds to the years 1992, 1995, 2001, 2003, and 2006; in Chile to 1992, 1996, 2000, 2003, and 2006; in El Salvador to 1991, 1995, 2000, 2003, and 2005; and finally in Mexico, to the years 1992, 1996, 2000, 2004, and 2006. The definition of ‘‘rural areas’’ by the surveys performed in each of these four countries is fairly similar.12 In each country, only households with complete information on all variables and consistent answers on income were considered.13

Multidimensional Analysis of Poverty in Latin America

Table 1. Dimension Command over resources Education

Housing

11

Selected Indicators and Deprivation Cut-Off Values. Indicator

Deprivation Cut-Off Value

Income

Having a per capita family income of US$ 2.

Child in school

Having all children between 7 and 15 years old attending school. Household head with at least five years of education. Having tap water in the dwelling. Having flush toilet in the dwelling. House with nonprecarious wall materials.

Education of HH Running water Sanitation Shelter

Table 1 presents the dimensions and indicators selected to perform the poverty estimates. The selected indicators can be seen as pertaining to three different dimensions (although other groupings are also possible): command over resources, education of the household, and housing conditions. For the income indicator (command over resources), the World Bank’s poverty line of US$2 per capita per day is used. It is acknowledged that this is a rather conservative poverty line for Latin America, but it guarantees full comparability across countries.14,15 Education of the household contains two indicators. One is whether children between 7 and 15 years old (inclusive) are attending school. This indicator belongs to the UBN approach. Households with no children are considered nondeprived in this indicator. The other education indicator refers to the educational level of the household head, with the threshold set at five years of education. Again this indicator is part of the UBN approach, although in that approach (a) the required threshold is second grade of primary school and (b) it is usually part of a composite indicator of ‘‘subsistence capacity’’ together with the dependency index of the household (considered to be deprived if there are four or more people per employed member). Two years of education seemed a very low threshold, so five years were used instead.16 Also, given that the income indicator is being included, the high dependency index seemed less relevant in this hybrid approach. Finally, there are three indicators related to the dwelling’s conditions. The first two, having proper sanitation (flush toilet) and living in a shelter with nonprecarious wall materials are typically included in the UBN approach.17 The third indicator is having access to running water in the dwelling. Although this is not usually included in the UBN approach, it is considered important. In the absence of comparable health data, it can also be seen as a proxy of this dimension, which is one of the most valued according to the participatory study performed in Mexico

12

MARI´A EMMA SANTOS ET AL.

‘‘Lo que dicen los pobres’’ (Sze´kely, 2003). The overcrowding indicator typically included in the UBN approach (more than three people per room) could not be incorporated into the analysis because it was not present in some databases in some years. In this chapter, all results correspond to estimates performed using equal weights for all the considered indicators, which does not mean that the dimensions are equally weighted. In fact, three of the indicators used refer to dwelling’s characteristics and two other indicators (children attending school and the education of the household head) refer to the dimension of education of the household. Therefore, the equal weights are implicitly weighting the dwelling conditions three times, and the education dimension twice, compared to the income dimension. In Battisto´n, Cruces, Lo´pez-Calva, Lugo, and Santos (2009), we performed estimations using an alternative weighting system derived from the mentioned participatory study performed in Mexico (Sze´kely, 2003). In this scheme, the income dimension receives the highest weight, being 1.3 times the weight received by the children’s education, 4 times the weight received by the education of the household head and access to running water, and 8 times the weight received by having access to sanitation and proper shelter. When appropriate, we mention the differences in results obtained with this alternative weighting structure.

4. RESULTS This section presents the estimation of poverty figures for the six countries, using the multidimensional poverty measure presented in the previous section (M0) for different values of the second cut-off k ¼ 1, 2, y 6, and using equal weights for all indicators.18 Given that in two of the countries (Argentina and Uruguay) data for rural areas is not available, we have maintained urban and rural areas separately for all countries. For reference, the proportion of individuals living in urban and rural areas in each country can be found in Table A2. We highlight two results: the first, related to the evolution of multidimensional poverty, the second regarding the urban– rural disparities.

4.1. Evolution of Multidimensional Poverty Fig. 1 presents the evolution of the adjusted headcount ratio M0 for rural and urban areas, in panels A and B, respectively. The number of deprivations

13

Multidimensional Analysis of Poverty in Latin America Panel A Adjusted Headcount Ratio M0 Rural areas

M0

Panel B Adjusted Headcount Ratio M0 Urban areas

M0 0.80

0.80 El Salvador 0.70

0.70 Mexico

El Salvador

0.60 0.50 0.40

Brazil

0.60 0.50

Chile

0.40

0.30

0.30 El Salvador

0.20

0.20 Brazil

Mexico

0.10

0.10

0.00 1991 1993 1995 1997 1999 2001 2003 2005 2007

Argentina Uruguay 0.00 1991 1993 1995 1997 1999 2001 2003 2005 2007 Year

Year

Chile

Fig. 1. Evolution over Time of M0 with k ¼ 2 and Equal Weights. Source: Authors’ calculation, based on SEDLAC database.

used as the second cut-off is two. In other words, a person is considered to be poor if she falls short of the adequate level in two or more dimensions. k ¼ 2 is chosen because it is the minimum k that requires an individual to be deprived in more than one indicator so as to be considered poor (i.e., it is ‘‘truly’’ multidimensional) and at the same time it is meaningful for all countries (for higher k values the aggregate M0 estimate becomes virtually zero in the urban areas of Chile, Argentina, and Uruguay). Table A3 includes the estimation of M0 for all possible values of k.19 A number of noteworthy points emerge from these graphs. First, in all countries considered multidimensional poverty decreased between 1992 and 2006. Although not presented here, this result is robust to the number of deprivations used as cut-offs. In most cases, the decrease was sizeable and uninterrupted throughout the period. In other cases, such as Argentina and Uruguay urban areas, multidimensional poverty either did not change significantly or was reduced only marginally.20 Second, multidimensional poverty in rural areas is significantly higher than in urban areas, especially at the beginning of the period. We come back to this point later in the section. Thirdly, and focusing on urban areas solely, one could categorize the countries in two groups: the countries belonging to the ‘‘Southern Cone’’ (Argentina, Chile, and Uruguay), with multidimensional M0 estimates below 0.10, on the one hand, and Brazil, El Salvador,

14

MARI´A EMMA SANTOS ET AL.

and Mexico with poverty measurements closer to 0.20.21 Finally, while the distinction between these two groups of countries is still apparent at the end of the period, there seems to be some convergence as the countries in the second group present a higher rate of reduction in multidimensional poverty than that of the first group. As explained in Section 3.1, the M0 measure is the product of two informative measures: the multidimensional headcount ratio H and the average deprivation share across the poor A. In order to better understand the drop in multidimensional poverty described in the previous paragraphs, we present in Fig. 2 the evolution of its two components (H and A) between 1992 and 2006, again for rural areas in panel A (excluding Argentina and Uruguay) and for urban areas in panel B. As before, k is set to two and all indicators are weighted ‘‘equally.’’ In both urban and rural areas for Brazil, Chile, El Salvador, and Mexico the reduction in multidimensional poverty M0 is the result not only of the fall in the proportion of people deprived in two or more dimensions (H ) but also of the fact that, on average, they became poor in fewer dimensions (A). However, the contribution of each of these components to the overall reduction of M0 differs by country and area. For instance, in rural areas of Chile and urban areas of Brazil, most of the reduction seems to be driven by a substantial decline in the proportion of the population classified as poor, more than by a decrease in the average number of deprivations (even though, these all fell). On the contrary, in both the rural and urban areas of El Salvador the proportional reduction in A is larger than that of H, whereas in rural Mexico, both the percentage of the poor and the average deprivation seem to be reduced in similar proportions. Finally, in Uruguay and Argentina there was no significant reduction in the average number of deprivations experienced by the poor across the period. The only significant change is increase in the proportion of poor individuals in 2003 in Argentina, which is reflecting the increase in income poverty after the 2001 crisis.22 A second way of disentangling the meaning of the observed decrease in multidimensional poverty is to look at the evolution of the relative contribution of each of the dimensions into the total deprivation. As explained in Section 3.1 one of the advantages of the multidimensional poverty indices proposed by Alkire and Foster is that they can be brokendown by indicators. Fig. 3 presents the decomposition of M0 with k ¼ 2 and equal weights, panel A includes rural areas and panel B, urban areas. Table A4 presents the values for these decompositions. Before looking at the evolution, it is worth examining which are the main contributors to total multidimensional poverty at the beginning of the

15

Multidimensional Analysis of Poverty in Latin America Multidimensional Headcount Ratio H Panel A Rural Areas H 1.00

El Salvador

H 1.00

0.90 0.80 0.70

Panel B Urban Areas

Mexico Brazil

0.90 0.80 0.70

Chile

0.60

0.60

0.50

0.50

0.40

0.40

0.30

0.30

Brazil

0.20

0.20

Chile

0.10

0.10

El Salvador

Mexico

Argentina Uruguay

0.00 1991 1993 1995 1997 1999 2001 2003 2005 2007

0.00 1991 1993 1995 1997 1999 2001 2003 2005 2007 Year

Year

Average deprivation share across the poor A Rural Areas A 0.75

A 0.75

0.70

0.70

0.65

Urban Areas

0.65 Mexico

El Salvador 0.60

0.60 Brazil

0.55

0.55 0.50

El Salvador 0.50 Chile

Mexico Brazil

0.45

0.45

0.40

0.40

Chile Argentina

Uruguay 0.35 1991 1993 1995 1997 1999 2001 2003 2005 2007 Year

Fig. 2.

0.35 1991 1993 1995 1997 1999 2001 2003 2005 2007 Year

Evolution over Time of Components of M0 with k ¼ 2 and Equal Weights. Source: Authors’ calculation, based on SEDLAC database.

period. While the composition differs slightly across countries and region, attendance of children to school (second bar from the bottom) contributes relatively little to aggregate poverty, not exceeding 10 percent in most cases. On the contrary, insufficient education of the household head and poor sanitation appear in almost all countries and regions as the main contributors to poverty, the first contributing between 19 and 32 percent

MARI´A EMMA SANTOS ET AL.

16

Panel A Rural Areas 100%

90%

Percentage contribution

80%

70%

60%

50%

40%

30%

20%

Income

Brazil

Chile

Children at school

Education of hh

Mexico

El Salvador Water

2006

1992

2006

1992

2006

1992

2006

0%

1992

10%

Sanitation

Shelter

Panel B Urban Areas 100%

Percentage contribution

90% 80% 70% 60% 50% 40% 30% 20%

Argentina Income

Brazil

Children at school

Chile

El Salvador

Education of hh

Water

Mexico Sanitation

2006

1992

2006

1992

2006

1992

2006

1992

2006

1992

2006

0%

1992

10%

Uruguay Shelter

Fig. 3. Contribution of Deprivation in Each Indicator to Overall M0 with k ¼ 2 and Equal Weights. Source: Authors’ calculation, based on SEDLAC database.

Multidimensional Analysis of Poverty in Latin America

17

and the second contributing 22 to 33 percent.23 Two exceptions are urban areas in Chile and Mexico, where access to adequate shelter also explains a sizeable proportion of total poverty (31 and 21 percent, respectively). Finally, as expected, in rural areas the deprivations seem to be more concentrated in the dimensions related to infrastructure such as water, shelter, and sanitation. For instance, in Chile the contribution of deprivation in sanitation and running water is about one and a half and three and a half times that in urban areas correspondingly. Similarly, in Brazil the contribution of water is more than twice that of urban areas. In terms of the evolution, two points are worth highlighting. First, in Argentina and Uruguay urban areas – where multidimensional poverty has not decreased as much as in the other countries – there is an increase in the contribution of income to aggregate poverty. In both countries the contribution of income to total poverty soared from 10 percent to about 24 percent. This is consistent with the idea that between 1992 and 2006 there was a shift from ‘‘structural poor’’ to ‘‘new poor.’’ Second, in the rest of the countries, most of the contributions of dimensions remain relatively constant. This, together with the previously observed reduction in aggregate poverty implies that there was a similar improvement in all dimensions of well-being included. Some exceptions include sanitation in urban areas of El Salvador and shelter in rural parts of Mexico that have seen their share rise considerably, from 23 to 29 percent and from 20 to 25 percent, respectively (Table A4 in the Appendix).

4.2. Persistence of Poverty in Rural Areas The second result of the present analysis regards the disparities between the urban and rural areas. The analysis is, therefore, restricted to the four countries where the information on both areas is available, that is, Brazil, Chile, El Salvador, and Mexico. Despite the progress experienced in the countries of the region, rural areas still present high levels of multidimensional poverty. People living outside the cities are not only more likely to be multidimensionally poor than those in urban areas but also they are more prone to experience multiple deprivations at the same time. This means that someone who falls short in one indicator of well-being (for instance, education) is quite likely to fall short also in another indicator (such as dwelling characteristics). Fig. 4 presents for each country, disaggregated by urban and rural areas, the percentage of individuals deprived in one or more dimensions (k ¼ 1),

MARI´A EMMA SANTOS ET AL.

18

0.93

Urban Rural

0.9 0.8

0.74

0.72 0.7 0.6 0.52 0.5

0.44

0.4

0.36

` 0.28

0.3

0.24

0.2 0.18

0.2 0.1

El Salvador

Mexico

Brazil

k=6

k=5

k=4

k=3

k=2

k=1

k=6

k=5

k=4

k=3

k=2

k=1

k=6

k=5

k=4

k=3

k=2

k=1

k=6

k=5

k=4

k=3

0.04 k=2

0

k=1

Proportion of multidimensioally poor individuals (H)

1

Chile

Fig. 4. Proportion of Individuals Deprived in k ¼ 1, 2 ... 6 Dimensions and Equal Weights (Multidimensional Headcount H) – Year 2006. Source: Authors’ calculation, based on SEDLAC database.

two or more (k ¼ 2), and so on at the end of the period of study. Table A6 in the appendix presents the values of H for each country and area for the year 2006. In all four countries, the measurement of poverty is higher in rural areas than in its respective urban area for all possible k chosen, most often at least twice as large. Let us focus for the moment on multidimensional poverty as defined as those who are deprived in two or more indicator. While the proportion of poor in urban areas of El Salvador is 44 percent, more than twice (93 percent) are poor in rural areas. Similarly, in Mexico 28 percent of the urban population is poor whereas 72 percent of the rural one. In Brazil, the percentages are 18 and 74, respectively, while in Chile are 4 and 36 percent, respectively. In addition, with the exception of Chile, the proportion of individuals suffering multiple deprivations is still significantly high in rural (above 20 percent) even when the required number of deprivations is four or more – out of six. At the extreme, we find El Salvador and Mexico, where the estimate of H falls below 5 percent only with the intersection approach, that is, when k ¼ 6. These results point to a pattern in which people living in rural areas of Latin American who are deprived in one indicator are more

Multidimensional Analysis of Poverty in Latin America

19

likely to fall below the minimum required in several other dimensions. This is also true in urban areas of El Salvador and, to a lesser extent, in urban Mexico. In contrast, people living in cities in Chile or Brazil who are deprived in an indicator are more likely to be deprived only in that single indicator (similarly for urban Argentina and Uruguay).

5. CONCLUSIONS This chapter provides an analysis of multidimensional poverty in Argentina, Brazil, Chile, El Salvador, Mexico, and Uruguay for the period between 1992 and 2006. We use an approach that intends to improve the UBN method used in the region in three ways: first, using a hybrid approach in the selection of indicators incorporated in the multidimensional measure, including variables normally used by both the UBN and those who prefer the income method; second, by using one of the measures of the family of multidimensional poverty indices proposed by Alkire and Foster (2007), we can account for both poverty incidence and the average proportion of deprivations that the poor experience. The index also allows alternative weighting schemes, although in this chapter we weigh all indicators equally. The overall picture from the present analysis is mixed. On the one hand, there has been an enormous improvement in all 6 countries and regions in the 14 years considered, with decreasing trends in multidimensional poverty due, in general, to both a reduction in the proportion of individuals that are poor and the number of deprivations that they have on average. With the exception of urban Argentina and Uruguay, the overall reduction seems to come from an equal improvement in all indicators. Instead, in Argentina and Uruguay, the increase in the importance of income poverty in the overall multidimensional poverty reflects the increase in deprivation in this dimension over the period. The second result of the chapter relates to the fact that multidimensional poverty estimates in rural areas are still considerably high. Even when we observe a trend toward convergence with urban areas rates, there seems to be a pattern: when people living outside the cities do not reach the adequate level of achievement in a given indicator, they are more likely to be deprived in several other indicators as well. In contrast, inhabitants of Chilean and Brazilian cities who fall short in a given indicator of well-being, they tend to be deprived only in that single indicator. This is true to a lesser extent in urban areas of Mexico and El Salvador. Such results suggest the need of a renewed effort to tackle poverty in rural areas.

MARI´A EMMA SANTOS ET AL.

20

The methodological improvements proposed in this chapter highlight the need for a thorough (public) discussion on the full set of dimensions, indicators, and weights to be included in a multidimensional poverty measure for the region. This discussion could contribute to the development of a tool that becomes relevant, meaningful, and useful to track progress in the Millennium Development Goals as well as in social policy in general. Such index will, most likely, demand improving the collection of information in several key dimensions of well-being.

NOTES 1. Household surveys did not become regular until the 1970s or even later in Latin American countries and even when they were performed, microdatasets were not publicly available for researchers (Gasparini, 2004). 2. Cruces and Gasparini (2008) illustrate these inclusion and exclusion effects by studying the targeting of cash transfer programs based on a combination of income and other UBN-related indicators. 3. Note that this is also the procedure followed when constructing the Human Development Index. 4. In the case of the six countries considered in this study, we found that the Spearman correlation of income with the other considered dimension does not exceed 0.5 in any case and it is decreasing over time. See these results in Battisto´n et al. (2009). 5. Several other unidimensional poverty measures have been proposed that satisfy monotonicity and transfer: the mentioned papers of Watts and Sen propose measures themselves, as well as Foster et al. (1984), Clark, Hemming, and Ulph (1981), Chakravarty (1983), among others. Foster (2006), Foster and Sen (1997), and Atkinson (1987) provide excellent surveys of unidimensional poverty measures and the desirable axiomatic framework. P P 6. The family is defined as M a ðX; zÞ ¼ ð1=ndÞ ni¼1 dj¼1 wj ðgij ðkÞÞa with  aZ0, where  gij ðkÞ is the censored poverty gap of individual i in dimension j: gij ðkÞ ¼ ðzj  xij Þ=zj , if xij ozj and P ci  k, and gij ðkÞ ¼ 0 otherwise; wj is the weight assigned to dimension j, such that dj¼1 wj ¼ d, and a is the parameter of dimension-specific poverty aversion. The Ma family is presented in Alkire and Foster (2007) as the mean of the censored matrix of normalized alpha poverty gaps. 7. In Battisto´n et al. (2009) we present results for other members of the Alkire and Foster family of measures which account for depth and severity (namely M1 and M2). In that paper, we acknowledge the technical problems of using such measures when not all the indicators are cardinal. 8. Given a population subgroup I, its contribution to overall poverty is given by C I ¼ ½ðnI =nÞM Ia =M a . P 9. Specifically, the contribution of dimension J is given by C J ¼ ½ðð1=ndÞ ni¼1 ðgiJ ðkÞÞa Þ=M a .

Multidimensional Analysis of Poverty in Latin America

21

10. On the meaning of dimension weights in multidimensional indices of wellbeing and deprivation and alternative approaches to setting them, see Decancq and Lugo (2010). 11. The SEDLAC database (CEDLAS & World Bank, 2009) will report multidimensional poverty indicators systematically starting in 2009. http://www. depeco.econo.unlp.edu.ar/sedlac/eng/index.php 12. In Chile, it corresponds to localities of less than 1,000 people or with 1,000– 2,000 people, of which most perform primary activities. In Mexico, it refers to localities of less than 2,500 people. In Brazil, rural areas are not defined according to population size but rather they are all those not defined as urban agglomerations by the Brazilian Institute of Geography and Statistics. In El Salvador, rural areas are all those outside the limits of municipalities heads, which are populated centers where the administration of the municipality is located. Again, this definition does not refer to any particular population size. 13. The statistical institute of each country has a criterion to identify invalid income answers (such as reporting zero income when working for a salary), which is incorporated in the SEDLAC dataset, as well as other types of invalid answers (such as reporting labor income when being unemployed). 14. The dollar-a-day lines are not without problems. It can be argued, for instance, that the national poverty lines computed by each country’s statistical institute tend to be more accurate and appropriate than the World Bank’s ones as the former are based on nationally relevant food baskets (Reddy & Pogge, 2010). Still, we use the World Bank lines for this study because they stem from a common methodology and provide benchmarks for international comparisons. Moreover, these are the lines considered by the monitoring indicators of the Millennium Development Goals. See Reddy and Pogge (2010) for critique and alternative approach, and Ravallion (2010) for a response. See also Deaton’s (2010) presidential address to the American Economic Association on the difficulties of dealing with PPP values for international comparisons of poverty. 15. In Brazil, household income is expressed in gross terms (rather than net terms, as in other countries). Fortunately, during the period under analysis there have not been major changes in the labor or income tax, hence the comparability across years is not compromised. 16. Five years of complete schooling is seen here as a good ‘‘compromise,’’ albeit arbitrary, between alternative thresholds used by countries in computing UBN and current demands of the labor market. This is set as a (lower bound) approximation to complete primary education. UNESCO considers that in Argentina, Mexico, El Salvador, Chile and Uruguay, the equivalent to the first level of the International Standard Classification of Education (1997) lasts 6 years, while it lasts 4 in Brazil. 17. This definition of the shelter indicator in this paper is slightly different than the one used by the UBN approach, which uses ‘‘adequate shelter.’’ Only in the case of Uruguay, we used ‘‘adequate shelter,’’ as the question on wall materials was not included after the year 2000. The SEDLAC database defines precarious house from Uruguayan EPH as a room inside a house, fully occupied as a residence by a single household. 18. As explained in Section 3.1, Battisto´n et al. (2009) also compute all combinations of multidimensional poverty using weights derived from the participatory study ‘‘Lo que dicen los pobres’’ (Sze´kely, 2003). The main conclusions of this section are

22

MARI´A EMMA SANTOS ET AL.

consistent with the results using this alternative weighting system, even though, as expected, the estimates follow more closely the evolution of the higher-weighted dimensions, that is, income and education. 19. Note that with k ¼ 2 someone deprived only in income (or in any other indicator) is not counted as poor. We purposely choose this approach since we are aiming at identifying people with coupled deprivations, that is, more than one. However, Tables A3 and A6 provide the M0 and H estimates for all possible k values. 20. To unambiguously assess the reductions in poverty, all estimates were bootstrapped using 200 replications. Results of the bootstraps can be found in a companion document to Battisto´n et al. (2009) ‘‘WP 17 Bootstrapped Estimates and Correlations’’ (http://www.ophi.org.uk/publications/ophi-working-papers/). 21. One should remember that these figures cannot be interpreted as the proportion of the population that is multidimensionally poor but rather as the combination of that proportion and the average share of deprivation suffered by this population. Therefore, a value of 0.10 could be the result 30 percent of individuals classified as poor with, on average, two (out of six) deprivations, or 15 percent of individuals with an average of four (out of six) deprivations. 22. See also Table A5 in the Appendix which reports the headcount ratios by indicator. 23. The high contribution of sanitation can be due to the threshold used: we require flush toilet to be nondeprived.

ACKNOWLEDGMENTS This study was supported by the United Nations Development Programme Regional Bureau for Latin America and the Caribbean, and the Oxford Poverty and Human Development Initiative, University of Oxford. M. A. Lugo kindly acknowledges the financial support from the Economic and Social Research Council.

REFERENCES Alkire, S. (2008). Choosing dimensions: The capability approach and multidimensional poverty. In: N. Kakwani & J. Silber (Eds), The many dimensions of poverty. Basingstoke: Palgrave Macmillan. Alkire, S., & Foster, J. E. (2007). Counting and multidimensional poverty measurement. OPHI Working Paper Series no. 07. Oxford. Amarante, V., Arim, R., & Vigorito, A. (2008). Multidimensional poverty among children in Uruguay 2004–2006. Evidence from panel data. Presented at the Meeting of the LACEA/IADB/WB/ UNDP Network on Inequality and Poverty, Universidad Cato´lica de Santo Domingo. Santo Domingo, Repu´blica Dominicana, June 13, 2008.

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Arim, R., & Vigorito, A. (2007). Un ana´lisis multidimensional de la pobreza en Uruguay. 1991–2005. Serie Documentos de Trabajo DT 10/06, Instituto de Economı´ a, Universidad de la Repu´blica, Uruguay. Atkinson, A. (2003). Multidimensional deprivation: Contrasting social welfare and counting approaches. Journal of Economic Inequality, 1(1), 51–65. Atkinson, A. B. (1987). On the measurement of poverty. Econometrica, 55(4), 749–764. Ballon, P., & Krishnakumar, J. (2008). Estimating basic capabilities: A structural equation model applied to Bolivia. World Development, 36(6), 992–1010. Battisto´n, D., Cruces, G., Lo´pez-Calva, L., Lugo, M. A., & Santos, M. E. (2009). Income and beyond: Multidimensional poverty in six Latin American countries. OPHI Working Paper Series no. 17. Oxford. Bourguignon, F., & Chakravarty, S. (2003). The measurement of multidimensional poverty. Journal of Economic Inequality, 1(1), 25–49. CEDLAS. (2009). A guide to the SEDLAC-socio-economic database for Latin America and the Caribbean. Centro de Estudios Distributivos, Laborales y Sociales, Universidad Nacional de La Plata. Available at http://www.depeco.econo.unlp.edu.ar/sedlac/eng/ index.php CEDLAS, & World Bank. (2009). Socio-economic database for Latin America and the Caribbean (SEDLAC). Available at http://www.depeco.econo.unlp.edu.ar/sedlac/eng/ index.php Chakravarty, S. (1983). A new index of poverty. Mathematical Social Sciences, 6(3), 307–313. Clark, S., Hemming, R., & Ulph, D. (1981). On indices for the measurement of poverty. Economic Journal, 91(362), 515–526. Conconi, A., & Ham, A. (2007). Pobreza multidimensional relativa: Una aplicacio´n a la Argentina. Documento de trabajo CEDLAS N. 57, CEDLAS, Universidad Nacional de La Plata, Argentina. Cruces, G., & Gasparini, L. (2008). Programas sociales en Argentina: Alternativas para la ampliacio´n de la cobertura. Documento de trabajo CEDLAS N. 77, CEDLAS, Universidad Nacional de La Plata, Argentina. Deaton, A. (2010). Price indexes, inequality, and the measurement of world poverty. American Economic Review, 100(1), 5–34. Decancq, K., & Lugo, M. A. (2010). Weights in multidimensional indices of well-being: An overview. CES Discussion Paper 10.06. Katholieke Universiteit Leuven, Belgium. Duclos, J.-Y., & Araar, A. (2006). Poverty and equity measurement, policy, and estimation with DAD. Berlin and Ottawa: Springer and IDRC. Duclos, J.-Y., Sahn, D., & Younger, S. (2006). Robust multidimensional poverty comparisons. The Economic Journal, 116(514), 943–968. Feres, J. C., & Mancero, X. (2001). El me´todo de las necesidades ba´sicas insatisfechas (NBI) y sus aplicaciones a Ame´rica Latina. Series Estudios Estadı´sticos y Prospectivos, CEPAL – Naciones Unidas. Foster, J. E. (2006). Poverty indices. In: A. de Janvry & R. Kanbur (Eds), Poverty, inequality and development: Essays in honor to Erik Thorbecke. New York: Springer ScienceþBusiness Media, Inc. Foster, J. E., Greer, J., & Thorbecke, E. (1984). A class of decomposable poverty indices. Econometrica, 52(3), 761–766. Foster, J. E., & Sen, A. (1997). On economic inequality: After a quarter century. Annex to the expanded edition of A. Sen. On economic inequality. Oxford: Clarendon Press.

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Gasparini, L. (2004). Poverty and inequality in Argentina: Methodological issues and literature review. CEDLAS, Universidad Nacional de La Plata. http://www.depeco.econo.unlp. edu.ar/cedlas/monitoreo/pdfs/review_argentina.pdf Lo´pez-Calva, L. F., & Ortiz-Jua´rez, E. (2009). Medicio´n multidimensional de la pobreza en Me´xico: Significancia estadı´ stica en la inclusio´n de dimensiones no monetarias. Estudios Econo´micos (Special Issue), 3–33. Lo´pez-Calva, L. F., & Rodrı´ guez-Chamussy, L. (2005). Muchos rostros, un solo espejo: Restricciones para la medicio´n multidimensional de la pobreza en Me´xico. In: M. Sze´kely (Ed.), Nu´meros que Mueven al Mundo: La Medicio´n de la Pobreza en Me´xico. Me´xico: Miguel A´ngel Porru´a. Maasoumi, E., & Lugo, M. A. (2008). The information basis of multivariate poverty assessments. In: N. Kakwani & J. Silber (Eds), Quantitative Approaches to Multidimensional Poverty Measurement. London: Palgrave MacMillan. Paes de Barros, R., De Carvalho, M., & Franco, S. (2006). Pobreza multidimensional no Brasil. Texto para discussa˜o n1 1227. IPEA, Brazil. Ravallion, M. (2010). How not to count the poor? A reply to Reddy and Pogge. In: S. Anand, P. Segal & J. Stiglitz (Eds), Debates on the Measurement of Poverty. Oxford: Oxford University Press. Reddy, S., & Pogge, T. (2010). How not to count the poor?. In: S. Anand, P. Segal & J. Stiglitz (Eds), Debates on the measurement of poverty. Oxford: Oxford University Press. Sen, A. (1976). Poverty: An ordinal approach to measurement. Econometrica, 44(2), 219–231. Sen, A. (1992). Inequality Reexamined. New York: Harvard University Press. Sen, A. (2009). The idea of justice. London: Allen Lane. Streeten, P., Burki, J. S., Haq, M. U., Hicks, N., & Stewart, F. (1981). First things first: Meeting basic human needs in developing countries. New York: Oxford University Press. Sze´kely, M. (2003). Lo que dicen los pobres. Cuadernos de desarrollo humano N1 13. Secretarı´ a de Desarrollo Social, Me´xico. Tsui, K. (2002). Multidimensional poverty indices. Social Choice and Welfare, 19(1), 69–93. Watts, H. W. (1969). An economic definition of poverty. In: D. P. Moynihan (Ed.), On understanding poverty. New York: Basic Books.

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APPENDIX Table A1. Country

Sample Size for each Country and Year, Rural, and Urban Areas. Household Survey

a

Argentina

Encuesta Permanente de Hogares (EPH)

Encuesta Permanente de Hogares Continua (EPH-C)

Year

Sample Size (People) Urban

Rural

1992 1995 2000 2003 2006

59,528 62,372 43,255 29,075 45,676

NA NA NA NA NA

Brazil

Pesquisa Nacional por Amostra de Domicilios (PNAD)

1992 1995 2001 2003 2006

244,473 266,287 316,860 322,839 337,509

55,544 57,859 52,753 53,932 65,372

Chile

Encuesta de Caracterizacion Socioeconomica Nacional (CASEN)

1992 1996 2000 2003 2005

86,179 94,925 142,029 150,156 153,234

46,698 32,500 89,441 80,411 86,058

El Salvador Encuesta de Hogares de Propositos Multiples (EHPM)

1991 1995 2000 2003 2005

49,243 20,989 40,940 35,622 34,127

39,235 18,009 29,843 35,708 35,517

Mexico

Encuesta Nacional de Ingresos y Gastos de los Hogares (ENIGH)

1992 1996 2000 2004 2006

27,913 39,974 26,402 68,016 58,760

20,265 21,840 13,989 21,907 23,140

Uruguay

Encuesta Continua de Hogares (ECH)

1992 1995 2000 2003 2005

28,658 64,177 51,913 54,750 53,738

NA NA NA NA NA

a

For the sake of comparability over time, the samples used correspond to the same 15 urban agglomerations.

MARI´A EMMA SANTOS ET AL.

26

Table A2.

Proportion of Urban Population.

Year

Brazil

Chile

El Salvador

Mexico

1992 1995 2001 2003 2006

0.8 0.81 0.85 0.86 0.85

0.84 0.85 0.86 0.87 0.87

0.52 0.59 0.63 0.62 0.63

0.76 0.77 0.78 0.77 0.78

Source: Authors’ calculations, based on SEDLAC database.

Table A3. Country

Multidimensional Poverty M0 for Alternative k and Equal Weights.

Year

Rural k¼1

k¼2

k¼3

k¼4

Urban k¼5

k¼6

k¼1

k¼2

k¼3

k¼4

k¼5

k¼6

0.080 0.070 0.071 0.088 0.070

0.046 0.038 0.039 0.052 0.037

0.021 0.017 0.015 0.019 0.016

0.005 0.006 0.003 0.003 0.005

0.001 0.000 0.000 0.001 0.001

0.000 0.000 0.000 0.000 0.000

Argentina

1992 1995 2000 2003 2006

Brazil

1992 1995 2000 2003 2006

0.487 0.509 0.463 0.429 0.395

0.472 0.491 0.442 0.402 0.364

0.388 0.424 0.357 0.313 0.267

0.245 0.298 0.229 0.190 0.145

0.093 0.138 0.075 0.056 0.039

0.018 0.031 0.008 0.004 0.004

0.227 0.200 0.166 0.154 0.129

0.173 0.143 0.109 0.098 0.072

0.108 0.087 0.059 0.049 0.031

0.056 0.044 0.026 0.020 0.011

0.020 0.016 0.007 0.004 0.002

0.004 0.003 0.001 0.000 0.000

Chile

1992 1995 2000 2003 2006

0.421 0.389 0.324 0.263 0.202

0.396 0.360 0.285 0.217 0.151

0.319 0.273 0.196 0.130 0.079

0.170 0.140 0.085 0.045 0.023

0.049 0.037 0.017 0.007 0.004

0.006 0.006 0.003 0.000 0.000

0.124 0.073 0.059 0.047 0.044

0.073 0.040 0.027 0.018 0.014

0.033 0.018 0.010 0.007 0.004

0.011 0.005 0.003 0.002 0.001

0.003 0.001 0.001 0.000 0.000

0.000 0.000 0.000 0.000 0.000

El Salvador

1992 1995 2000 2003 2006

0.732 0.706 0.654 0.605 0.590

0.729 0.703 0.648 0.597 0.582

0.711 0.682 0.613 0.549 0.527

0.641 0.600 0.513 0.435 0.403

0.463 0.401 0.314 0.246 0.234

0.180 0.153 0.095 0.065 0.057

0.311 0.297 0.267 0.252 0.251

0.278 0.264 0.231 0.213 0.214

0.230 0.213 0.171 0.154 0.153

0.166 0.137 0.105 0.092 0.085

0.086 0.065 0.047 0.037 0.034

0.024 0.019 0.007 0.005 0.007

Mexico

1992 1995 2000 2003 2006

0.622 0.592 0.532 0.427 0.405

0.613 0.581 0.517 0.394 0.378

0.573 0.539 0.464 0.325 0.300

0.476 0.426 0.338 0.224 0.178

0.291 0.246 0.161 0.100 0.068

0.096 0.084 0.030 0.016 0.013

0.243 0.230 0.174 0.180 0.164

0.206 0.192 0.131 0.137 0.125

0.145 0.137 0.083 0.091 0.075

0.080 0.077 0.043 0.044 0.034

0.033 0.027 0.015 0.014 0.010

0.007 0.004 0.003 0.003 0.001

Uruguay

1992 1995 2000 2003 2006

0.078 0.073 0.056 0.050 0.053

0.040 0.035 0.024 0.019 0.022

0.016 0.014 0.008 0.006 0.009

0.005 0.004 0.002 0.002 0.001

0.001 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

Source: Authors’ calculations, based on SEDLAC database.

0.057 0.041

0.156 0.167

0.140 0.129

1992 2006

1992 2006

1992 2006

1992 2006

Brazil

Chile

El Salvador 1992 2006

1992 2006

Argentina

Mexico

Uruguay

Rural

Urban

0.078 0.046

0.088 0.048

0.031 0.012

0.086 0.021

0.205 0.196

0.192 0.198

0.199 0.209

0.295 0.305

0.137 0.098

0.189 0.186

0.211 0.204

0.091 0.190

Source: Authors’ calculations, based on SEDLAC database.

0.181 0.121

0.240 0.278

0.218 0.265

0.311 0.349

0.299 0.324

0.199 0.254

0.157 0.136

0.191 0.185

0.048 0.039

0.097 0.225

0.106 0.114

0.170 0.170

0.146 0.137

0.195 0.174

0.103 0.248

0.137 0.111

0.079 0.052

0.065 0.043

0.035 0.055

0.084 0.033

0.118 0.044

0.259 0.224

0.235 0.204

0.207 0.212

0.222 0.252

0.322 0.349

0.272 0.189

0.094 0.085

0.097 0.073

0.180 0.175

0.049 0.056

0.095 0.089

0.122 0.069

0.335 0.318

0.272 0.311

0.230 0.292

0.238 0.207

0.276 0.328

0.332 0.372

0.079 0.037

0.212 0.247

0.149 0.109

0.310 0.294

0.029 0.027

0.053 0.079

Income Children Education of Water Sanitation Shelter Income Children Education of Water Sanitation Shelter at school household at school household head head

Year

Decomposition of Multidimensional Poverty M0 for k ¼ 2 and Equal Weights (Proportional Contribution of Each Indicator).

Country

Table A4.

Multidimensional Analysis of Poverty in Latin America 27

1992 1995 2000 2003 2006

1992 1995 2000 2003 2006

1992 1995 2000 2003 2006

1992 1995 2000 2003 2006

1992 1995 2000 2003 2006

1992 1995 2000 2003 2006

Argentina

Brazil

Chile

El Salvador

Mexico

Uruguay

0.515 0.650 0.537 0.369 0.301

0.683 0.612 0.640 0.603 0.584

0.140 0.136 0.093 0.080 0.045

0.514 0.393 0.390 0.352 0.270

Income

0.290 0.240 0.169 0.097 0.111

0.385 0.303 0.233 0.192 0.168

0.075 0.054 0.035 0.020 0.014

0.253 0.189 0.062 0.053 0.046

Children at school

0.771 0.686 0.658 0.558 0.495

0.844 0.822 0.774 0.720 0.697

0.504 0.483 0.436 0.329 0.282

0.885 0.863 0.838 0.796 0.750

Education of household head

Rural

0.506 0.407 0.263 0.275 0.234

0.829 0.826 0.698 0.649 0.654

0.505 0.499 0.412 0.307 0.226

0.272 0.606 0.512 0.452 0.419

Water

0.907 0.887 0.849 0.661 0.684

0.964 0.974 0.980 0.962 0.962

0.818 0.796 0.688 0.592 0.434

0.898 0.870 0.863 0.829 0.798

Sanitation

0.738 0.674 0.721 0.628 0.609

0.688 0.696 0.594 0.507 0.476

0.485 0.395 0.321 0.275 0.233

0.162 0.155 0.122 0.095 0.085

Shelter

0.029 0.029 0.027 0.049 0.060

0.146 0.286 0.128 0.118 0.103

0.321 0.257 0.221 0.267 0.256

0.096 0.056 0.053 0.046 0.030

0.233 0.145 0.165 0.166 0.106

0.042 0.075 0.094 0.221 0.091

Income

0.050 0.042 0.037 0.031 0.029

0.108 0.083 0.060 0.059 0.050

0.122 0.098 0.068 0.062 0.065

0.021 0.018 0.015 0.010 0.011

0.094 0.071 0.029 0.023 0.018

0.057 0.047 0.012 0.010 0.017

Children at school

Urban

0.203 0.213 0.145 0.131 0.127

0.388 0.307 0.257 0.255 0.222

0.431 0.430 0.369 0.332 0.347

0.198 0.154 0.138 0.104 0.113

0.552 0.527 0.446 0.414 0.368

0.174 0.126 0.121 0.106 0.096

Education of household head

Headcount Rates by Dimension and Countries.

Source: Authors’ calculations, based on SEDLAC database.

Year

Country

Table A5.

0.028 0.025 0.026 0.019 0.017

0.121 0.078 0.046 0.078 0.060

0.307 0.340 0.253 0.254 0.239

0.022 0.013 0.010 0.007 0.007

0.100 0.114 0.075 0.061 0.041

0.032 0.031 0.016 0.012 0.017

Water

0.123 0.105 0.081 0.063 0.068

0.383 0.369 0.300 0.290 0.303

0.407 0.404 0.487 0.451 0.448

0.136 0.090 0.061 0.049 0.030

0.341 0.317 0.261 0.244 0.223

0.143 0.111 0.158 0.154 0.176

Sanitation

0.032 0.024 0.018 0.010 0.014

0.313 0.259 0.253 0.281 0.251

0.281 0.251 0.197 0.145 0.153

0.270 0.108 0.081 0.069 0.074

0.031 0.028 0.017 0.014 0.013

0.019 0.020 0.013 0.012 0.024

Shelter

28 MARI´A EMMA SANTOS ET AL.

29

Multidimensional Analysis of Poverty in Latin America

Table A6.

Multidimensional Headcount Rates with Equal Weights Year 2006.

Country

Rural

Urban

k¼1 k¼2 k¼3 k¼4 k¼5 k¼6 k¼1 k¼2 k¼3 k¼4 k¼5 k¼6 Argentina Brazil Chile El Salvador Mexico Uruguay

0.93 0.67 0.99 0.88

0.74 0.36 0.93 0.72

0.45 0.14 0.77 0.49

0.20 0.03 0.52 0.24

0.05 0.00 0.27 0.08

0.00 0.00 0.06 0.01

0.29 0.52 0.22 0.66 0.51 0.24

Source: Authors’ calculations, based on SEDLAC database.

0.10 0.18 0.04 0.44 0.28 0.06

0.03 0.06 0.01 0.25 0.13 0.02

0.01 0.02 0.00 0.12 0.05 0.00

0.00 0.00 0.00 0.04 0.01 0.00

0.00 0.00 0.00 0.01 0.00 0.00

CHAPTER 2 MULTIDIMENSIONAL POVERTY AMONG CHILDREN IN URUGUAY Vero´nica Amarante, Rodrigo Arim and Andrea Vigorito ABSTRACT The multidimensional nature of well-being is now widely recognized. However, multidimensional poverty measurement is still an expanding field of research and a consensus about the ‘‘best’’ composite indicator has not yet emerged. In this chapter, we provide an empirical analysis using three existing methodologies: Bourguignon and Chakravarty (2003), Alkire and Foster (2007), and Lemmi (2005); Chiappero Martinetti (2000). We present an empirical study of the convergence and divergence of poverty profiles for children in Uruguay considering the following dimensions: nutritional status, child educational achievement, housing condition, and household income. Our data gather information of 1,185 children attending public schools in Montevideo and the surrounding metropolitan area, and were specially gathered to carry out a multidimensional analysis of poverty. Our results indicate that the three families of indexes yield very different cardinalizations of poverty. At the same time, the correlation coefficients among the three groups of measures for the generalized headcount ratio also highlight important differences in the children labeled as ‘‘more deprived.’’ For the generalized severity and intensity indexes the correlation coefficients increase significantly suggesting a high level of Studies in Applied Welfare Analysis: Papers from the Third ECINEQ Meeting Research on Economic Inequality, Volume 18, 31–53 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2010)0000018005

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VERO´NICA AMARANTE ET AL.

concordance among the three measures, particularly among the Bourguignon and Chakravarty methodology and the Alkire and Foster one.

1. INTRODUCTION During the past decade researchers on well-being seem to have come to an agreement about its multidimensional nature. The extension of this idea to the field of poverty analysis leads to the consideration of deprivation taking into account not only income, but also other dimensions or capabilities, following Sen’s seminal approach. On the empirical side, this implies the challenge of building composite indexes that result from well-being assessments in different dimensions. Multidimensional poverty measurement is still an expanding field of research and several methods have recently been developed. As argued by Atkinson (2003), the definition of an aggregation procedure, the weight assigned to each component, and the rates of substitution among dimensions are key issues that must be addressed in order to create multidimensional poverty measures. This chapter aims at comparing three existing methodologies: two of them are variants of generalized Foster Greer and Thorbecke indexes, one developed by Bourguignon and Chakravarty (2003) and the other by Alkire and Foster (2007). Additionally, we consider aggregate indexes based on fuzzy sets (Lemmi, 2005; Chiappero Martinetti, 2000). Our aim is to measure the degree of convergence of these different multidimensional measures in a study of child poverty.1 In Uruguay, poverty analysis has been mainly done on the basis of income measures. From these poverty profiles, a clear fact has emerged: there is a very strong association between age and income poverty. This is the result of various factors: labor market performance, the characteristics of a social protection system that is mainly focused in the elder population, and differences in fertility rates, among others (UNDP, 2005). Children constitute a deprived group, that has attracted the attention of researchers and policy makers both in terms of income and multidimensional poverty (Arim & Vigorito, 2007). Income poverty incidence in households with children is more than five times higher than its value in households with elder adults (Table 1). In this chapter we aim at providing new evidence on child poverty in Uruguay, based in a multidimensional framework and exploring a new data, Encuesta de Situacio´n Nutricional de los Nin˜os 2006 (ESN06),

33

Multidimensional Poverty among Children in Uruguay

Table 1.

1991 1994 1998 2002 2003 2004 2005 2006

Income Poverty Incidence by Age Group (1991–2006). 0–5

6–12

13–17

18–64

65þ

41.1 30.5 34.7 46.5 56.5 56.5 54.1 46.0

39.8 28.6 29.2 41.9 50.2 53.7 51.0 43.8

33.0 24.0 26.7 34.6 42.8 45.0 42.8 37.8

19.1 11.9 13.1 20.3 27.8 28.7 25.8 21.3

12.5 4.1 5.4 5.4 9.7 10.8 9.2 7.5

Source: Elaborated by the authors based on Encuestas Continuas de Hogares.

specially built with this purpose. ESN06 contains data on dimensions that are relevant to child development and are not usually included in standard household surveys. The analysis of other domains than income, can highlight aspects of poverty not usually considered. At the same time, if the correlation among the dimensions is not high, multidimensional poverty indexes could yield to very different criteria in terms of identifying poor households. This finding can have strong policy implications for the targeting of social transfers. The chapter is organized as follows: Section 2 contains information on the dataset we used, the variables we constructed, and the different methodologies selected for computing multidimensional poverty indexes; Section 3 contains our main results; and Section 4 presents some final remarks.

2. CAPABILITIES, POVERTY, AND CHILDREN The capability approach developed by Sen provides a normative framework for the evaluation of individual well-being, which is conceived as the freedom of individuals to lead lives that are valued. The multidimensional nature of well-being is explicitly recognized in this approach. The approach implies a shift away from monetary indicators and toward a focus on nonmonetary factors in the evaluation of well-being. In this framework, poverty is defined as capability deprivation, or failure to achieve certain minimal or basic capabilities. These basic capabilities are the ability to satisfy certain crucially important functionings up to a certain minimally adequate levels (Sen, 1993). This approach emphasizes people’s ability to enjoy various sets of alternative beings and doings, and implies considering human freedom. In strict terms,

34

VERO´NICA AMARANTE ET AL.

poverty is capability deprivation, not functioning deprivation. But it is not possible to capture this freedom component in a poverty measure, as most of the time it is not possible to extrapolate from an achieved functioning set to the capability set associated with this achieved functionings. For this reason, empirical applications of Sen’s capability approach have argued that capability measurement is empirically very difficult or impossible to construct with existing data, and have been based on functionings. Our research undertakes the same approach, focusing on functionings or achievements of children. Capabilities, choices, and living conditions during childhood influence children’s positions and capabilities as adults (Sen, 1992). The relevance of this phase of life and the consideration of children as a subject with identifiable capabilities underlies our application of the capability approach to the study of child well-being. Biggeri, Libanora, Mariani, and Menchini (2006) argue that there are five important issues regarding children’s capabilities that should be considered. The first one refers to the fact that parents’ capability may directly or indirectly affect the capabilities of the child. Another important aspect is the fact that the possibility of converting capabilities into functionings depends on parents’ and teachers’ decisions. Child’s conversion factors from resources and commodities into capabilities and functionings is subject to further constraints, given that the child is neither an autonomous nor a passive actor, and the balance between these two states varies with age. A third issue refers to the more intense relationship between different capabilities and functionings at this age, as each capability has an intrinsic value but is also instrumental for other capabilities. The fourth issue is related to the fact that relevant capabilities for children can vary according to age, and even gender. Finally, the authors argue that children can have an important role in changing their conversion factors, and, through participation with others, modify external conversion factors, being resources for a better future. Research that concentrates on children’s well-being within the capability approach is not abundant. Klasen (2001) combines Sen’s capability approach and rights-based approach to analyze childhood social exclusion in OECD countries. He focuses on education as a relevant dimension, and presents information on dropouts and distribution of educational outcomes. Di Tomasso (2006) defines the concept of capabilities for children, and identifies relevant capabilities for Indian children following the methodology proposed by Robeyns (2003). He uses factor analysis to construct an aggregate measure of child well-being, based on four functionings corresponding to some of the relevant capabilities identified (Table 2).

35

Multidimensional Poverty among Children in Uruguay

Table 2. Author

Studies of Children Well-Being and Capabilities.

Method for Selecting Capabilities

List of Capabilities

Di Tommaso (2006)

The author selects seven out of ten central capabilities from Nussbaum’s (2000) list, following the procedure suggested by Robeyns (2003)

Life Bodily health Bodily integrity Senses, imagination, and thought Leisure activity Emotions Social interaction

Biggeri et al. (2006)

Participatory approach through public reasoning and scrutiny

Love and care Mental well-being Bodily integrity and safety Social relations Freedom of exploitation Shelter and environment Leisure activities (considered very important by children below 11 years)

Measurement of Well-Being Aggregate measure including four functionings: height for age, weight for age (both related to bodily health), enrolment (senses, imagination and thought), and work status (leisure activity) No empirical application

Source: Own elaboration.

Using a different methodological approach, Biggeri et al. (2006) identify a list of relevant capabilities for children. In their research, the selection of capabilities is understood as a task of a democratic process, so children were asked to define their capabilities. The authors conclude that children can conceptualize capabilities, and they identify a list of relevant capabilities. They highlight the fact that levels of relevance of different capabilities vary according to the age of the child (Table 2). To employ the capability approach to child well-being we must choose a list of capabilities that reflect what children are effectively able to do and to be, concentrating on their potential functionings. As discussed above, the inability to capture the freedom component leads most empirical studies, including this, to concentrate on actual functionings, which reflect achievements and outcomes. To operationalize the capability approach, we consider four functionings that are mentioned in Sen’s writings and that can be seen in most of the capabilities literature. Two of these dimensions refer directly to children: their nutritional status and their educational achievement. The other two dimensions reflect their environment and are relevant for the generation of children’s functionings: housing conditions and the level of income.

36

VERO´NICA AMARANTE ET AL.

3. METHODOLOGY In the process of measuring multidimensional poverty, important decisions have to be taken about many aspects: the selection of the dimensions to be considered and the indicators that reflect these dimensions, the weighting scheme attached to the selected dimensions, the setting of a threshold to identify deprived population, and the way of aggregating the results in each dimension. In this section we present our decisions related to these aspects: first we present our data and discuss the main dimensions, variables and thresholds of our analysis (Section 3.1), and then we discuss the methods selected to compute multidimensional measures (Section 3.2). These methods may imply different weighting schemes and aggregation criteria.

3.1. Data and Variables The analysis presented in this chapter is based on a dataset of a special survey, ESN06, based on a representative sample of children attending the third year of primary school in public institutions in Montevideo and the metropolitan area. Eighty-five percent of the children living in these areas attend public schools, so our analysis is representative of this population and probably is not considering the richer income strata. The sampling frame of this survey is the 2002 Height Census undertaken in all public schools in Uruguay. A first round of this survey was undertaken in 2004 (ESN04), when these children were attending the first year of primary schools. Information corresponding to the second round of the survey, used in this chapter, was collected in 2006, and includes 1,185 children. Fieldwork included collecting information for each child at school and then interviewing an adult in charge of the child. Table 3 presents the main characteristics of the sample. The identification of the dimensions and variables to include in a multidimensional analysis of poverty is a crucial step. We note that there is no consensus on the list of capabilities that must be considered; this is known in the literature as the problem of ‘‘horizontal vagueness’’ (Qizilbash, 2003). At the empirical level, this problem can be expressed as a trade-off between redundancies because of overlapping variables and the risk of not including relevant variables. Despite these problems, there is a ‘‘hard core’’ of dimensions that can be found in most empirical applications of this approach. We chose to include the following variables: child health, child educational attainment, housing conditions, and income. For each dimension, we specify

Multidimensional Poverty among Children in Uruguay

Table 3.

37

ESN06 Sample Characteristics.

Children in the sample % of boys Distribution of children by age 8 years 9 years 10 years þ 10 years Distribution of children by last schooling year approved First year Second year Third year

1,185 51.4 100.0 28.69 61.27 8.78 1.27 100.0 14.7 67.0 18.3

a threshold level; given the innate arbitrariness of this choice this problem is known as ‘‘vertical vagueness.’’ For the health (nutritional) dimension, we use the height for age index. Specifically, mild malnutrition is detected through the z-score (for height for age). Following the National Center for Health Statistics (NCHS) criteria, we consider that a child suffers malnutrition when his z-score is higher than 2 standard deviations from the mean. Educational attainment is measured considering school lag. A child that has repeated at least one grade is considered deprived in this dimension. The dimension of housing conditions is reflected by a crowding variable that considers that a household is deprived if the number of people sleeping in one room is greater than two. Children living in crowded households are defined as deprived in this dimension. Our dimensions try to reflect achievements, in order to be consistent with Sen’s capability approach. Nevertheless, we decided to include income that reflects a mean and not an achievement, and so is not strictly comparable with the other dimensions. Our decision is based on the fact that income is a central variable, particularly in a developing country emerging from a deep economic crisis. In other words, income provides information about the evolution of household well-being in the short run. In this case, the threshold is given by the national poverty line (INE, 2002). Table 4 describes the dimensions and variables, as well as their respective thresholds. 3.2. Methods In this section, we discuss the three main approaches to the study of multidimensional poverty: the Bourguignon and Chakravarty (2003)

VERO´NICA AMARANTE ET AL.

38

Table 4.

Dimensions, Variables, and Thresholds for Measuring Poverty.

Dimension

Variable

Threshold

Health Education Housing Income

Height for age (z-score) Repetition Crowding Per capita household income

Malnutrition: more than 2 standard deviations Repeated at least one academic year Two or more persons sleeping in the same rooma. Poor household: per capita income lower than the national poverty line

a

A room is defined as the total number of separate spaces in the house not considering kitchen and toilet.

approach, the Alkire and Foster (2007) approach, and the ‘‘fuzzy sets’’ approach. 3.2.1. Bourguignon and Chakravarty (2003) Generalized FGTs Bourguignon and Chakravarty (2003) propose a multidimensional poverty measure based on a specific threshold or deprivation line for each of the multiple dimensions of poverty. A person is poor if she falls beneath at least one of the deprivation lines. They depart from a vector Xi ¼ (xi1, xi2, y , xim) containing the quantity of each attribute j (with j ¼ 1, y , m) that a certain person or household i possesses, and a vector Z ¼ (z1, z2, y , zm) that reflects the deprivation thresholds. As mentioned before, person i is deprived in terms of attribute j if xijozj. In this context, a multidimensional poverty index can be defined as a nonconstant function P(X; Z) that gives the extent of poverty associated to attributes X and thresholds Z. This P index satisfies a set of properties as strong and weak poverty focus, symmetry, monotonicity, continuity, principle of population, scale invariance, and subgroup decomposability. Bourguignon and Chakravarty (2003) also consider two transfer properties that deal with the redistributive criteria involving two attributes. To illustrate these postulates, they assume that there are two persons, i and t, and a bidimensional poverty space associated with attributes j and k. Person i has more of k but less of j. If the two persons interchange an amount of attribute j and, after that, person i, who had more of k, has now more of j too, there is an increase in the correlation of the attributes within the population. It is reasonable to expect that such a switch will not decrease poverty if the two attributes correspond to similar aspects of poverty. The poorer person cannot compensate the lower quantity of one attribute by a

Multidimensional Poverty among Children in Uruguay

39

higher quantity of the other. On the same token, it is reasonable to expect that such a switch will not increase poverty if the two attributes correspond to different aspects of poverty. The nondecreasing poverty under correlation increasing switch (NDCIS) postulate indicates that poverty cannot decrease with such correlation increasing switches. The converse property is denoted NICIS (nonincreasing poverty under correlation increasing switches). The similarity or difference of attributes can be expressed in terms of substitutability or complementarities. If attributes are considered substitutes, then the marginal utility of one attribute decreases when the quantity of other increases. So a decrease in poverty due to the increase in one attribute is less when attributes are substitutes than when they are complements. The first postulate (NDCIS) holds for attributes that are substitutes, whereas the second one (NICIS) holds for attributes that are complements in the individual poverty function. The authors propose a full specification of a poverty multidimensional measure based on the FGT index and derived from a CES (constant elasticity substitution) function. For the two dimensions case, the P(X; Z) index becomes: "   y   y #a=y n X 1 x x i1 i2 a1 max 1  ;0 þ a2 max 1  ;0 Pya ðX; ZÞ ¼ n i¼1 z1 z2 where a1 and a2 are positive weights attached to the attributes, aW0 is a parameter that reflects ‘‘poverty aversion’’ and y is the parameter of substitution between the shortfalls of the attributes. In this way, multidimensional poverty is defined as the average of aggregate shortfalls, raised to the power a, over the whole population, and reflects a generalization of the FGT index for the multidimensional case. When a ¼ 0, the index becomes the multidimensional headcount. When a ¼ 1, the index becomes a multidimensional poverty gap, obtained by some particular averaging of the poverty gaps of the included dimensions. As in the one-dimensional case, higher values of a reflect more aversion toward extreme poverty. As stated by the authors, this measure has the property that it satisfies NDCIS or NICIS depending on the relation between a and y. If attributes are substitutes (aWy), then a transfer that increases correlation of attributes among individuals does not decrease poverty, whereas if attributes are complements (ary) such a transfer does not increase poverty. This implies that the drop in poverty due to a unit decrease in income is less important for people who have an educational level close to the education poverty threshold than for persons with very low education, if income and education

VERO´NICA AMARANTE ET AL.

40

are considered substitutes. On the contrary, the drop in poverty is larger for persons with higher education if these two attributes are supposed to be complements. One of the limitations of the generalization of this index for more than two dimensions implies assuming the same elasticity of substitution between attributes. 3.2.2. Alkire and Foster Multidimensional Index Alkire and Foster (2007) propose a class of measures to address multidimensional poverty based on the generalized FGT index. This new measure is sensitive to deprivation in each dimension and to the number of deprivations that each household or person experiences. In this sense, the index resembles the basic needs criteria, and is known as the dimension-adjusted FGT. In order to define the index, a poverty threshold for each dimension is required (zi). At the same time, a threshold reflecting the minimum number of deprivations per household or person must be defined (k). The extreme values for this threshold are the intersection criteria, where a person will be labeled as poor only if she is deprived in all the dimensions considered, and the union criteria, where a person will be labeled as poor if she is deprived in any one dimension. This methodology allows checking for the robustness of the poverty measure as k varies; in this chapter we let k vary between 1 and 4. Those persons or household that show a number of deprivations (ci) greater than k will be labeled as poor. The resulting poverty measure Ma represents the sum of the normalized poverty gaps in relation to the maximum value that that sum can assume: M a ¼ m½ga ðkÞ

for a  0

where m represents the average operator, a is a poverty aversion parameter, and g is a matrix whose elements represent the distance to each poverty threshold for those individuals with achievements lower than the threshold and whose number of deprivations is lower or equal to k (poor) and zero in the remaining cases (nonpoor). If zj represents the deprivation threshold for each dimension and z is the set of poverty thresholds, n is the number of persons or households included in the dataset, d is the number of dimensions, and Yij is the data matrix, n  d, the generalized poverty gaps matrix, ga, can be represented as:   zj  yij a gaij ¼ if yij ozj zj

Multidimensional Poverty among Children in Uruguay

gaij ¼ 0

41

if yij  zj

In the particular case of a equal to zero, the matrix will be composed of zeros and ones and the index represents the proportion of persons or households whose number of deprivations is greater or equal to k. Ma results from two components: a poverty measure (H, G, or S) and a measure of the average number of deprivations observed in the population under study (A). A can be written as: A¼

jcðkÞj qd

where |c(k)| is the total number of deprivations experienced by those persons with ciWk and qd is the total number of deprivations observed in the population under study. This index reflects the average proportion of deprivations faced by poor people: M 0 ¼ HA

if a ¼ 0

M 1 ¼ HGA

if a ¼ 1

M 2 ¼ HSA

if a ¼ 2

where H represents the proportion of persons identified as poor. The interpretation of the three indexes is very similar to the unidimensional FGTs with parameters 0,1, and 2. M1 is the sum of the normalized poverty gaps over the number of potential deprivations (nd). G reflects the average poverty gap. M2 considers the squared poverty gap and S reflects the average severity of poverty. Although this measure accepts both continuous and discrete variables, M1 and M2 are hybrid measures when the two types of variables are considered in the index. The measures combine poverty gaps and squared poverty gaps in the case of continuous variables and maintain the cardinalization 0/1 for discrete ones. Ma satisfies the properties of subgroup decomposability, population invariance, symmetry, and poverty and deprivation focus. Ma can also be broken down into dimension subgroups. In effect, this measure can be expressed as: Ma ¼

n m½ga ðkÞ X j i¼1

d

42

VERO´NICA AMARANTE ET AL.

where gaj is the column j of the censored matrix ga ðkÞ. For each j, fm½gaj ðkÞ=dg=M a can be interpreted as the post-identification contribution of dimension j to overall multidimensional poverty (Santos & Ura, 2008). Moreover, for a greater than zero they also satisfy unidimensional and multidimensional monotonicity, normalization, and the weak principles of transfers and reranking. This family of indexes evaluates each individual’s achievements in each dimension, independent of the achievements in the other dimensions and of others’ achievements. This means that it considers each dimension as independent. As stated in Santos and Ura (2008), the proponents of the index explain that the measure could be converted in order to consider either all dimensions as substitutes or all dimensions as complements, as it is the case in Section 3.2.1. Such transformation would be at the cost of losing the possibility of breaking it down into dimensions. 3.2.3. Fuzzy Sets This measurement procedure entails a very different concept of poverty than the previous two approaches. Both at the unidimensional and multidimensional levels, poverty measurements face the problem of vertical vagueness; that is, the arbitrariness involved in the specification of the threshold level, for one or many dimensions. The set up of threshold levels and the dichotomization of the population in two excluding groups hides the fact that deprivation is a matter of the degree, not a clear condition free of ambiguity (Betti, Cheli, Lemmi, & Verma, 2005). The fuzzy set theory replaces the traditional approach to the demarcation of poverty through of a binary function that assigns people to two nonsuperposed sets (poor and nonpoor) by a generalized function, which varies between zero and one. This function is named as membership function and larger values indicate higher degrees of membership. In more formal terms, we can denote as X the population, A the fuzzy set, and mA the membership function. Then, mA is defined as: mA : X ! ½0; 1 If mA(x) ¼ 1, the person x belongs completely to the set A whereas if mA(x) ¼ 0 the person does not belong to A at all. In the intermediate cases, a person may belong partially to A. Therefore, the application of this approach to analyze well-being requires specifying the following three aspects: (1) definition of indicators with an appropriate ordering of its values that reflect different degrees of well-being in each dimension; (2) identification of extreme conditions that allow

43

Multidimensional Poverty among Children in Uruguay

considering that either the person belongs completely to the dimensionpoverty set or she does not belong at all to that set; (3) specification of the membership function. Cheli and Lemmi (1995) propose a membership function directly derived from the distribution function, which presents the following form: ( mðxj Þ ¼

0 if j ¼ 1 j1

mðx

Þþ

Fðxj ÞFðxj1 Þ 1Fðx1 Þ

where F(x) is the sampling distribution function of the x and j indicates the rank of the observation in an increasing ordering of x. However, this membership function is a totally relative approach2, since it assigns extreme values (zero and one) only to the lowest and highest positions in the rank. If the researcher thinks that there are both an upper threshold (zu) over which a given functioning is fully achieved by a person and a lower threshold (zl) below which she does not achieve an acceptable functioning at all, then the membership function can be reformulated in order to capture this situation:

hðxj Þ ¼

8 > < > :

mðxj1 Þ þ

1 if xj ozl Fðxj ÞFðxj1 Þ 1Fðx1 Þ xj

2 ðzl ; zu Þ

0 if xj 4zl

The last expression combines a relative fuzzy approach in the cases that belong to the interval (zl,zu) with absolute thresholds to identify the situations of extreme deprivation or complete fulfillment of basic functionings. In this chapter, we use a TFR approach presented previously to compute the membership function for the social participation dimension. In the remaining dimensions, the mixed approach represented by h is used. Table 5 shows the upper and lower thresholds for each dimension. After computing the membership function for each dimension, we use the Bourguignon and Chakravarty (2003) operator to obtain a composite index. Table 6 summarizes the information discussed above about the three methodologies for poverty measurement used in this chapter. In the three families of indexes computed in this chapter, we use the same weight for each dimension (0.25). By doing so, we are avoiding the problem of assigning different relevance to each dimension, although it is an arbitrary weighting scheme.

VERO´NICA AMARANTE ET AL.

44

Table 5.

Upper and Lower Threshold by Dimension.

Dimension

Lower Threshold

Upper Threshold

Nutrition Education Housing Income

2 Standard deviations Repeated at least one academic year More than three persons sleeping in the same room Poor household: per capita income lower than the national food basket

1 Standard deviation No upper threshold No upper threshold No upper threshold

Table 6.

Main Characteristics of Methods Used for Poverty Measurement. Bourguignon and Chakravarty

Theoretical referent Type of variables admitted Relation among variables Weights

Vaguely SCA Continuous

Alkire and Foster

Fuzzy Sets

SCA Continuous and discrete

SCA Continuous and discrete Independence Substitution or complementation Defined by the researcher Defined by the researcher Ordinal Cardinal Yes Yes

Substitution or complementation Defined by the researcher Type of poverty ranking Cardinal Decomposability by Yes groups Decomposability by Only in the special case After identification, the Only in the special dimension when a ¼ y weight of each case (dimensions (dimensions are dimension on total are independent) independent) poverty can be assessed

4. MAIN RESULTS In the first part of this section we present a brief analysis of poverty in each of the dimensions considered in this study (Section 3.1). After that, we compare the results obtained from the computation of the three families of multidimensional indexes (Section 3.2).

4.1. Poverty by Dimension The magnitude and degree of deprivation in each of the dimensions considered in this study varies significantly (Table 7). Income poverty is

45

Multidimensional Poverty among Children in Uruguay

Table 7.

Income Crowding Repetition Nutrition

Table 8.

Income poverty Crowding Education Nutrition

FGT Indexes by Dimension (2006). FGT0

FGT1

FGT2

0.725 0.262 0.149 0.024

0.395 0.095 0.075 0.005

0.263 0.044 0.038 0.002

Correlation Coefficients among Dimensions. Income

Crowding

Education

Nutrition

1.000 0.374 0.211 0.050

1.000 0.230 0.041

1.000 0.017

1.000

Significant at 90%; Significant at 95% or more.

prevalent among almost three out of four children in our sample. Recall that income poverty presents a pattern strongly associated with age and is mainly concentrated in households with children. At the same time, as the children included in our sample attend to public schools, the upper tail of the distribution is probably lost and hence poverty is higher compared to figures for the overall population of children aged 9 years (66% in 2006). The intensity and severity of income poverty is also higher among this population. Crowding affects only one of four children in our sample and its deepness is considerably lower than income poverty. Repetition affects 15% of the children included in our sample and it is one of the main problems at the primary school system in Uruguay together with absenteeism (UNDP, 2008). Severe nutritional problems only affect 2.4% of Uruguayan children attending primary schools. Its intensity and severity is also very low. In comparison with most Latin American countries, the performance of Uruguay in terms of nutrition is quite good. Uruguay’s main nutritional problems is obesity rather than emaciation. The correlation coefficients among the different dimensions are low (Table 8). The highest association refers to the negative relation between household average income and crowding, depicting the access to resources of the adults in charge of the children.

VERO´NICA AMARANTE ET AL.

46

This relatively low association among the dimensions included in the index suggest that identifying poor children in terms of one dimension, usually income, can potentially neglect children that show deficits in other important aspects of life. 4.2. Aggregate Indexes We now turn to the aggregate multidimensional poverty measures. We first comment on the results obtained for the three families of aggregate measures, considering variations in the different parameters used in the computation and then assess the extent to which the three measures identify and produce the same orderings in the deprivation of children. In the three cases, we built the indexes for values of the poverty aversion parameter (a) equal to 0, 1, and 2. In the Alkire and Foster version of generalized FGTs, we computed the calculations considering k equal to 1, 2, 3, and 4, representing the two extreme cases – the union and intersection criteria. In the case of the generalization of FGT proposed by Bourguignon and Chakravarty (2003) – used to compute this class of indexes and to aggregate the membership functions of the fuzzy sets – we considered values of the scale substitution parameter (y) equal to 1 and 2. This was done in order to assess whether the results vary under different assumptions of substitutability and complementarity among dimensions. The assumption of substitutability (aWy) holds in this empirical application only for the cell highlighted in gray in Table 9. The three families of indexes yield very different cardinalizations of multidimensional poverty. The significant differences between the two versions of Table 9.

Composite Indexes by Methodology and Parameter.

Multidimensional Poverty among Children in Uruguay

47

generalized FGTs follow to the fact that the aggregation procedures and the unit of analysis are different. For example, the value of the Alkire and Foster index when a ¼ 0 and k ¼ 1 is showing that there are 49.5% deprivations among all the plausible deprivations that can be observed in the sample. As long as k grows, the number of deprivations falls as the conditions for a child to be considered poor become more demanding. Recall that the number of deprivations in this index is computed only for the cases where the number of deprivations observed for a child is greater or equal to k. In the Bourguignon and Chakravarty generalized FGTs case, the headcount index (a ¼ 0) reflects the proportion of children who are poor in any of the dimensions. These results are showing that a very high percentage of children in our sample are deprived in at least one of the dimensions considered. As expected, the magnitude of deprivation is higher when substitutability between attributes falls (higher y). This responds to the fact that low substitutability between attributes gives more weight for each observation to attributes with the largest shortfalls, leading to a higher index. The fuzzy set methodology aggregated using the Bourguignon and Chakravarty (2003) operator yields results that indicate higher poverty than the Bourguignon and Chakravarty measures. This is reasonable as long as these indexes can be interpreted as vulnerability indexes rather than poverty ones. This observation comes from the fact that information from the observations whose achievements are higher than the lower poverty threshold. Hence, the level of vulnerability is high as long as it affects almost 4 out of 5 of the children included in the sample. To assess the extent to which the rankings of children provided by these indexes vary we computed correlation coefficients among them and built quintiles for each multidimensional measure (Table 10). The highest value of the correlation coefficients between Alkire and Foster’s measure and Bourguignon and Chakravarty’s measure is near 0.6 in the case of the multidimensional headcount ratio (a ¼ 0) considering that the dimensions are complementary (ary) and when k is lower than 4. Although the Alkire and Foster measure does not assume a parameter for modeling the substitutability or complementation relation among attributes, these results suggest that the measure is closer to a complementation relation. The correlations of the Alkire and Foster measure (k ¼ 0) with the fuzzy sets aggregate indexes are lower (around 0.5) although the same comments apply. The Bourguignon and Chakravarty measure and the fuzzy sets measure present a correlation of 0.6 or more when the same assumptions hold with regards to complementation among attributes.

A&F 0 (k ¼ 1) A&F 0 (k ¼ 2) A&F 0 (k ¼ 3) A&F 0 (k ¼ 4) A&F 1 (k ¼ 1) A&F 1 (k ¼ 2) A&F 1 (k ¼ 3) A&F 1 (k ¼ 4) A&F 2 (k ¼ 1) A&F 2 (k ¼ 2) A&F 2 (k ¼ 3) A&F 2 (k ¼ 4) B&C (a ¼ 0; y ¼ 1) B&C (a ¼ 1; y ¼ 1) B&C (a ¼ 2; y ¼ 1) B&C (a ¼ 0; y ¼ 2) B&C (a ¼ 1; y ¼ 2) B&C (a ¼ 2; y ¼ 2) Fuzzy set (a ¼ 0; y ¼ 1) Fuzzy set (a ¼ 1; y ¼ 1) Fuzzy set (a ¼ 2; y ¼ 1) Fuzzy set (a ¼ 0; y ¼ 2) Fuzzy set (a ¼ 1; y ¼ 2) Fuzzy set (a ¼ 2; y ¼ 2)

Pearson Correlation Coefficients among Poverty Measures.

0.486

0.504

0.571

0.455

0.486

0.468

0.365

0.491

0.49

0.469

0.433

0.463

0.481

0.586

0.472

0.496

0.48

0.377

0.47

0.494

0.481

0.447

0.477

0.496

0.456

0.473

0.484

0.437

0.398

0.499

0.511

0.491

0.603

1 0.925 0.407 0.487 0.527 0.767 0.376 0.332 0.354 0.53 0.328 0.575

1 0.986 0.966 0.431 0.502 0.525 0.801 0.398 0.345 0.359 0.554 0.347 0.585

1 0.264 0.52 0.524 0.843 0.244 0.367 0.372 0.588 0.213 0.595

0.143

0.135

0.126

0.205

0.174

0.132

0.048

0.077

0.086

0.094

0.124

1 0.101 0.102 0.148 0.925 0.037 0.038 0.075 0.806 0.121

0.726

0.707

0.675

0.625

0.652

0.568

0.833

0.908

0.836

0.909

0.968

1 0.994 0.85 0.159 0.905 0.906 0.876 0.174 0.874

0.724

0.706

0.673

0.626

0.659

0.591

0.827

0.904

0.836

0.903

0.966

1 0.847 0.158 0.895 0.902 0.869 0.173 0.877

0.678

0.653

0.614

0.606

0.617

0.526

0.752

0.823

0.722

0.843

0.899

1 0.182 0.756 0.758 0.896 0.183 0.766

0.19

0.174

0.157

0.272

0.217

0.148

0.085

0.123

0.125

0.139

0.17

1 0.084 0.084 0.114 0.95 0.151

0.542

0.525

0.498

0.453

0.477

0.41

0.971

0.946

0.745

0.971

0.916

1 0.999 0.947 0.115 0.706

0.544

0.528

0.501

0.456

0.484

0.424

0.97

0.948

0.748

0.971

0.919

1 0.947 0.115 0.712

0.562

0.54

0.507

0.489

0.503

0.425

0.955

0.939

0.729

0.975

0.931

1 0.136 0.703

0.182

0.167

0.15

0.248

0.202

0.137

0.112

0.151

0.143

0.151

0.179

1 0.153

0.649

0.635

0.608

0.584

0.649

0.623

0.664

0.818

0.97

0.733

0.875

1

0.689

0.67

0.637

0.601

0.642

0.577

0.901

0.971

0.871

0.957

1

0.574

0.553

0.52

0.503

0.524

0.449

0.977

0.969

0.766

1

0.583

0.569

0.544

0.534

0.612

0.601

0.732

0.868

1

0.588

0.573

0.546

0.512

0.566

0.526

0.961

1

0.471

0.455

0.43

0.403

0.433

0.384

1

0.678

0.677

0.665

0.77

0.904

1

0.777

0.764

0.737

0.957

1

0.744

0.721

0.684

1

0.971

0.991

1

0.994

1 1

B&C Fuzzy Fuzzy Fuzzy Fuzzy Fuzzy Fuzzy B&C B&C B&C B&C A&F 0 A&F 0 A&F 0 A&F 0 A&F 1 A&F 1 A&F 1 A&F 1 A&F 2 A&F 2 A&F 2 A&F 2 B&C Set Set Set Set Set Set (k ¼ 1) (k ¼ 2) (k ¼ 3) (k ¼ 4) (k ¼ 1) (k ¼ 2) (k ¼ 3) (k ¼ 4) (k ¼ 1) (k ¼ 2) (k ¼ 3) (k ¼ 4) (a ¼ 0; (a ¼ 1; (a ¼ 2; (a ¼ 0; (a ¼ 1; (a ¼ 2; y ¼ 1) y ¼ 1) q ¼ 1) y ¼ 2) y ¼ 2) y ¼ 2) (a ¼ 0; (a ¼ 1; (a ¼ 2; (a ¼ 0; (a ¼ 1; (a ¼ 2; y ¼ 1) y ¼ 1) y ¼ 1) y ¼ 2) y ¼ 2) y ¼ 2)

Table 10.

Multidimensional Poverty among Children in Uruguay

49

Correlations between the Alkire and Foster measure and the Chakravarty and Bourguignon measure increase significantly and are above 0.9 when considering the generalized poverty gaps (a ¼ 1) suggesting that the rankings are almost the same. In this case, the substitutability assumption among dimensions yields to different results in the value of the correlation coefficient although it is always very high. Now, assuming k ¼ 1 or k ¼ 2 produces almost no difference but for k ¼ 3 or k ¼ 4 the correlation falls significantly. In the case of the pairwise correlation among the Alkire and Foster measure and the fuzzy sets measure, the correlation increases but in a less substantial proportion than in the former case. In this case, the different assumptions on the value of the scale parameter produce significant variations in the correlation coefficient; the value y ¼ 2 increases the correlation significantly. The previous comments on the values of k apply for this case. In regard to the correlation between the Bourguignon and Chakravarty measure and the fuzzy sets measure, there are not significant changes in the correlation coefficients in the case of the generalized poverty gap. The generalized severity measure (a ¼ 2) increase even more the correlations between Alkire and Foster and Chakravarty and Bourguignon measures reaching values of 0.97 when k is lower than 4, showing again that the strict intersection criterion is too demanding. The value of the scale parameter makes no difference now but in the two cases we are assuming complementarity. In the case of the fuzzy sets measure, the correlation with the Alkire and Foster index falls again and the results are very similar to what happens in the case of the generalized headcount ratio. There are no changes in the correlation coefficient, which keeps at a significantly lower value for the fuzzy sets and the Chakravarty and Bourguignon measures. Finally, for the case in which correlations were more significant (a ¼ 1) we divided the children in quintiles of each composite measure, to illustrate about the degree of overlapping between the different measures (Table 11). Results show again a high degree of overlapping among the Bourguignon and Chakravarty measure and the fuzzy sets index on one hand and among the Bourguignon and Chakravarty measure and the Alkire and Foster index on the other. Since the indicators are increasing with the deprivation level, the fifth quintile represents the poorest children. The 84% of children that belong to the last quintile according to Bourguignon and Chakravarty index also belong to that quintile according to Alkire and Foster index.

VERO´NICA AMARANTE ET AL.

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Table 11. Distribution of the Children by Quintile of Aggregate Poverty Measure. (Richest Quintile ¼ 1; Poorest Quintile ¼ 5). Quintiles B&C (a ¼ 1, y ¼ 2)

Quintiles Fuzzy Sets (a ¼ 1, y ¼ 2)

1 2 3 4 5 Total

1

2

3

4

5

Total

8.3 21.7 10.2 0.0 0.0 15.9

41.7 13.3 15.3 1.7 0.0 19.2

35.0 31.7 28.8 10.0 1.7 21.3

13.3 28.3 27.1 16.7 1.7 21.8

1.7 5.0 18.6 71.7 96.6 21.7

100.0 100.0 100.0 100.0 100.0 100.0

Quintiles fuzzy sets (a ¼ 1, y ¼ 2)

Quintiles Alkire and Foster (a ¼ 1, k ¼ 2)

1 2 3 4 5 Total Quintiles B&C (a ¼ 1,y ¼ 2)

1 2 3 4 5 Total

1

2

3

4

5

Total

11.8 53.6 39.0 27.4 5.5 27.5

27.9 14.8 8.5 9.3 2.1 12.5

59.5 19.8 14.4 5.5 0.4 20.0

0.8 10.6 22.9 41.8 24.2 20.0

0.0 1.3 15.3 16.0 67.8 20.0

100.0 100.0 100.0 100.0 100.0 100.0

Quintiles Alkire and Foster (a ¼ 1, k ¼ 2) 1

2

3

4

5

Total

85.0 10.0 0.0 1.7 0.6 27.5

15.0 23.3 0.0 0.0 0 12.5

0.0 38.3 18.6 0.0 0 20.0

0.0 26.7 39.0 18.3 15 20.0

0.0 1.7 42.4 80.0 84.4 20.0

100.0 100.0 100.0 100.0 100.0 100.0

5. FINAL REMARKS This chapter studies child poverty in Uruguay using the multidimensional approach first advocated by Sen. Putting Sen’s idea into practice can be problematic. The literature addresses these problems using different methodologies that attempt to preserve the informative and conceptual contents of the approach. Basically, the problem consists on aggregating, for each person, the deprivations in the different attributes in a first stage. In a second stage, it is necessary to determine the way in which the aggregate deprivations of the individuals are combined. In this chapter, we considered

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three of the proposals that address this problem: two of them were built on generalized FGTs, and the other one based on fuzzy sets. To operationalize our study of multidimensional child poverty in Uruguay we consider four attributes – income, nutrition, household crowding, and education. We find relatively low correlations among these dimensions suggesting that identifying poor children in terms of one dimension, usually income, can potentially neglect children that show deficits in other important aspects of life. We consider three families of indexes in this chapter, the Bourguignon and Chakravarty measure, the Alkire and Foster index, and the fuzzy set approach. The cardinalizations of the three families of indexes computed in this research yield very different results. Orderings for the generalized headcount ratio are also different for each family, especially in the Alkire and Foster index. For the generalized severity and intensity measures, the orderings are very close, particularly when considering the generalized Bourguignon and Chakravarty measure and the Alkire and Foster measure. This is especially relevant as long as these multidimensional measures can be used for focalization of public transfers. The results obtained using the fuzzy sets methodology yield to different results and this can be related to the fact that this methodology uses information from a broader proportion of the distribution. Rather than a poverty measure, fuzzy sets-based indexes resemble vulnerability indexes. The magnitude and degree of deprivation in each of the dimensions considered in this study varies significantly, income being the more extended one. This finding illustrates that income poverty is not necessary mechanically translated into the remaining domains. The low degree of correlation among the different dimensions considered shows that the election of the domains in which poverty assessments are to be done has strong policy implications. Finally, this exercise shows that although the cardinal values of the indexes are very different they produce very similar orderings among the extreme cases whereas in intermediate situations variations in rankings are more significant.

NOTES 1. The multidimensional approach to poverty is not new in Latin America, as the basic needs approach has been widely used in the region for almost three decades. This approach was fostered in the region by the Economic Commission for Latin America

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and the Caribbean, since the 1980s. It has been subject to important criticisms due to the lack of a clear conceptual framework. Recently, Battiston, Cruces, Lo´pez Calva, Lugo, and Santos (2009) calculate multidimensional poverty measures, for six countries in the region, including Uruguay. 2. This approach has been named Totally Fuzzy and Relative (TFR).

REFERENCES Alkire, S., & Foster, J. (2007). Counting and multidimensional poverty measurement. OPHI Working Paper no. 7. OPHI Working Paper Series. University of Oxford. Arim, R., & Vigorito, A. (2007) Un ana´lisis multidimensional de la pobreza en Uruguay. 1991– 2005. Documento de Trabajo 10/06, Instituto de Economı´ a, UDELAR. Atkinson, A. B. (2003). Multidimensional deprivation: Contrasting social welfare and counting approaches. Journal of Economic Inequality, 1, 51. Battiston, D., Cruces, G., Lo´pez Calva, L. F., Lugo, A., & Santos, M. E. (2009). Income and beyond: Multidimensional poverty in six Latin American countries. OPHI Working Paper no. 17, University of Oxford, Oxford, UK. Betti, G., Cheli, B., Lemmi, A., & Verma, V. (2005). The fuzzy approach to multidimensional poverty: The case of Italy in the 90’s. Paper presented at the conference The Multiple Dimensions of Poverty, International Poverty Centre, Brazilia. Biggeri, M., Libanora, R., Mariani, S., & Menchini, L. (2006). Children conceptualizing their capabilities: Results of a survey conducted during the first children’s world congress on child labour. Journal of Human Development and Capabilities, 7(1), 59–83. Bourguignon, F., & Chakravarty, S. (2003). The measurement of multidimensional poverty. Journal of Economic Inequality, 1, 25–49. Cheli, B., & Lemmi, A. (1995). A ‘totally’ fuzzy and relative approach to the measurement of poverty. Economic Notes, 24(1), 115–134. Chiappero Martinetti, E. (2000). A multi-dimensional assessment of well-being based on Sen’s functioning theory. Rivista Internationale di Scienzie Sociali, CVIII, 207–231. Di Tomasso, M. L. (2006). Measuring the well being of children using a capability approach. An application to Indian data. CHILD Working Paper no. 6/2006, Max Planck Institute for Demographic Research, Rostock, Germany. INE (2002). Evolucio´n de la pobreza en el Uruguay por el me´todo del ingreso. 1986–2001. Available at: http://www.ine.gub.uy/biblioteca/pobreza Klasen, S. (2001). Social exclusion, children and education: Implications of a rights based approach. European Societies, 3(4), 413–445. Lemmi, A. (2005). The fuzzy approach to multidimensional poverty. The case of Italy in the 90s. Paper presented at the conference The Many Dimensions of Poverty, Centre for Poverty Analysis, Brazilia. Nussbaum, M. (2000). Women and human development. The capabilities approach. Cambridge: Cambridge University Press. Qizilbash, M. (2003). Vague language and precise measurement: The case of poverty. Journal of Economic Methodology, 10(1), 41–58. Robeyns, I. (2003). Sen’s capability approach and gender inequality: Selecting relevant capabilities. Feminist Economics, 9(2-3), 61–92.

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Santos, M. E., & Ura, K. (2008). Multidimensional poverty in Bhutan: Estimates and policy implications. OPHI Working Paper no. 14, University of Oxford, Oxford, UK. Sen, A. (1992). Inequality re-examined. Cambridge: Cambridge University Press. Sen, A. (1993). Capability and well-being. In: M. Nussbaum & A. K. Sen (Eds), The quality of life. Oxford: Clarendon Press. UNDP. (2005). El Desarrollo Humano en Uruguay. National Report. Montevideo. UNDP. (2008). Polı´tica, polı´ticas y desarrollo Humano en Uruguay. National Report. Montevideo.

CHAPTER 3 EXPLORING INTERGENERATIONAL EDUCATIONAL MOBILITY IN ARGENTINA$ Ana Ine´s Navarro ABSTRACT This chapter estimates the degree of intergenerational educational mobility in Argentina, focusing on the mobility differences between teenagers and young adults. Based on a new database, the Survey of Employment and Education of Youth (CEDLAS-INDEC) nonbiased mobility estimators for children older than teenagers is obtained. Our robust estimations reveal a lower degree of intergenerational mobility for young adults than for teenagers. Furthermore, young adult immobility is not uniform across parents’ education level. Finally, gender differences also affect mobility.

1. INTRODUCTION To evaluate social fairness requires knowing not only the degree of (dynamic) income inequality, but also an accurate measure of social mobility.

$ This Chapter is based on Chapter III of my PhD Dissertation at Universidad de San Andre´s. Argentina.

Studies in Applied Welfare Analysis: Papers from the Third ECINEQ Meeting Research on Economic Inequality, Volume 18, 55–78 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2010)0000018006

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In a high-mobility society, the prospects of the less-favored individuals are not severely hampered by their initial conditions of life. In such an ‘‘equal opportunity’’ society, family background like parents’ education and households’ income will not be relevant in determining a child’s future socioeconomic level. In Roemer’s view, childhood circumstances will not impinge on their success in later life. There is a vast empirical evidence indicating that Latin America is one of the most unequal regions of the world (Bourguignon, Ferreira, & Leite, 2002; De Ferranti, Perry, Ferreira, & Walton, 2004; Perry, Lo´pez, Maloney, & Arias, 2005; IPES, 2008). Although the high degree of Latin American inequality is clearly established, few systematic quantitative analyses of mobility across generations exist. The vast data requirement of those estimations is, undoubtedly, a major drawback for the researchers. To overcome the lack of appropriate longitudinal data, some researchers have attempted to measure intergenerational mobility in Latin American countries using cross-sectional surveys (e.g., Behrman, Birdsall, & Sze´kely, 1998; Dahan & Gaviria, 2001; Andersen, 2001; Conconi, Cruces, Olivieri, & Sa´nchez, 2008). Focusing on young children who still live with their parents, their strategy consists on estimating the degree to which family background determines the schooling outcomes of children. These studies actually measure educational mobility assuming a close connection between children’s schooling and children’s future opportunities. This methodology is advantageous in that it allows for comparisons of social mobility among the Latin American countries using standard data sets from existing household surveys. These papers focus on teenagers’ school attainment, revealing practically nothing about young adults’ mobility. The aim of this chapter is to investigate the degree of educational mobility in Argentina including young adults as well as teenagers. I use a new and original data source, The Survey of Employment and Education of Youth (Encuesta de Educacio´n y Empleo de los Jo´venes, EEEJ) collected by the Instituto Nacional de Estadı´stica y Censos (INDEC) and the Centro de Estudios Distributivos, Laborales y Sociales (CEDLAS) of Argentina. This chapter contributes to the ongoing discussion on the suitable data to measure social mobility in developing countries, where there exist no long panel data sets. In addition, this study is one of the very few for Argentina that study intergenerational mobility (see Ferna´ndez, 2006; FIEL, 2008). The rest of the chapter is organized as follows: Section 2 describes the methodology used to estimate educational mobility. Section 3 describes the data used for this project. Section 4 summarizes the main results. Section 5 explores mobility patterns for young adults and, Section 6 concludes.

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2. METHODOLOGY 2.1. Schooling and Social Mobility The intergenerational transmission of social status is a complex process that involves many links among family incomes, parental investments in children’s human capital, family tastes, children abilities, schooling attainment, and future incomes of children in later life (Becker & Tomes, 1979, 1986; Behrman et al., 1998; Han & Mulligan, 2001; Bowles & Gintis, 2002). In the context of perfect capital markets, without unobserved differences between low- and high-income households, there would be no differences in schooling investments associated with income after controlling for any observed differences in household characteristics. In this scenario, educational accomplishment of children would be independent of their households’ socioeconomic characteristics. Therefore, intergenerational mobility would be high. Instead, if household income and the unobserved innate ability of the children were positively correlated, associations would appear between household income and investments in schooling. The causal role of household income on child’s schooling appears with imperfect capital markets as well. Following Behrman, Gaviria, and Sze´kely (2001), measuring social mobility entails the estimation of a dynamic linear model linking a relevant socioeconomic indicator for entity i in period t (Sit) with the value of that indicator in the previous period (Sit1) and a stochastic term (wit) that is independent of the preceding period indicator and that is independently distributed across individuals and across periods: S it ¼ a þ bS it1 þ wit

(1)

The standard interpretation of b suggests a very limited intergenerational mobility whenever b is close to unity, whereas the estimates of b close to zero suggest that outcomes, say earnings, incomes or schooling, are not closely related across the generations. In this standard model, b is interpreted as a measure of the degree to which family background affects children’s socioeconomic outcomes, and thus as a measure of (in)equality of opportunities. However, considering equality of opportunities as synonymous with a zero intergenerational correlation could be misleading, particularly when taking public policies into account to enhance fairness. Parents influence children not only through the genetic transmission of ability but also through their social connections, culture, beliefs, and motivation. b value close to zero would imply that all these sources of inheritance are irrelevant. As Roemer

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(2004) points out, this is ‘‘a view that only a fraction of those, who consider the issue would, upon reflection, endorse’’ (Roemer, 2004, p. 49). Applying the linear model in Eq. (1) to the transmission of schooling from parents to children, Sit refers to the educational children achievement and Sit1 to the educational attainment of each parent or the education level of the most educated parent. Therefore, a b coefficient close to unity indicates that schooling achievements between generations are closely related: children belonging to a family where parents have a high educative level will have a high level as well; conversely, those unfortunate children with uneducated or less-educated parents will achieve a low-schooling level.

2.2. Data Requirements The estimation of intergenerational mobility is a challenging task due to the detailed data requirements it necessitates. Actually, ideal data sets for intergenerational studies rarely exist even in developed countries (Corak, 2006). This is because analyzing the linkage of earnings or incomes across generations requires a longstanding longitudinal survey that follows people from their early years when living in their parental home to their adulthood. The survey needs also to be based on a representative sample of individuals. Parental incomes need to reflect a measure of their permanent income, not merely the annual income for a limited number of years. In developing countries, these requirements largely surpass the longitudinal existent data. Not only are sample sizes often too small but also the temporal length of the surveys. In addition, household panel studies usually do not follow people who moved out of their original households, so the samples are not truly representative (Jenkins & Siedler, 2007). In fact, very little information is available on how family background affects the socioeconomic outcomes and equality of opportunity in Latin America. Behrman et al. (1998), Dahan and Gaviria (2001), Andersen (2001), and Conconi et al. (2008) offer an alternative approach to measure social mobility. They use standard household surveys information on parental and children’s schooling. The authors focus on young children still coresiding with their parents, thus overcoming the lack of longitudinal data. The basic assumption underlying their strategy is that schooling and future opportunities are highly correlated for young people. At the same time, this approach contemplates the common belief that whenever family background exerts a

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strong influence on children’s success in adulthood, the principle of equality of opportunities is infringed (Conconi et al., 2008; Andrews & Leigh, 2009). This approach defines children’s schooling gap as ‘‘the disparity between the years of education that a teenager or young adult would have completed had she entered school at normal school starting age and advanced one grade each year, on the one hand, and the actual years of education, on the other hand’’ (Andersen, 2001, p. 8). This concept is a very simple indicator of future opportunities, often very well suited to study social mobility for teenagers or young adults. For example, a 17-year-old teenager who has completed 9 years of schooling will register a schooling gap of (1796) ¼ 2 years, if he lives in a country where children begin schooling only at the age of 6 years. Hence, schooling gap is defined as the average years of missing schooling time. Using children’s schooling gap as the independent variable requires modifying the specification in Eq. (1) to the specification in Eq. (2): SGc ¼ a þ bSp þ wc

(2)

Herein, SGc refers to children’s years of missing education, Sp to the parental education level, and wc is an error term. The modified linear model in Eq. (2) estimates how strong is the influence of parental educational level (Sp) on the children schooling achievement (SGc). Because we define SGc as children’s years of missing education; b is expected to be negative. A b coefficient far from zero suggests a high generational schooling dependence as long as better educational parental level prevent children for missing years of school (low educational mobility), whereas a b close to zero suggests that schooling outcomes are not closely related across generations (high educational mobility). Following Conconi et al. (2008), the schooling gap has several advantages as compared with measures based on earnings or years of education. First, income measures are notoriously inaccurate, highly dependent on the season for large groups of the population, and often subject to measurement error bias. Second, using the number of years of education has not proved to be a good measure of educational attainment for young people, as many of them are still in school. A drawback to this approach is that the differences in school quality are not considered. A limitation of previous research using this approach is that although it allows for estimating intergenerational school mobility for teenagers still living in parental households, a very large group of children who are young adults are not included in the analysis. Studying mobility for young adults using the standard household surveys involves a substantial loss of

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information and probably a bias, as those who leave home relatively early may differ significantly from those who leave home later. (Behrman et al., 2001). Restricting the sample to teenagers who still coreside with their parents solves the data problem but tells only part of the story. Cross-sectional surveys containing retrospective questions on parental background are an alternative approach to overcome data restrictions. In retrospective surveys, individuals are typically interviewed only once and they provide retrospective information using recall. Parental income information from such a source is often unreliable. However, retrospective information on parental education and occupation are usually quite good. The retrospective questions allow us to estimate mobility for young adults as well as teenagers.1 Retrospective surveys not only enlarge the sample but also allow the researcher to detect potentially different links between children’s schooling and parental background across the successive educational levels. Educational persistence can be low for children at the secondary level when secondary schooling has been expanded or when it becomes mandatory. But, because tertiary or university studies did not expand, educational persistence could be large for individuals deciding to pursue their studies beyond secondary schooling. This advantage is more valuable given the close link between educational attainment and later incomes, and the fact that demand in the labor market for high-skilled workers has been steadily growing. Including young adults allows the inspection of the intergenerational link for those who have reached or would have reached tertiary studies or university. This chapter is based on the concept that when parents’ education and household income are both important determinants of the offspring’s opportunities, social mobility will be low. Conversely, when the opportunities children encounter are not strongly determined by family background, their social mobility would be high. Thus, this methodology recognizes a strong relationship between children future incomes and schooling. This is a sound hypothesis for Argentina where empirical evidence reveals that returns to education increases with the schooling level, and that the overall rate of return to an additional year of schooling is above the average for middle income countries (Lo´pez Bo´o, 2007). Hence, with convex returns to schooling, which implies that big differences in schooling eventually translate into big differences in earnings, failing to consider educational mobility for children approaching university studies could produce a misleading measurement of the real intergenerational mobility.

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3. DATA Data for this study comes from the Survey of Employment and Education of Youth (Encuesta de Educacio´n y Empleo de los Jo´venes, EEEJ ) collected in June 2005 by Instituto Nacional de Estadı´stica y Censos (INDEC) and Centro de Estudios Distributivos, Laborales y Sociales (CEDLAS) of Argentina (INDEC-CEDLAS). This is a new and original source of information that focuses on labor and educational issues in young people. The survey was designed to collect more complete and relevant information to study labor participation and labor experience of the youth. It provides additional data on the educational performance of the youth and the characteristics of the school – whether private or public, single or double shift type – that they have attended or are currently attending. It also offers information on early job experiences of the youth and their social environment and family background. This chapter is particularly interested in the respondents’ response in the survey, on their parents’ educational level. This is highly valuable, as the household survey regularly collected in Argentina omits any question on the respondents’ family background. The EEEJ was carried out in the Greater Buenos Aires, Argentina, on 807 young men and women between 15 and 30 years of age, from 526 households, who had been earlier interviewed by the Permanent Household Survey (Encuesta Permanente de Hogares, EPH ). This strategy links both surveys, substantially improving the Survey of Employment and Education of Youth by including the demographic, socioeconomic, and labor information of all household’s members. Unfortunately, at the time of this study, there was no way to link both surveys as the code necessary to accomplish the matching purposes had not been provided by INDEC (Marchionni, Bet, & Pacheco, 2007). However, the INDEC has already added several households’ variables in the EEEJ database. Table 1 shows that the three age groups are quite similar in size and a little less in gender composition. It also reveals that about 95% of all teenagers live in the parental home. However, for young adults, the percentage of coresiding children drops to 73% for those between 20 and 25 years of age, and to 61% for the older group. School gap, as defined in Andersen (2001), requires first calculating the number of schooling years per individual. In the survey, the respondent declares the highest level of schooling (e.g., primary and secondary) that the child is currently in or has been in, whether the child has completed it or not, and the last grade the child passed. This information makes it possible to calculate each individual’s years of schooling. In cases where some questions

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Table 1. Age Group 15–19 20–25 26–30 Total observations

Data Descriptive Statistics.

Total

Male (%)

Female (%)

Still Living at Home (%)

278 296 233 807

51 47 45 375

49 53 55 432

95 73 61 587

Source: Own calculations based on EEEJ. Note: This data come from the Survey of Employment and Education of Youth (Encuesta de Educacio´n y Empleo de los Jo´venes) (EEEJ) collected by the Instituto Nacional de Estadı´stica y Censos (INDEC) and Centro de Estudios Distributivos, Laborales y Sociales (CEDLAS) of Argentina (INDEC-CEDLAS) in June 2005.

are left unanswered, the variable years of schooling cannot be defined. However, in cases where children did not respond about the last grade they had passed but did declare that the highest level of education achieved was primary or secondary, it was possible to calculate their years of schooling.2 Considering parents’ education, the survey computes the highest education level that they have really achieved (e.g., none, primary, and secondary), but does not specify how many years of education they have completed. This is a limitation of the EEEJ and does potentially affect the regression coefficients. Hertz et al. (2007) found that regression coefficients were sensitive to these coding decisions. However, this chapter aims to measure the social mobility differences between teenagers and young adults, and not to establish its overall degree. Thus, it appears that this deficit would not seriously affect the main purpose of the study.3 Several definitions of household income are included. Herein, the per capita household income is included in the estimation of conditional education mobility. However, the income variables included in the EEEJ database correspond to parents’ home only for children who still coreside with them. For those living on their own, those magnitudes are their own contemporaneous incomes. Fortunately, the EEEJ includes a set of qualitative questions pertaining to which school the children attended or are still attending (e.g., whether it is public or private, single or double shift, teaching foreign languages or not). Those characteristics were analyzed to select the best proxy to the household’s income level at the time the children attending primary school. The survey includes a question on the age at which the individuals began their first job. The data reveals that in many cases this happened during the child’s school age. This early entrance to the labor market could undoubtedly

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be significant in explaining schooling gaps. Therefore, two additional controls were added. One of them is a dummy to detect whether the individual was previously employed or not. The other control is a variable that registers the age at the time of first employment. Table 2 shows that on average, the group of young adults (20–25 years of age) has achieved more schooling, whereas the teenage group presents a much lower schooling gap. For teenagers, the average schooling gap is 1.08 grades, indicating that on the average, a 16-year-old who should have completed 11 grades of schooling in reality has completed a little less than 10 grades. Likewise, a 20-year-old who could have completed 15 grades of schooling, in fact, has completed a little less than 10 grades. These data suggest that the average schooling gap is substantial for young adults. By gender, females have a slightly higher education level than males, and exhibit less disparity between the expected and actual years of education. Regarding the educational level of parents, Table 2 reveals that mothers possess a higher educational level than the fathers. The average educational level of the parents is around 3.5, indicating that they have achieved an education level between the primary and secondary levels. Although it is not

Table 2. Respondent

Whole sample 15–19 20–25 26–30 Male Female

Mean on Parental and Respondents’ Schooling Characteristics.

Average Schooling

11.60 10.79 12.31 11.74 11.37 11.80

Average Father Average Schooling Education Gap Level

3.66 1.08 5.09 5.25 3.81 3.52

3.40 3.31 3.58 3.31 3.52 3.31

Mother Average Education Level

Household’s Average Income Decile

Private Primary (%)

3.55 3.50 3.70 3.42 3.66 3.46

5.61 5.19 5.99 5.67 5.68 5.55

27.70 29.50 28.72 23.46 25.80 29.30

Source: Own calculations based on EEEJ. Note: This data come from the Survey of Employment and Education of Youth (Encuesta de Educacio´n y Empleo de los Jo´venes) (EEEJ) collected by the Instituto Nacional de Estadı´stica y Censos (INDEC) and Centro de Estudios Distributivos, Laborales y Sociales (CEDLAS) of Argentina (INDEC-CEDLAS) in June 2005. Average schooling includes 1 grade of preschool. In the survey, parent’s education is collected by attainment level. An average education level ¼ 3 means parents have achieved around 9–10 years of school. Average schooling gap for respondents older than 25 is computed assuming 17 years of education as a top. Parent’s education level is achieved assuming that all of them have completed the level.

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possible to precisely compare the number of years of education between the parents and children, it appears that education level does not differ too much between the two generations. However, the effective schooling level of parents would in reality fall below the level showed in Table 2. Respondents declared that 36–40% of both parents had not completed the education level computed in the survey when their children were attending school, or at least the children could not recall. Restricting the sample to those parents whose children could recall for certain that they had completed their education level (e.g., completed primary, completed secondary) reveals average education levels that are practically the same as those obtained with the full sample of parents. The table also reveals that the percentage of children who attended primary private schools ranges between 24% and almost 30%, indicating the major role of the private schools in Argentina’s educational system.

4. ESTIMATION RESULTS 4.1. Educational Mobility In this section, the results of applying the linear model in Eq. (2) are reported. Several different specifications are estimated and the results are shown in Tables 3–7 to detect differences in educational mobility between teenagers and young adults; the model is estimated for each group separately, in addition to the whole sample. All estimations are shown to be robust to specification changes of the basic model, and the sign and the significance of the coefficients do not change when adding progressively additional independent variables. Table 3A shows the results of estimating model in Eq. (2) using schooling gap as the dependent variable and the parents’ maximum educational level (from now on, ‘‘Max-schoolparents’’) as the independent. This exercise reveals the unconditional mobility (Fields, Puerta, Herna´ndez, & Rodriguez, 2007), documenting the degree of generational convergence on educational level. This is relevant because children’s convergence on educational achievement is an indicator of equality of opportunity in the economy. The results show that mobility is large for teenagers (b1519 ¼ 0.24) but not for young adults (b2025 ¼ 0.73 and b2630 ¼ 0.91), all the coefficients being statistically significant at a 5%. For example, for young adults older than 25 years, the findings indicate that an improvement of one education level for their most educated parent (e.g., from primary to secondary level) results in a

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Table 3.

Educational Mobility.

Schooling Gap

Panel A: All individuals Max-schoolparents Cohort-annual observations Adjusted R2

Whole sample

Teenagers

Young adults (20–25)

Young adults (26–30)

0.634 (0.0650)

0.236 (0.0610)

0.728 (0.0955)

0.907 (0.1091)

682 0.121

251 0.052

247 0.188

184 0.271

Panel B: Individuals whose parents completed some educational level 0.239 0.676 Max-schoolparents 0.581 (0.0897) (0.0767) (0.1238) Cohort-annual observations Adjusted R2

294 0.122

99 0.081

119 0.196

Panel C: All individuals with parents’ educational level reestimated 0.230 0.720 Max-schoolparents 0.599 (0.0643) (0.0596) (0.0927) Cohort-annual observations Adjusted R2

682 0.112

251 0.052

247 0.297

1.030 (0.1711) 76 0.319

0.907 (0.1112) 184 0.263

Source: Own calculations based on EEEJ. Note: For Panel A – Education level for parents is achieved assuming that all of them have completed the level. Note: For Panel B – The sample includes only the records for which educative level for parents was achieved before children attend media school. Note: For Panel C – Education level for parents whose children do not recall if they had completed the level they attended is reduced by one level. 0.05opo0.10. po0.05.

drop of almost a year in their schooling gap; for the youngest group, a onelevel improvement of their most educated parent amounts to a meager decrease of their schooling gap. Even so, as the schooling gap is quite different between teenagers and young adults, these differences in the absolute magnitudes of b do not necessarily indicate the relative differences in the family educational background influence over both groups. Considering that the schooling gap for teenagers is about 1 year, and about 5 years for young adults (25–30), their respective b coefficient entails about a 20%

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Table 4.

Intergenerational Mobility by Parent’s Level of Education.

Schooling Gap

Father schooling level Mother schooling level Sex Age Household income decile Cohort-annual observations Adjusted R2

Model 1 Whole sample

Teenagers

Young adults (20–25)

Young adults (26–30)

0.207 (0.0898) 0.262 (0.0916) 0.638 (0.2391) 0.455 (0.0263) 0.199 (0.0506)

0.007 (0.0882) 0.278 (0.0887) 0.062 (0.2248) 0.351 (0.0814) 0.022 (0.0538)

0.266 (0.1518) 0.173 (0.1492) 1.018 (0.4113) 0.914 (0.1148) 0.341 (0.0863)

0.489 (0.1752) 0.315 (0.1903) 0.927 (0.4745) 0.329 (0.1671) 0.384 (0.0895)

492 0.444

190 0.151

172 0.399

130 0.377

Source: Own calculations based on EEEJ. Note: Education level for parents is achieved assuming that all of them have completed the level. 0.05opo0.10. po0.05.

decrease in the average educational gap. Considering both these results together, it is evident that the parents’ education level is quite decisive on the generational transmission of social status for the young adults who take marginal decisions on education but not for those for whom education is at the compulsory stage. It is also evident that parents’ educational influence is proportionally the same for both groups, the average delay of their cohort taken into account. To check the robustness of the results obtained earlier, Table 3B shows the results of estimating unconditional educational persistence using a smaller subsample for which the children recall that both parents had completed their educational level by the time the children were at school. Despite the large drop in the sample size, the results are quite similar to those seen in Table 3A, corresponding to the full sample. Similar results are obtained with the whole sample, when assigning a lower education level to those parents whose children do not recall whether they had completed the educational level they had attended. The results are shown in Table 3C. Considering conditional mobility, Table 4 shows the results of estimating a model (Model 1) with the same independent variable but using both

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parents’ educational level as covariates, and the deciles of per capita household income, as family background. Standard controls such as sex and age are also included. When the whole sample is considered, the educational link between parents and children indicates high intergenerational mobility. Both coefficients are quite below 0.5, being statistically significant at the 1% level. The results also indicate that the schooling gap tends to diminish as the family income rises higher, being the coefficient statistically significant at the 1% level. The age coefficient is positive and statistically significant at the 1% level, indicating an increase in the years of missing education as the youth get older. The gender dummy has a negative and significant effect in the mean schooling gap, suggesting that the expected performance of boys is higher than that of girls. Table 4 also shows the results for each group of children estimated separately. The estimates of the intergenerational link on education for both parents grow markedly from teenagers to young adults (26–30 years of age). The father’s coefficient is neither statistically nor economically noted to be significant for teenagers. Following Behrman’s approach, Table 5 shows the results of applying the linear model in Eq. (2) to the transmission of schooling from the parents to Table 5.

Intergenerational Mobility by Parent’s Maximum Level of Education.

Schooling Gap

Max-schoolparents Sex Age Household income decile Cohort-annual observations Adjusted R2

Model 2 Whole sample

Teenagers

Young adults (20–25)

Young adults (26–30)

0.387 (0.0742) 0.623 (0.2365) 0.453 (0.0260) 0.213 (0.0489)

0.221 (0.0775) 0.0514 (0.2268) 0.379 (0.0820) 0.053 (0.0519)

0.322 (0.1318) 1.082 (0.4101) 0.888 (0.1150) 0.349 (0.0863)

0.692 (0.1287) 0.858 (0.4607) 0.303 (0.1622) 0.410 (0.0849)

499 0.447

193 0.148

173 0.394

133 0.395

Source: Own calculations based on EEEJ. Note: Education level for parents is achieved assuming that all of them have completed the level. 0.05opo0.10. po0.05.

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children, where Sp refers to the educational attainment of the most educated parent and the rest of explanatory variables are the same as in Table 4. The results in Table 5 are broadly similar to those found in Table 4, showing that the estimations are robust to changing the definition of the main independent variable. Here again differences are detected between the groups in the absolute magnitude of the intergenerational link, but the relative influence of parents’ education level over each group of young ones is similar. For the whole sample, the gender dummy shows a negative and significant effect in the mean schooling gap. For each sample subgroup separately, the effect of being a male is still advantageous but it is not statistically significant for teenagers. Regarding the effect of household incomes on children’s schooling performance, these results suggest that this is greater the older the youth group. For the purpose of this study, for those who are no longer living with their parents, the income variable included by INDEC in the EEEJ database refers to their own income, not to their parents’ income. Hence, for young adults who do not coreside any longer with their family, the income variable does not reveal the family background at all. For this reason, it seems helpful to select a variable that better reflects the family’s financial capacity to educate their children. The survey poses several questions regarding the type of primary school the students had attended, which reveals the socioeconomic level of the family during children’s early schooling age. Specifically, it focuses on whether the primary school they had attended was public or private, single or double shift, gender-exclusive or not, and bilingual or not. All these features also reveal the quality of the school. Thus, the type of primary school (public or private) selected becomes an indicator of parents’ income level and the results of the estimations including it are shown in Table 6 (Model 3). The negative sign for the estimated coefficient on ‘‘private primary school’’ implies that better family’s economic background has a diminishing effect on schooling gap. The results show that this indicator of family background is strongly associated with the schooling gap of the young, suggesting that having attended a private primary school reduces about 1 year the average schooling gap of the whole sample. But considerable variation occurs across the age groups. For the older group (26–30), Table 6 shows that the estimated coefficient on ‘‘private primary school’’ amounts to more than 1.5 years. This estimate largely doubles for the teenagers, denoting that the economic position of the family has a greater effect on the schooling gap of those individuals beyond the age of compulsory education. Furthermore, in Table 6 (Model 3), replacing incomes by a dummy for

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Table 6.

Economic Family Background and Mobility.

Schooling Gap

Max-schoolparents Sex Age Private primary school Cohort-annual observations Adjusted R2

Model 3 Whole sample

Teenagers

Young adults (20–25)

Young adults (26–30)

0.488 (0.0546) 0.541 (0.1958) 0.402 (0.0211) 0.928 (0.2279)

0.169 (0.0618) 0.165 (0.1984) 0.376 (0.0708) 0.683 (0.2300)

0.687 (0.0839) 1.032 (0.3244) 0.814 (0.0911) 0.976 (0.3636)

0.779 (0.1127) 0.436 (0.4052) 0.254 (0.1396) 1.639 (0.5068)

682 0.442

251 0.174

247 0.415

184 0.323

Source: Own calculations based on EEEJ. Note: Education level for parents is achieved assuming that all of them have completed the level. 0.05opo0.10. po0.05.

private school enlarges the intergenerational transmission of schooling for the whole sample and reveals more pronounced differences between teenagers’ educational mobility and that of young adults. The gender effect is negative but it is statistically significant only for the whole sample and the young adults (20–25) group. Finally, Table 7 shows the negative impact of this early entrance into the labor market, which enlarges the schooling gap. The effect on the schooling gap of working at an early age is more pronouncedly evident in young adults.

4.2. Gender Differences Tables 4–7 consistently reveal that being a boy results in a drop in the schooling gap. Considering that girls showed better educational performance than boys, seen both in the attained years of schooling and in the schooling gap, it is interesting to study gender differences more thoroughly. Tables 8A–9B show the estimated results of Models 3 and 4, by gender. Table 8a–8b shows that the parents’ education level has a greater influence on males than on females irrespective of whether they were teenagers or young adults. However, the family’s economic background has

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Table 7.

Intergenerational Mobility and Early Entrance in Labor Market.

Schooling Gap

Max-schoolparents Sex Age Private primary school Employment Age at first employment Cohort-annual observations Adjusted R2

Model 4 Whole sample

Teenagers

Young adults (20–25)

Young adults (26–30)

0.425 (0.0552) 0.371 (0.1964) 0.419 (0.0240) 0.835 (0.2248) 3.260 (0.6703) 0.189 (0.0381)

0.147 (0.0622) 0.078 (0.1996) 0.394 (0.0793) 0.667 (0.2278) 2.349 (0.9155) 0.139 (0.0601)

0.624 (0.0845) 0.866 (0.3280) 0.868 (0.0907) 0.889 (0.3575) 3.262 (1.297) 0.224 (0.0665)

0.620 (0.1119) 0.179 (0.3904) 0.281 (0.1338) 1.330 (0.4863) 3.453 (1.5329) 0.277 (0.0581)

682 0.460

251 0.178

247 0.440

184 0.396

Source: Own calculations based on EEEJ. Note: Education level for parents is achieved assuming that all of them have completed the level. 0.05opo0.10. po0.05.

the greatest significance in explaining schooling gap in females. Attending private primary school reduces about a year and a half of schooling gap for females. However, for young adult men (26–30), primary private school coefficient is greater than for females. Table 9A–9B shows the strong negative effect on the schooling gap for young adult males who worked early in life; the results indicate an enlargement of the gap, in about 7 years. For females, the employment coefficient is statistically significant only for teenagers.

5. INTERGENERATIONAL IMMOBILITY PATTERNS The results obtained above broadly suggest that, in absolute magnitude, the transmission of educational level between parents and children is higher for young adults than for teenagers. The intergenerational persistence of

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Table 8.

Intergenerational Mobility by Gender.

Schooling Gap

Panel A: Males Max-schoolparents Age Private primary school Cohort-annual observations Adjusted R2 Panel B: Females Max-schoolparents Age Private primary school Cohort-annual observations Adjusted R2

Model 3 Whole sample

Teenagers

Young adults (20–25)

Young adults (26–30)

0.530 (0.0794) 0.416 (0.0321) 0.448 (0.3491)

0.292 (0.914) 0.345 (0.1043) 0.444 (0.3348)

0.757 (0.1278) 0.716 (0.1463) 0.523 (0.5763)

0.707 (0.1569) 0.078 (0.1906) 1.726 (0.8288)

317

117

118

0.421

0.165

0.323

0.316

0.439 (0.0758) 0.396 (0.0282) 1.336 (0.3011)

0.060 (0.0840) 0.417 (0.0965) 0.913 (0.3157)

0.557 (0.1103) 0.915 (0.1109) 1.549 (0.4580)

0.872 (0.1627) 0.434 (0.2032) 1.557 (0.6511)

365

134

129

102

0.461

0.168

0.515

0.327

82

Source: Own calculations based on EEEJ. Note: Education level for parents is achieved assuming that all of them have completed the level. 0.05opo0.10. po0.05.

education level between parents and their adult offspring possibly varies with the parents’ education level. To add to the understanding of the young adults’ low mobility, transition matrices are estimated to determine who moves where, within a generation. To construct those matrices, schooling years for young adults were grouped in three levels, and similarly, the former educational level for each parent was grouped in three levels too. Table 10 presents nonconditional transition matrices for young adults (20–30 years of age) for the whole sample and also differentiating it by gender. Each element pjk of the matrix provides an estimate of a child’s nonconditional probability of being in educational level k, given that his

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Table 9.

Intergenerational Mobility and Early Entrance in Labor Market by Gender.

Schooling Gap

Panel A: Males Max-schoolparents Age Private primary school Employment Age at first employment Cohort-annual observations Adjusted R2 Panel B: Females Max-schoolparents Age Private primary school Employment Age at first employment Cohort-annual observations Adjusted R2

Model 4 Whole sample

Teenagers

Young adults (20–25)

Young adults (26–30)

0.439 (0.0803) 0.402 (0.0362) 0.383 (0.3416) 4.486 (1.0424) 0.223 (0.0605)

0.259 (0.0942) 0.341 (0.1158) 0.462 (0.3340) 2.162 (1.2113) 0.127 (0.0808)

0.622 (0.1293) 0.663 (0.1433) 0.202 (0.5614) 7.564 (2.2979) 0.348 (0.1110)

0.556 (0.1438) 0.023 (0.1705) 1.510 (0.7412) – – 0.389 (0.0850)

317 0.450

117 0.175

118 0.374

82 0.455

0.391 (0.0764) 0.435 (0.0.323) 1.271 (0.2989) 2.234 (0.8699) 0.155 (0.0490)

0.048 (0.0841) 0.463 (0.1116) 0.867 (0.3153) 2.550 (1.4319) 0.155 (0.0927)

0.517 (0.1089) 1.008 (0.1126) 1.577 (0.4438) 1.472 (1.4967) 0.164 (0.0787)

0.700 (0.1682) 0.510 (0.2015) 1.263 (0.6460) 2.337 (1.8475) 0.223 (0.0794)

365 0.474

134 0.176

129 0.547

102 0.372

Source: Own calculations based on EEEJ. Note: Education level for parents is achieved assuming that all of them have completed the level. 0.05opo0.10. po0.05.

parents’ maximum educational level achieved was j. For example, the top left-handed element of the first matrix indicates that a child, whose father or mother has achieved only primary schooling, has 24.8% probability of ending up with a similar low level of education. The diagonal elements of

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Table 10. Transition Matrices (Young Adults 20–30 Years Old): Nonconditional Probabilities of Child’s Schooling Level Given Parents Schooling Level. Primary

Secondary

Parent Primary Secondary University

24.8 5.0 0.0

18.2 47.5 82.8

100.0 100.0 100.0

15.4 39.4 81.3

100.0 100.0 100.0

20.3 53.9 84.3

100.0 100.0 100.0

57.0 47.5 17.2 Male children

28.6 5.6 0.0

Parent Primary Secondary University

Total

Children

Parent Primary Secondary University

University

56.0 54.9 18.7 Female children

22.0 4.5 0.0

57.7 41.6 15.7

Source: Own calculations based on EEEJ. Note: Primary includes preschool, primary, and EGB; Secondary includes secondary and polimodal; university includes tertiary, university, and postgrades.

each matrix represent the nonconditional probability of a child remaining at the same schooling level as that of their parents. The results show diminishing mobility from low to high school level, and reveal that for children older than teenagers the educational marginal decision depends largely on the education level achieved by their parents. Elements in the upper triangle of the matrix represent the probabilities of a child achieving a higher education level than their parents. In the three matrices studied, it is evident that children raised in households where parents have, at most, achieved primary school, have a higher probability of attending secondary school, but a very low probability attending university or tertiary studies. For those children whose family educational background is at the secondary level, the probability of attaining the same educational level as their parents is already the same as that of achieving university level. The elements in the lower triangle represent the nonconditional probabilities of a child achieving a lower education level than their parents. Those probabilities are quite low in all the matrices. .

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The transition matrices for sons and daughters show different patterns of intergenerational persistence, suggesting that different mechanisms of intergenerational education transmission exist, by gender. Educational mobility seems quite large for daughters whose parents have achieved a low education level. Conversely, it appears that having parents with low education levels largely indicates that their sons achieve a low education level as well. However, a female child whose parents’ maximum education level is at the secondary school level also has a greater probability of getting a university degree than a male one. But, intergenerational persistence is higher for daughters over sons when the parents have achieved a university or tertiary degree. Table 11 shows the results obtained by employing the multinomial probit model to assess the conditional probabilities of transition from the maximum parents’ educational level to the child’s. It is shown that, both child age and private primary school diminish the intergenerational educative persistence for low educational level (primary or less). This drop

Table 11. Transition Matrices (Young Adults 20–30 Years Old): Conditional Probabilities of Child’s Schooling Level Given Parents Schooling Level. Primary

Secondary

Parent Primary Secondary University

14.0 3.0 0.0

21.5 44.0 80.3

100.0 100.0 100.0

16.9 42.6 89.1

100.0 100.0 100.0

0.0 33.1 66.8

100.0 100.0 100.0

64.5 52.8 19.6 Male children

55.1 17.9 0.0

Parent Primary Secondary University

Total

Children

Parent Primary Secondary University

University

27.9 39.4 10.8 Female children

1.4 84.3 14.3

2.7 47.3 50.0

Source: Own calculations based on EEEJ. Note: Primary includes preschool, primary, and EGB; Secondary includes secondary and polimodal; university includes tertiary, university, and postgrades.

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of the intergenerational link at the lowest educational level suggests that the family’s income plays a significant role in explaining the persistence of this low educative level. Differentiating by children’s gender, it appears that this effect is substantial for daughters but rather low for sons. Conditioning on families’ economic background markedly increases the probability of obtaining a secondary degree for daughters from poorly educated families. These results suggest that girls from poor families would have less opportunity to climb the social ladder than boys. Conditioning reveals another interesting result. For those parents who had achieved the maximum education level, a decline of persistence was evident when adding the controls. Although the effect here is quite small, this goes in the same direction, suggesting that family background, specifically their income levels, plays a role in explaining intergenerational mobility. These results are quite different by gender. For male children, conditioning on family’s income slightly increases the intergenerational persistence of attaining a university degree; for females, the intergenerational persistence diminishes, largely showing that the family income background indeed plays an important role on the daughter’s highest educative level of success.

6. CONCLUDING REMARKS This chapter explores the differences in the degree of intergenerational mobility in Argentina between teenagers and young adults. A new database with retrospective questions, the Survey of Employment and Education of Youth (Encuesta de Educacio´n y Empleo de los Jo´venes, EEEJ ) allows unbiased educational mobility estimations for young people older than teenagers. Applying several variants of Behrman et al. dynamic linear model, the degree to which family background determines schooling outcomes of children is estimated. The estimations unveil quite low intergenerational mobility for young adults relative to teenagers, a result that is robust for several specifications of the model. These findings suggest that when deciding whether to continue studying or not beyond compulsory schooling, the young are strongly influenced by their parents’ educative background. The results also reveal that generational convergence of young adults and teenagers is quite similar when considering the average schooling gap of each group. This means, belonging to a family where the highest education level achieved by the parents’ increases by one level (e.g., from primary to secondary), the

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education delay of any child drops by around 20%. However, while for teenagers this implies a very short delay, for young adults it involves more than a year. This result indicates that in Argentina, beyond compulsory studies, the schooling attainment of children is highly correlated with that of their parents. But does this result imply a low intergenerational mobility in terms of earnings or even of socioeconomic status? Although we do not know for certain the answer to this question, because there is some evidence of convexity in the returns to education in Argentina, this result may suggest high intergenerational correlations of incomes and earnings. Discriminating by gender, the influence of the parents’ education level seems greater on males than on females, irrespective of whether they were teenagers or young adults. However, it is evident that the family’s economic background has the utmost importance on the schooling gap of females. It was also found that young adult immobility is not uniform across parents’ education level. Specifically, it appears that immobility is quite large at the higher (university and tertiary) educative levels. Besides, the findings show different patterns of intergenerational mobility by gender. This result also cautions against studying intergenerational mobility solely by using only cross-section data, which allows estimating the mobility for teenagers, although not for young adults. In light of these findings, it appears necessary to complement the diagnosis obtained from regular household surveys with other surveys that include retrospective information on parental background, to overcome the limitations imposed by the former data.

NOTES 1. This is also the strategy followed by FIEL (2008), which designed and collected a specific survey on the socioeconomic life conditions and family background in the Greater Buenos Aires area. 2. For example, if a child declares having completed secondary studies, she/he would have attended 7 years of primary plus 5 years of secondary; thus having had 12 years of schooling. In the rest of the educational levels, it was not possible to calculate the years of schooling as the duration of the tertiary or university education studies is not fixed. Similarly, it was not possible to calculate the schooling years for those children for whom Educacio´n General Ba´sica was the highest level achieved. This level includes 9 years of basic education composed of three consecutive levels, which are not distinguishable in the data. In addition, although the last year of preschool was not mandatory until 1993, attending 1 year of kindergarten was quite the norm earlier. Therefore, here it is assumed that all children had attended it. Hence, in the above scenario, the entire span of schooling attainment would amount to 13 years. Hence, schooling gap is computed as the age at the time of the interview

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minus the years of schooling, minus the normal school starting age, where the last set at 5 years corresponded to a child entering school to attend 1 year of kindergarten. For example, for a 15-year-old boy who has achieved eight grades of education, 9 years of schooling are computed; thus assuming 5 years of age as mandatory to attend preschool implies 1 year of schooling gap. For people above 25 years of age, the maximum number of schooling years they may have achieved is assumed to be 17. This would mean assuming that the duration of university studies on an average is not more than 4 years. This assumption avoids spurious schooling gap based on the respondent’s age. 3. There are 74 records that do not recall their parents’ education level, and 14 declared that either parent had no education at all. Here, those zero values are treated as the true value, although a few could have acquired some informal education. The survey also questions if this parental level of education had been completed at the time the respondent was attending secondary school. Restricting the sample to those respondents whose fathers had completed their education level by the time the child was at school, diminishes the sample in 40% or in 36% when considering the mothers. The survey does not consider whether parents ended their studies after that.

ACKNOWLEDGMENTS I would like to thank John Bishop, Ismael Ahamdanech Zarco, anonymous referee, and the participants of the Third Meeting of the Society for the Study of Economic Inequality, July 2009, for a number of helpful comments; particularly Gary Fields, for his valuable suggestions. I am also grateful to the attendants to the XLIII Reunio´n Anual de la Asociacio´n Argentina de Economı´a Polı´tica and to the NIP, November 2008. I also thank CEDLAS specially Mariana Marchionni for giving me the database corresponding to the Survey of Employment and Education of Youth; Ana Pacheco for kindly helping me to understand some aspects of it. I am also grateful for the helpful suggestions of Walter Sosa Escudero and Guillermo Cruces; and to the Universidad Austral for its financial support.

REFERENCES Andersen, L. (2001). Social mobility in Latin America: Links with Adolescent Schooling. IADB Research Network Working Paper no. R-433, Washington, DC. Andrews, D., & Leigh, A. (2009). More inequality, less social mobility. Applied Economic Letters, 16, 1489–1492. Becker, G., & Tomes, N. (1979). An equilibrium theory of the distribution of earnings and intergenerational mobility. Journal of Political Economy, 87(6), 1153–1189. Becker, G., & Tomes, N. (1986). Human capital and the rise and fall of families. Journal of Labor Economics, 4(s3), S1–S39.

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Behrman, J., Birdsall, N., & Sze´kely, M. (1998). Intergenerational schooling mobility and macro conditions and schooling policies in Latin America. Working Paper no. 386. InterAmerican Development Bank. Research Department, Washington, DC. Behrman, J., Gaviria, A., & Sze´kely, M. (2001). Intergenerational mobility in Latin America. Working Paper no. 452. Inter-American Development Bank. Research Department, Washington, DC. Bourguignon, F., Ferreira, F. H. G., & Leite, P. G. (2002). Beyond Oaxaca-Blinder: Accounting for differences in household income distributions across countries. William Davidson Institute Working Papers Series 478, William Davidson Institute at the University of Michigan Stephen M. Ross Business School. Bowles, S., & Gintis, H. (2002). The inheritance of inequality. The Journal of Economic Perspectives, 16(3), 3–30. Conconi, A., Cruces, G., Olivieri, S., & Sa´nchez, R. (2008). E Pur si Muove? Movilidad, Pobreza y Desigualdad en Ame´rica Latina. Econo´mica, LIV(1–2), 121–159. Corak, M. (2006). Do poor children become poor adults? Lessons from a cross country comparison of generational earnings mobility. IZA. Discussion Paper no. 1993. Dahan, M., & Gaviria, A. (2001). Sibling correlations and social mobility in Latin America. Economic Development and Cultural Change, 49(3), 537–554. Ferna´ndez, A. G. (2006). Alternative measures of intergenerational social mobility in Argentina. Anales de la Asociacio´n Argentina de Economı´ a Polı´ tica. De Ferranti, D., Perry, G., Ferreira, F., & Walton, M. (2004). Inequality in Latin America and the Caribbean. Breaking with history. Washington, DC: The World Bank. Fundacio´n de Investigaciones Econo´micas Latinoamericanas (FIEL). (2008). La Igualdad de Oportunidades en la Argentina: Movilidad Intergeneracional en los 2000. Buenos Aires: Temas Grupo Editorial. Fields, G., Puerta, M. L. S., Herna´ndez, R. D., & Rodriguez, S. F. (2007). Intragenerational earnings mobility. Economı´a, 7(2), 101–154. Han, S., & Mulligan, C. B. (2001). Human capital, heterogeneity and estimated degrees of intergenerational mobility. The Economic Journal, 111(470), 207–243. Hertz, T., Jayasundera, T., Piraino, P., Selcuk, S., Smith, N., & Verasnchagina, A. (2007). The inheritance of educational inequality: International comparisons and fifty-year trends. The B.E. Journal of Economic Analysis and Policy, 7(2). IPES. (2008). Outsiders? The changing patterns of exclusion in Latin America and the Caribbean. Washington, DC: Inter-American Development Bank. Jenkins, S. P., & Siedler, T. (2007). The intergenerational transmission of poverty in industrialized countries. Discussion Papers of DIW Berlin 693, DIW Berlin, German Institute for Economic Research. Lo´pez Bo´o, F. (2007). The evolution of returns to education in Argentina. Argentina: Anales de la Asociacio´n Argentina de Economı´ a Polı´ tica. Marchionni, M., Bet, G., & Pacheco, A. (2007). ‘‘Empleo, Educacio´n y Entorno Social de los Jo´venes: Una Nueva Fuente de Informacio´n’’. CEDLAS. Universidad nacional de la Plata. Documento de Trabajo no. 61. Perry, G., Lo´pez, J. H., Maloney, W. F., & Arias, O. S. (2005). Virtuous circles of poverty reduction and growth. Washington, DC: World Bank. Roemer, J. (2004). Equal opportunity and intergenerational mobility: Going beyond intergenerational income transition matrices. In: M. Corak (Ed.), Generational income mobility in North America and Europe. Cambridge: Cambridge University Press.

CHAPTER 4 ARE INFORMALITY AND POVERTY DYNAMICALLY INTERRELATED? EVIDENCE FROM ARGENTINA Francesco Devicienti, Fernando Groisman and Ambra Poggi ABSTRACT Poverty and informal employment are often regarded as correlated phenomena. Many empirical studies have shown that informal employment has a causal impact on household poverty, mainly through low wages. Yet other studies focus on the reverse causality from poverty to informality, arising from a range of constraints that poverty poses to jobholders. Only recently have empirical researchers tried to study the simultaneous twoway relationship between poverty and informality. However, existing studies have relied upon cross-sectional data and static econometric models. This chapter takes the next step and studies the dynamics of poverty and informality using longitudinal data. Our empirical analysis is based on a bivariate dynamic random-effect probit model and recent panel data from Argentina. The method used provides a means of assessing the persistence over time of poverty and informal employment at the individual level, while controlling for both observed and unobserved determinants of the two processes. The results show that both poverty and informal employment are Studies in Applied Welfare Analysis: Papers from the Third ECINEQ Meeting Research on Economic Inequality, Volume 18, 79–106 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2010)0000018007

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highly persistent processes. Moreover, positive spillover effects are found from past poverty on current informal employment and from past informality to current poverty status, corroborating the view that the two processes are also shaped by interrelated dynamics in segmented labor markets.

1. INTRODUCTION The persistence of high levels of informality and poverty in Argentina is a feature shared by many developing countries in the Latin American region. The nature of informality is a matter of controversy, which is partly related to the heterogeneity of the economic activities typically included in the informal sector (ILO, 2002). One of the most debated features of informality lies in the role it can have in economic development, and within this debate a primary place is occupied by the study of the relationship between informal jobs and poverty (Perry, Lopez, Maloney, Arias, & Serven, 2006). The fact that a large part of the informal workers are poor, and vice versa, supports the view that poverty and informality are connected. Poverty comprises those households below a certain income line. Informality, on the other side, includes a large fraction of workers with low earnings. Hence, low incomes appear as the link relating informality and poverty. Although there is some consensus around this asseveration there is still scarce evidence about the interactions between the two phenomena. The analysis of the relationship between poverty and informality allows disentangling the reasons that lead certain individuals to engage in informal employment. According to the supply-led view, workers voluntarily opt for jobs in the informal sector given their preferences and productivity (e.g., Heckman & Sedlacek, 1985). According to an alternative demand-led view, work in the informal sector is instead the consequence of the lack of opportunities for accessing to a formal job. A host of institutional barriers and labor regulations may segment labor markets into a formal sector offering high-quality jobs and an informal sector offering second-choice jobs, typically characterized by lower wages, poorer working conditions and poorer career prospects than in the formal sector (e.g., Fields, 1975). In this perspective, employment in the informal sector is mainly driven by firms’ demand for a cheaper form of labor input and workers’ need to work to sustain their household necessities. Unlike the supply-led view, the demandled view emphasizes the involuntary nature of informality, rather than workers’ preference for this type of employment. In this respect, the inability

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to cover minimum household food, clothing, shelter, and fuel requirements, coupled with the difficulty to get a job in the formal sector, may explain individual’s decision to look for a job in the informal sector (AmuedoDorantes, 2004). In other words, the involuntary view of informality opens the way for household poverty to be a crucial determinant of informality, rather than simply being one of its unpleasant implications. The identification of poverty as one of the negative consequences of informality is quite widespread in the literature. Informality may be one of the causes of poverty if informal jobs are associated with low incomes. Therefore, an important bulk of the research has centered on the empirical evaluation of the existence of an earnings gap between formality and informality. Another strand of research has instead focused on the link between household poverty and informality, and this is the focus of the present chapter. Modeling the interrelated dynamics between poverty and informality can be quite challenging, especially because household poverty is related to the labor market choices and outcomes of each household member. Our strategy to circumvent this complexity is to restrict our empirical analysis to a sample of household heads. Our main justification for doing so is that household heads’ earnings comprise a rather significant fraction of household incomes. Note that, even with this restricted focus, the poverty implications of an informal sector job can come about as a combination of low unit wages and high instability of the household head’s job. Less explored is the inverse relationship, from poverty to informality. Indeed, the fact that the head of a poor household faces a greater chance to engage in informal employment with respect to a nonpoor head sustains a vision that emphasizes the involuntary nature of informality. The impossibility of getting a formal job (in a dual labor market well-paid jobs are scarce), usually with high and more stable incomes comparing to those from informal jobs, may lead to enter into informality. This is regularly the only alternative to unemployment in the absence of generalized social security networks, a characteristic of Latin American countries. Poor household heads usually cannot afford the entry costs in the formal sector and cannot wait until a formal job offer materializes; their household immediate necessities make the take-up of an informal job a survival, albeit second-best, choice. A number of factors associated with the condition of poverty (i.e., residential segregation, spatial labor mismatch, labor discrimination) may make the prospects of a formal job even less likely (Groisman, 2008). Hence, informality would be the result of some poverty attributes. This chapter contributes to the analysis of the interaction between informality and poverty by providing evidence for the Argentinean case.

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Only recently have empirical researchers tried to study the simultaneous two-way relationship between poverty and informality. However, existing studies have relied upon cross-sectional data and static econometric models. This chapter takes the next step and studies the dynamics of poverty and informality using longitudinal data. The interconnection between informality and poverty that we aim at exploring is arguably dynamic in nature. In particular, we are interested in whether having been employed in an informal job in the past may lead to poverty in the future. Similarly, we would like to know whether having suffered episodes of poverty may lead to episodes of informality hereafter. Our empirical analysis is based on a bivariate dynamic random-effect probit model and panel data covering the period 1996–2003. The results show that both poverty and informal employment are highly persistent processes at the individual level. Moreover, positive spillover effects are found from past poverty on current informal employment and from past informality to current poverty status, corroborating the view that the two processes are also shaped by interrelated dynamics in segmented labor markets. The chapter is organized as follows. The next section describes previous research and Section 3 the definitions and data used. Section 4 provides descriptive evidence about informality and poverty in Argentina. Section 5 presents the econometric approach. Section 6 analyzes the results and Section 7 discusses additional methodological issues. The conclusions are presented in Section 8.

2. PREVIOUS RESEARCH Several studies confirm the persistence of high rates of informal employment and poverty in Latin America (Perry et al., 2007). However, research about the connection between informality and poverty is insufficient. Analyses of poverty dynamics are usually concerned with patterns and determinants of transitions and persistence. In general, results are that movements in and out of poverty are frequently associated with changes in employment status. Gasparini and Tornaroli (2007) found that on average the difference in the poverty headcount ratio between informal and formal workers is around four times in the region. Amuedo-Dorantes (2004) using cross-section data for Chile concludes that household poverty increases the likelihood of employment in the informal sector. Also, it is shown that having an informal job raises the probability of becoming poor. Beccaria and Groisman (2008) explore if informality is the main cause of Argentinean poverty and find

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that, although relevant, the later one is not limited to households with members working in the informal sector. There is a second group of research interests that is related to informality and low labor incomes. As it was mentioned above, these studies are oriented to test the existence of segmented labor markets. The usual methodological approach consists on isolating the effect of informality from those derived from other income determinants. Both parametric and semiparametric methods can be found in various studies for different countries: Maloney (1999) for Mexico, Packard (2007) for Chile, Pratap and Quintı´ n (2006) and Beccaria and Groisman (2008) for Argentina, and Perry et al. (2007) for various Latin American countries. From a different methodological perspective Sosa Escudero and Arias (2008) analyze the interrelation between labor informality and relative informal/formal wages in Argentina. Although most of the papers find a formality premium, especially among wage earners, the reasons for being informal remain polemical. As for the research on the dynamics of poverty, many different methodological approaches have been used in the literature (Aassve, Burgess, Propper, & Dickson, 2005). First, some papers have used components of variance models to capture the dynamics of income using a complex error structure (i.e., Lillard & Willis, 1978; Stevens, 1999 and Devicienti, 2001). These models are able to predict the fraction of population likely to be in poverty for different lengths of time. One shortcoming of these models is that they typically assume that the dynamics of the income process is identical for all individuals in the sample, rich and poor, which does not seem to match reality (Stevens, 1999; Devicienti, 2001). Another strand of research has focused on the estimation of Markovian transition poverty models (first order Markov models) taking simultaneously into account that individuals are not randomly distributes either within the poor at first interview – initial conditions problem – or within the effectively observed at second interview – attrition problem – (i.e., Capellari & Jenkins, 2004). Alternatively, if the first-order Markov assumption is violated, a longstanding approach to model poverty transitions has been the use of duration models in a hazard rate framework (i.e., Kalbfleisch & Prentice, 1980; Allison, 1982; Ducan, 1984; Bane & Ellwood, 1986; Jenkins, 1995; Devicienti, 2001; Hansen & Wahlberg, 2004; Biewen, 2006; Arranz & Canto, 2008). Third, some recent approaches to the analysis of poverty transitions have used binary-dependent variable dynamic random-effects models where an individual’s poverty status at time t is assumed to depend on the same individual’s poverty status at t1, a list of covariates and some unobserved individual effects. In this case, state dependence is summarized by the

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coefficient estimated for lagged poverty. In this type of models the distribution of the unobserved effects is conditional on the initial value of poverty (initial conditions) and a group of exogenous variables. See Wooldridge (2005) and for examples of this approach to the analysis of poverty see Poggi (2007) and Biewen (2009). Moreover, this type of models can be extended to analyze the relationship between two supposedly related concepts (i.e., poverty and informal employment). Devicienti and Poggi (in press) propose the use of a dynamic random-effects bivariate probit following the recent contribution by Stewart (2007) who generalizes Wooldridge (2005) to the bivariate case.

3. DATA The data used in this document come from the household survey from Argentina (Permanent Household Survey – PHS – carried out by National Statistics and Census Institute (INDEC)). Panel data are obtained from the rotating panel used to define the sample of households interviewed. The survey covers urban areas and collects information on labor market variables, income, and other social dimensions. Household and individual data are collected. Interviews are held two times a year (in May and in October) and households are visited in four successive moments or waves through a period of 18 months. Data for all urban areas surveyed are available from 1996. The questionnaires and methods of the PHS were modified in 2003, therefore we focus on the period 1996 to 2003. The universe under analysis is the group of household heads at least once employed.

4. INFORMALITY AND POVERTY TRENDS From mid-1970s to 1990 Argentina experienced a 15-year period of macroeconomic instability and productive stagnation. In 1989 and 1990 Argentina suffered two severe hyperinflationary crises and in 1991 a stabilization program was implemented. The economy recovered until the Mexican crisis – in late 1994 – impacted on Argentina. From 1996 to 1998 it resumed a rapid growth path; however, by the end of 1998 there was a new downswing in GDP. By the end of this year a deep recession started and the economy collapsed three years later. Following the great crisis of 2001, the macroeconomic regime changed. In the new economic context, Argentina experienced a steady and lasting economic recovery until 2008.

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Since the early 1990s there was a marked increase in the rate of open unemployment: 6% in 1991, 9% in 1993, 18% in 2001, and 16% in 2003. Between 1991 and 1994 the employment grew at a rate of 0.2% per annum, which was very small compared to the strong increase of GDP at 6%. During the recession associated with the Mexican debt crisis the employment fell 2% and in the recovery from 1996 to 1998 job creation rose at 3.6% per year. Finally, during the recession that began in 1998 and lasted until 2001, total employment contracted 3.2%. All along the period, informality (including self-employed and precarious salary workers) did not modify markedly its relative size. The main feature shown by occupational structure was the important advance of nonregistered salaried workers (controlling the incidence of domestic service workers and beneficiaries of employment plans) which entirely explains the modest expansion of informal employment. It has to be taken into account that although unemployment insurance was introduced early in the 1990s, its coverage was very low. About 5% to 7% of the unemployed had access to the benefit. Basically, this was due to the reduced share of registered wage earners in the occupational structure, a clause to have access to the unemployment insurance. Changes in the labor regulations also seem to have exerted some influence on the persistence of a high informality rate. During the period under analysis, many changes have been implemented with the purpose of altering the functioning of the labor market. Four national laws were passed from 1991 to 2000 and most of them were inspired by the paradigm of labor flexibilization. Nevertheless, as it was mentioned, the employment growth was largely concentrated in nonregistered jobs. This evolution suggests that more flexible labor norms (such as the reduction in payroll taxes in 19941 or the time extension of the trial period for new recruitments in19952) would have facilitated not the registration but instead the noncompliance with labor laws. According to the International Labour Office, informality can be seen as the incapacity of economy to create enough jobs compared to the labor force. Thus, informal jobs are sometimes self-employment jobs and, in other cases, wage earners working in small units. According to this perspective, the informal unit is characterized by a no clear separation between capital and labor and usually acts in easy-entry activities and registers very low productivity. Alternatively, informality can be defined with the noncompliance of labor regulations (mainly, evading taxes). However, the two definitions clearly overlap. In this chapter, we combine both definitions as it is usual in the recent literature (see Hussmanns, 2005): workers in informal employment are those nonwage earners in small firms (i.e., firm with less than five workers), those evading taxes (i.e., nonregistered wage earners), and workers in domestic

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Table 1.

Employment Structure and Poverty: All Urban Areas.

% Heads of Households

May 1996

October 1998

October 2001

May 2003

2.0 42.1 6.8

1.6 40.5 6.3

1.6 38.9 6.0

1.1 34.0 5.5

30.2 6.9 7.4 4.5

29.1 9.3 8.6 4.8

30.2 7.8 9.2 4.8 1.5

29.4 8.6 8.4 5.2 7.7

Informal employmenta Overall Poor households

49 62.5

51.7 70.1

52 71.6

51.7 69.9

Absolute poverty incidence Overall Head of household informal

22.1 25

21.8 27.2

28.3 35.8

42.8 54.9

Formal employment Nonwage earners (firm size W5) Registered wage earners (firm size W5) Registered wage earners (firm size r5) Informal employment Nonwage earners (firm size r5) Nonregistered wage earners (firm size W5) Nonregistered wage earners (firm size r5) Domestic service Employment plans

a

Excluding employment plans.

service.3 The distinction between formal and informal employments refers to the main occupation (information about secondary employments are not available). About 51.7% of the household heads are employed in the informal sector in 2003 (see Table 1). During the period 1996–2003, informality incidence increased 2.7 percentage points on average. This aggregate performance combined an increase of nonregistered wage earners and sharp decrease of registered workers in firms with more than five workers. During the same period poverty increased dramatically.4 Besides, informality rate among poor households was systematically higher indicating that those with informal jobs had lower probabilities of becoming nonpoor. Similarly, the poverty rate among informal heads was also higher. From 1996 to 1998, overall poverty decreased although among informal households poverty incidence rose slightly. Poor households augmented markedly from 1998, both in informal and formal households. Poverty and informality trends reflect different dynamics: informality showed greater stability than poverty during the period under analysis. Indeed, the latter rose sharply accompanying macroeconomic fluctuations. Secondly, the higher value of the rate of informal jobs (compared to that of

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poverty) anticipates that a significant proportion of the former correspond to nonpoor households. Both pieces of evidence justify the research about interconnection of these phenomena. Table 2 gives information about the sample composition showing differences in attributes between male and female household heads. The share of female heads is 17%. Women show an average age slightly larger than males’ and run smaller households. The educational level was higher for women than for males while the hours worked and the employment rates were lower. The employment structure shows that male heads were highly concentrated in two categories: registered salary workers in firms with at least six employed people (44.6%) and nonwage earners in small firms (31.8%). Female heads also show high concentration in same categories than male but with high incidence of domestic service as well (23.9%). Finally, the probability of working in the informal sector was really greater for women than for men, while the risk of falling below the poverty line was somewhat lower. Consequently, households headed by women showed higher rates of both poverty and informality.

5. ECONOMETRIC MODEL Our aim is to analyze the relationship between two supposedly related concepts: poverty and informal employment. To do so, we use a dynamic random-effect bivariate probit model for the joint probability of experiencing the two states. The model allows for correlated unobserved heterogeneity and accounts for the initial conditions of the two processes. For an individual i, the risk of being in poverty at time t is expressed in terms of a latent variable y1it , as specified as in Eq. (1), while the risk of working in the informal sector in t is expressed by the latent variable y2it , specified in Eq. (2). y1it ¼ x0it b1 þ y1i;t1 g11 þ y2i;t1 g12 þ c1i þ u1it

(1)

y2it ¼ x0it b2 þ y1i;t1 g21 þ y2i;t1 g22 þ c2i þ u2it

(2)

j k yjit ¼ 1 yjit 40 ;

j ¼ 1; 2;

t ¼ 2; . . . ; T

(3)

The dependent variables are the dummy indicators y1it (equal to one if the individual is at risk of poverty in t, and zero otherwise) and y2it (equal to one if individual i is employed in the informal sector in t, and zero otherwise).

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Table 2.

Sample Composition: All Urban Areas.

Heads of Households (Ever Employed) Sex (%) Age (mean) Household size

Female

Male

Total

17 47 2.9

83 44 4.1

100 44 3.9

60 25 14 100 79.8 33

65 26 10 100 89.6 44

64 26 10 100 87.9 43

0.5 39.0 5.2

1.9 44.6 6.2

1.7 43.7 6.1

19.4 6.5 5.6 23.9

31.8 7.7 7.4 0.5

29.9 7.5 7.1 4.1

Informal employment (%) Obs. 1 Obs. 2 Obs. 3 Obs. 4

55.4 54.8 55.1 55.6

47.3 47.5 48.0 48.5

48.6 48.6 49.1 49.6

Poverty incidence (%) Obs. 1 Obs. 2 Obs. 3 Obs. 4

29.5 31.2 32.6 32.7

30.1 32.5 34.0 34.8

30.0 32.3 33.7 34.4

15.6 15.8 17.9 17.4 4,844

14.1 14.9 15.9 16.5 23,621

14.4 15.1 16.2 16.7 28,465

Education (%) Low Medium High Total Employment rate (%) Hours worked a week (mean) Employment structure (obs. 1) (%) Formal employment Nonwage earners (firm size W5) Registered wage earners (firm size W5) Registered wage earners (firm size r5) Informal employmenta Nonwage earners (firm size r5) Nonregistered wage earners (firm size W5) Nonregistered wage earners (firm size r5) Domestic service

Both informal and poor (%) Obs. 1 Obs. 2 Obs. 3 Obs. 4 N a

Employment plans are included in the category ‘‘nonregistered wage earners.’’

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89

In the model represented by Eqs. (1)–(3), xit is a vector of independent variables, assumed to be strictly exogenous, and b ¼ (b1, b2) is the corresponding vector of parameters to be estimated. The error terms u1it and u2it are assumed to be independent over time and to follow a bivariate normal distribution, with zero means, unit variances and cross-equation covariance r. The model also includes individual random effects, c1i and c2i, assumed to be bivariate normal with variances s2c1 and s2c2 and covariance sc1sc2rc. We also assume that (c1i, c2i), (u1it, u2it; t ¼ 1, y ,T), and (xit; t ¼ 1, y ,T) are independent (implying that xit is strictly exogenous). The dynamics of the model is here assumed to be first order for simplicity. This dynamic random-effects model is well suited to tackle the issue of ‘‘true state dependence’’ and to study dynamic spillover effects from poverty to informal employment and from informal employment to poverty. Therefore, we can establish the causal impact of past poverty on current poverty of past experiences in the informal sector on current probability of working in the informal sector, once the confounding impact due to unobserved heterogeneity is accounted for. To disentangle between unobserved heterogeneity and true state dependence, the lagged dependent variable, y1i,t1, is included in the poverty Eq. (1) and the lagged dependent variable y2i,t1 is included in the informal employment Eq. (2). Moreover, to take into account the spillover effects, the model also includes cross-effect lagged variables: lagged informal employment y2i,t1 is included in the poverty equation and lagged poverty y1i,t1 is included in the informal employment equation. This way it may be possible to understand whether the correlation observed in the data between, say, y1,t1 and y2t is due to correlated unobserved heterogeneity (i.e., rc6¼0) or rather to state dependence across poverty and informal employment (i.e., the spillover effects g12 and g21 are nonzero). In dynamic panel data models with unobserved heterogeneity, the treatment of the initial observations is an important theoretical and practical problem, particularly so in short panels. In the univariate case (e.g., Eq. (1) with g12 ¼ 0), Heckman (1981) suggests to replace the equation for t ¼ 1 by a static equation with different regression coefficients and a linear function of the random-effect ci1. This can be seen as a linearized approximation to the reduced form for the latent variable in the initial period. Alessie, Hochguertel, and van Soest (2004) propose to generalize Heckman’s approach to the bivariate model represented by Eqs. (1)–(3) and introduce two static equations for t ¼ 1, containing a linear combination of ci1 and ci2. The approach followed by Alessie et al. (2004) provides a flexible characterization of the sample initial conditions in terms of the observable

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covariates and unobserved individual effects, and represents a straightforward generalization of the univariate Heckman estimator. In this chapter, we will follow a different route for the treatment of the initial conditions, extending to the bivariate case the simple approach proposed by Wooldridge (2005) for univariate dynamic random-effects probit models. We do so because the resulting estimator is simpler, and results in substantial savings in computer time, while the performance of the two estimators is otherwise comparable (for the univariate case see Arulampalam and Stewart, 2007, and Orme, 1997, 2001). For the bivariate case similar to model (1)–(3), see Devicienti and Poggi (in press).5 Wooldridge (2005) proposed a conditional maximum likelihood (CML) estimator that considers the distribution conditional on the initial values and the observed history of strictly exogenous explanatory variables. To generalize this approach in the context of our bivariate probit model, we specify the individual specific effects ci1 and ci2 given the initial conditions (y1i1 and y2i1) and the time-constant explanatory variables x i , as follows: c1i ¼ a10 þ a11 y1i1 þ a12 y2i1 þ x 0i a13 þ a1i c2i ¼ a20 þ a21 y1i1 þ a22 y2i1 þ x 0i a23 þ a2i where aj0, aj1, aj2, and aj3 (j ¼ 1,2) are parameters to be estimated, (a1i, a2i) are normally distributed with covariance matrix Sa: ! s2a1 s2a1 s2a2 ra Sa ¼ : s2a2 Then after inserting in model (1) and (2) we obtain: y1it ¼ x0it b1 þ y1i;t1 g11 þ y2i;t1 g12 þ a10 þ a11 y1i1 þ a12 y2i1 þ x 0i a13 þ a1i þ u1it y2it ¼ x0it b2 þ y1i;t1 g21 þ y2i;t1 g22 þ a20 þ a21 y1i1 þ a22 y2i1 þ x 0i a23 þ a2i þ u2it (4) Consistent estimates of the model’s parameters can be obtained by conditional maximum simulated likelihood methods. The contribution of individual i to the likelihood may be written as: LW ¼

Z

þ1 1

Z

T þ1 Y

1

F2 ðy~1it m1it ; y~2it m2it ; y~1it y~2it rjy1;t1 ; y2;t1    xit ; x i Þ

t¼1

 gða1 ; a2 ; Sa Þda1i da2i

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91

where m1it and m2it are the right-hand sides of equations in model (4) excluding the error terms u1it and u2it, and y~jit ¼ 2yjit  1 for j ¼ 1,2. Finally note that, according to Wooldridge (2005), the model needs to be estimated on a balanced panel. Accordingly, one may be worried that the estimator could potentially exacerbate attrition and sample selection present in the data. In fact, this is not the case, since Wooldridge’s method has some advantages in facing selection and attrition problems. In particular, as explained in Wooldridge (2005, p. 44), it allows selection and attrition to depend on the initial conditions and, therefore, it allows attrition to differ across initial levels of poverty and informality status. In particular, individuals with different initial statuses are allowed to have different missing data probabilities. Thus, we consider selection and attrition without explicitly modeling them as a function of the initial conditions.6 As a result, the analysis is less complicated and it compensates for the potential loss of information from using a balanced panel.7

6. RESULTS In this section, we present the estimates of the dynamic model for poverty and informality discussed in the previous section. In order to make the interpretation of the results easier, standard bivariate probit estimates are also presented. Results for male and female household heads are presented separately (see Tables 3–4). Covariates included in the vector xit refer to individual-level characteristics: gender, age (linear), dummies for high and medium education (low education is the reference category), marital status ( ¼ 1 if married) and job attributes as tenure (linear), occupation (blue or white collars), sector dummies, area dummies, and firm size. These variables are treated as time-constant variables.8 Household-level characteristics are also included in xit: the number of the household members and the number of working members of the household. Only the latter varies over our period of analysis (thus, in the specification of the dynamic random-effects model we also include the corresponding time-average variable in order to allow for correlation between the individual specific effects and the time varying variable). A set of period dummies is also included in the specification to capture the macroeconomic environment. In both equations, the same explanatory variables are used.9 While in principle a wider set of influences may be considered, we have maintained our reduced-form specifications relatively parsimonious because (i) we are already controlling for (correlated) unobserved heterogeneity, (ii) the estimation of our model is

Poor in t1 Informal in t1 Poor at t0 Informal at t0 Age Married Household size No. of working members in the household High education Medium education Tenure Blue collar Construction Domestic service and commerce Transport Modern services Social services

Only Male Household Heads Robust SE 0.0269 0.0286 No No 0.0012 0.0551 0.0079 0.0183 0.0607 0.0264 0.0001 0.0312 0.0343 0.0375 0.0429 0.0486 0.0392

1.3257 0.2896 No No 0.0022 0.0198 0.2703 0.5872 0.9426 0.5201 0.0009 0.1194 0.2552 0.0764 0.0908 0.0883 0.1042

Poverty

0.1700 2.1882 No No 0.0068 0.0476 0.0106 0.0552 0.1013 0.0034 0.0007 0.0778 0.3153 0.0725 0.1764 0.0690 0.4216

Coeff. 0.0284 0.0299 No No 0.0012 0.0475 0.0069 0.0152 0.0393 0.0242 0.0001 0.0307 0.0366 0.0351 0.0402 0.0436 0.0407

Robust SE

Informality

0.3446 0.2074 1.4299 0.2031 0.0042 0.0153 0.3790 0.8814 1.3405 0.7221 0.0012 0.1978 0.3378 0.1144 0.1071 0.1483 0.1458

0.0472 0.0748 0.0658 0.0783 0.0020 0.0849 0.0146 0.0404 0.0925 0.0450 0.0002 0.0531 0.0580 0.0624 0.0729 0.0820 0.0660

Robust SE

Poverty Coeff.

Male Household Heads: Estimates.

Coeff.

Table 3.

0.1088 0.5859 0.2150 3.0479 0.0108 0.0619 0.0199 0.0234 0.1419 0.0268 0.0009 0.0741 0.5227 0.1063 0.2913 0.2617 0.7279

Coeff.

0.0670 0.0634 0.0737 0.1438 0.0026 0.1018 0.0155 0.0472 0.0810 0.0538 0.0002 0.0677 0.0805 0.0792 0.0904 0.1000 0.0905

Robust SE

Informality

92 FRANCESCO DEVICIENTI ET AL.

Public sector 0.1349 Firm size: small 0.1030 Area is Pampeana 0.1422 Area is Cuyo 0.3066 Area is Noa 0.4493 Area is Pantagonia 0.1990 Area is Nea 0.5651 Period dummies Yes No. of working members in the household No (longitudinal average) Constant 1.6071 r Log-pseudo likelihood No. of obs. No. of clusters sa1 sa2 ra 0.0941

0.0573 0.0285 0.0342 0.0395 0.0352 0.0412 0.0395 Yes No 1.7895 0.1673

0.0475 0.6832 0.0360 0.0006 0.0911 0.0746 0.1001 Yes No 0.0899 0.0195 17,440 29,763 9,921

0.0465 0.0236 0.0311 0.0372 0.0342 0.0357 0.0400 Yes No 2.1600

0.2259 0.1383 0.1688 0.3987 0.6168 0.3075 0.7826 Yes 0.0548 0.1465

0.0968 0.0504 0.0572 0.0653 0.0599 0.0667 0.0688 Yes 0.0492

1.3319 0.9664 0.1467

2.8160 0.2081

0.1503 0.7928 0.1020 0.0470 0.1271 0.1032 0.1863 Yes 0.0980 0.1887 0.0417 16,532 29,763 9,921 0.0634 0.0414 0.0474

0.1035 0.0646 0.0695 0.0820 0.0748 0.0792 0.0877 Yes 0.0603

Correlation between Informality and Poverty 93

Poor in t1 Informal in t1 Poor at t0 Informal at t0 Age Married Household size No. of working members in the household High education Medium education Tenure Blue collar Construction Domestic service and commerce Transport Modern services Social services

Only Female Household Heads Robust SE 0.064 0.083 No No 0.003 0.064 0.022 0.052 0.108 0.067 0.000 0.160 0.440 0.151 0.232 0.157 0.148

1.279 0.653 No No 0.004 0.011 0.346 0.700 0.762 0.464 0.001 0.038 0.209 0.156 0.381 0.363 0.069

Poverty

0.407 2.397 No No 0.007 0.030 0.055 0.103 0.035 0.241 0.001 0.232 0.053 0.216 0.079 0.262 0.462

Coeff. 0.083 0.082 No No 0.003 0.066 0.020 0.051 0.087 0.067 0.000 0.143 0.575 0.128 0.234 0.133 0.131

Robust SE

Informality

0.4248 0.6033 1.2271 0.2754 0.0052 0.0480 0.4767 1.0570 1.0717 0.7544 0.0011 0.0206 0.2299 0.1439 0.3755 0.4778 0.0701

Coeff.

0.1105 0.2094 0.1467 0.2186 0.0047 0.1014 0.0410 0.1098 0.1607 0.1141 0.0005 0.2693 0.5477 0.2411 0.4038 0.2589 0.2496

Robust SE

Poverty

Female Household Heads: Estimates.

Coeff.

Table 4.

0.4284 0.8679 0.5281 2.8748 0.0128 0.0027 0.1467 0.1800 0.0919 0.3951 0.0020 0.2100 0.1317 0.1551 0.1026 0.7262 0.8639

Coeff.

0.1818 0.1803 0.2085 0.4142 0.0065 0.1391 0.0469 0.1422 0.1688 0.1430 0.0007 0.3441 0.8350 0.2910 0.4297 0.3107 0.3051

Robust SE

Informality

94 FRANCESCO DEVICIENTI ET AL.

Public sector 0.230 Firm size: small 0.009 Area is Pampeana 0.304 Area is Cuyo 0.387 Area is Noa 0.477 Area is Pantagonia 0.161 Area is Nea 0.619 Period dummies Yes No. of working members in the household No (longitudinal average) Constant 1.287 r Log-pseudo likelihood No. of obs. No. of clusters sa1 sa2 ra 0.228

0.205 0.089 0.098 0.110 0.098 0.111 0.106 Yes No 1.444 0.331

0.319 0.685 0.228 0.169 0.145 0.400 0.104 Yes No 0.252 0.053 2673.21 5,718 1,906

0.156 0.068 0.082 0.107 0.090 0.091 0.105 Yes No 1.7797

0.3104 0.0782 0.4464 0.6159 0.6812 0.2408 0.8567 Yes 0.1000 0.3811

0.3096 0.1410 0.1488 0.1750 0.1528 0.1654 0.1731 Yes 0.1342 0.4748 0.1309

0.1803 0.0958 0.1290

1.2198 0.8911 0.1660

0.3439 0.1807 0.1766 0.2146 0.1915 0.2072 0.2341 Yes 0.1829

2.3376 0.4957

0.6723 0.9602 0.3909 0.2193 0.3341 0.6037 0.1776 Yes 0.5380

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already computationally demanding. More importantly, the variables included in xit do not constitute the main focus of the analysis: this lies instead in the interrelated dynamics of poverty and informal sector employment, which is reflected in the estimates of the lagged indicators for both dependent variables. The joint estimation of the model equations is necessary: r is positive and statistically significant in all the specifications (both for male and female household heads). Therefore, the myriad of idiosyncratic shocks that, at any given time period, drive people into poverty and into informal sector employment have common elements. The estimates of the pooled bivariate probit models do not control for individual unobserved heterogeneity and assumes that the initial conditions are exogenous. One would then expect that this estimator overestimate the importance of state dependence, as the coefficient of the lagged dependent variable absorbs part of the effect that is instead due to (uncontrolled) unobserved heterogeneity. A quick glance at Tables 3–4 confirms that this is indeed the case. Therefore, the randomeffects bivariate probit model has to be preferred to the standard bivariate model.

6.1. Male Household Heads: Random-Effect Bivariate Probit Model Estimates for male household heads are reported in Table 3.10 For poverty, after controlling for unobserved heterogeneity, the lag coefficient is still statistically significant and it is estimated at 0.3; the lag cross-effect is also sizeable: it is estimated at 0.2 in the poverty equation. For informal sector employment, own lag estimate is even higher, at 0.6, and the lag cross-effect is estimated at 0.1. In both equations the initial values are also very important, and this implies that there is substantial correlation between the initial condition and the unobserved heterogeneity. For poverty, the coefficient on initial poverty (1.4) is much larger that the coefficient on the lag (0.3), while the coefficient on initial informal sector employment is statistically not different from the coefficient on the cross-lag (0.2). For informal sector employment, the coefficient on initial informal sector employment (3.05) is much larger than the coefficient on the lag (0.6); the coefficient on initial poverty (0.2) is also larger than the coefficient on the cross-lag (0.1).

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The standard deviations of the random effects are statistically significant and positive for both poverty and informal sector employment. This means that unobserved heterogeneity plays a role in explaining the observed persistence in poverty and informal sector employment. High values of coefficients (of initial condition and on the lag event) in the informality equation reveal the segmented nature of labor market. These figures show that probabilities of leaving informality are very low for male heads. Instead, the values of similar coefficients in poverty equation may be interpreted as indicative of a more flexible pattern. This is an expected evolution since poverty transitions in the short run usually derive from income changes that are closely related to macroeconomic fluctuations. We have already mentioned that the informality rate showed great stability all along the period while the incidence of poverty fluctuated according to the economic performance. Consequently the observed cross-lag effects were of low intensity although robust. Individuals with high-medium education have a lower risk of being in poverty than those with low education. Age, entered linearly for simplicity, has a small negative and statistically significant effect on income poverty, reflecting the increased command on economic resources as the individual ages. However, age has an opposite effects on informal sector employment indicating that older workers have more possibilities of working in the informal sector than younger workers. This is consistent with two complementary hypotheses. Firstly, firms would prefer to register younger workers. Secondly, older people would exhibit a larger entrepreneurial spirit that younger workers. The number of working members in the household decreases the probability of being in poverty, while the average number of working members increases the probabilities of working in the informal sector. A possible explanation for this correlation is the presence of barriers that limit access to formal jobs for spouses and other members. It may also reflect strong social networks in the informal sphere. Conversely, the risk of poverty increases with the number of the household members. Blue-collar workers have higher probabilities of being poor than white-collar workers, while workers with long tenure have low probabilities of being poor. Individuals working in small firms have both high probabilities of being poor and being employed in the informal sector. This is a common feature of Latin American labor markets where small firms tend to have low productivity and concentrate a great proportion of nonregistered workers. Finally, differences in the probabilities of being poor and/or employed in the informal sector are observed across individuals working in different sectors and different regions.

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6.2. Female Household Heads: Dynamic Bivariate Random-Effect Model Estimates for female household heads are reported in Table 4.11 For poverty, after controlling for unobserved heterogeneity, the lag coefficient is estimated at 0.4 and the lag cross effect is estimated at 0.6. For informal sector employment, own lag estimate is at 0.8, and the lag cross-effect is estimated at 0.4. In both equations, the coefficients on the lag and the crosslag are larger than the ones observed in Table 3 for male household heads. In both equations the initial values are, once again, very important pointing to the existence of substantial correlation between the initial condition and the unobserved heterogeneity. For poverty, the coefficient on initial poverty is estimated at 1.2 and the coefficient on initial informal sector employment is estimated at 0.3. For informal sector employment, the coefficient on initial informal sector employment is estimated at 2.8 and the coefficient on initial poverty at 0.5. Thus, we find some evidence that cross-lag effects are stronger for female household heads than for males. High concentration of female heads in domestic service may be part of the explanation. It must be emphasized that this activity has low barriers to entry/exit and low monthly income. In the same direction, the fact that female heads (usually in charge of little children) face low opportunities of getting high-quality jobs (i.e., more stable) might also influence this stronger relationship. The standard deviations of the random effects are statistically significant and positive for both poverty and informal sector employment. However, unobserved heterogeneity seems to play a slightly smaller role in explaining the observed persistence in poverty and informal sector employment for female household heads than for male household heads. In facts, in both equations, the standard deviations result to be slightly smaller for female household heads than for male household heads. Individuals with high-medium education have a lower risk of being in poverty than those with low education. Unexpected, age does not have a significant effect on poverty. The number of working members in the household decreases the probability of being in poverty, while the average number of working members increases the probabilities of working in the informal sector (again one point in the direction of presence of barriers that limit access to a formal jobs and/or the existence of strong social networks). Conversely, the risk of poverty increases with the number of the household members. There are no significant differences in the probabilities of being poor between blue and white-collar workers, and between female workers in small- or large-sized firms. But, females working in small firms have high probabilities of being employed in the informal sector. Workers with long

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tenure have low probabilities of being poor, while employment in informal sector is associated with shorter tenures than employment in the formal sector. Finally, differences in the probabilities of being poor and/or employed in the informal sector are observed across females working in different sectors and different regions.

6.3. Predicted Probabilities: Male Versus Female Probabilities For both equations, the lagged dependent variables concerning poverty and informal sector employment are significantly positive. To evaluate the relevance of the dynamics in the model, we estimate the predicted probabilities of being in poverty, and for working in the informal sector, for various lagged statuses of poverty-informal sector employment (Table 5). As suggested by Wooldridge (2005), predicted probabilities are first computed at individual characteristics, keeping lagged dependent variables at specified values, and then averaged in the sample. The estimated parameters corresponding to each variable in Xit ¼ (xit, y1i,t1, y2i,t1, y1ji,t2, y2ji,t2) are multiplied by ð1 þ s^ 2j Þ1=2 , for j ¼ 1, 2, so as take into account the estimated distribution of unobserved heterogeneity, and the corresponding linear predictions are inserted into the cumulative standard normal distribution function, separately for each equation. For male household heads, the probability of being poor in t is about 0.25 for those who were nonpoor and were not employed in the informal sector in t1. This same probability is 0.18 for female heads. However, if we look at heads who were poor the year before, albeit not employed in the informal sector, the probability of being currently poor increases at 0.29 for male heads and at 0.23 for female heads. The chances of being poor in t raise further, at about 0.32 for both male and female heads, if the individual was both poor and employed in the informal sector in t1. For male and female heads, the probabilities of working in the informal sector are about 0.42, if the individual was nonpoor and not employed in the informal sector in t1. These probabilities are higher (respectively, at 0.49 and 0.51) if the household head was employed in the informal sector the year before, albeit not poor. For those both poor and employed in the informal sector in t1, the chances of working in the informal sector in t slightly increase, respectively, at about 0.51 and 0.55. These results are compatible with the presence of barriers along the informal/formal line in labor market. In contrast, past episodes of poverty did not have similar effects (on future events) in the case of households ruled

0.145 0.475 0.2 0.562

0.138 0.173 0.802 0.844

0.0957 0.3584 0.203 0.555

0 0 1 1

0 1 0 1

0.1299 0.2151 0.7942 0.8765

Informal sector

Female

Poor

Males

Pooled bivariate probit model

Poor Informal Poor Informal sector sector

t1

Probability of Status in t1

Probability of Status in t

Probabilities.

0.24836 0.29644 0.2767 0.32775

Poor

0.42711 0.4396 0.4946 0.5073

Informal sector

Males

0.1787 0.2308 0.2554 0.3205

Poor

0.4248 0.4662 0.5085 0.5513

Informal sector

Female

Random-effect bivariate probit model

Table 5.

0.24543 0.29546 0.27833 0.33197

Poor

0.42696 0.44273 0.49494 0.51099

Informal sector

Males

0.17723 0.25123 0.23077 0.31789

Poor

0.42276 0.45188 0.50726 0.53704

Informal sector

Female

Random-effect bivariate probit model controlling for sample selection

100 FRANCESCO DEVICIENTI ET AL.

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by male heads. This suggests that income fluctuations of households located above/below the poverty line were widespread. Female heads, instead, showed positive effects compared to past poverty episodes. Cross-lag effects from poverty to informality are observed and give support to the hypothesis of informality as nonvoluntary (as an alternative to unemployment). In contrast, informal cross-lag effects were obtained only for households headed by women in line with the explanation already mentioned about the higher difficulties they face in labor market.

7. SAMPLE SELECTION ISSUES In this section, we discuss the impact of eventual sample selection problems deriving from the fact that the model is estimated only on working individuals. The male population (aged 16–65) is composed as following: 81.42% employed, 7.92% unemployed, and 10.66% inactive (7.25% retired, 0.3% renter, 0.87% student; 0.65% disables, 1.59% others). The female population (aged 16–65) is composed as following: 56.71% employed, 6.43% unemployed, and 36.86% inactive (18.79% retired, 0.92% renter, 2.78% student; 13.67% housewife, 0.31% disables, 0.39% others). To correct for any selection bias in moving from the entire population to the working individuals, we compute a Mills ratio using a selection variables that equals 1 at period t if the individual is observed over the entire period of analysis and works in period t (Clark & Etile´, 2002). We estimated selection equations separately on the male and female populations, as function of age, education, a dummy variable indicating whether the respondent is attending school, marital status, household size, number of working household members, number of children younger than 6 years old, regional dummies, and period dummies. The selection equations are identified by the exclusion of the dummy variable indicating whether the respondent is attending school and by the exclusion of the number of children under the age of 6 in the household from the structural equations (similar instruments have been previous used by Amuedo-Dorantes, 2004). Both of these variables are not significant in the structural equations once we account for the worker’s skill through occupation dummies and for family size: a test for the null hypothesis that the two instruments are jointly ignorable in Eqs. (1) and (2) is not rejected at standard significance levels for both men heads (w2 ¼ 5.16, p-value ¼ 0.272) and female heads (w2 ¼ 0.413, p-value ¼ 0.981). On the other hand, the two instruments enter in a highly significant way in the selection equation: a joint test for the null hypothesis that both instruments are not significant is

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comfortably rejected both in both the case of male heads (w2 ¼ 183, p-value ¼ 0.0) and in the case of female heads (w2 ¼ 169, p-value ¼ 0.0). The full results from the selection regressions and from the random-effect bivariate models including selection terms are available upon request. In the interest of brevity, here we limit ourselves to show the implications of controlling for sample selection in our random-effect bivariate models: the last two columns of Table 5 reports the predicted probabilities of being in poverty, and for working in the informal sector, for various lagged statuses of poverty-informal sector employment. As the table shows, the results are robust with respect to the ones presented in the previous section.

8. CONCLUSIONS In this chapter we have studied the determinants of poverty and informal employment using recent panel data from Argentina. In particular, we aimed at uncovering a mutually causal relationship between household poverty and household heads’ employment in the informal sector, a relationship that has attracted the interest of both academic researchers and policy makers. The analysis uses a bivariate dynamic random-effect probit model to account for the endogeneity of household poverty and household heads’ employment in the informal sector. Our model provides a means of assessing the persistence over time of poverty and informal employment at the individual level, while controlling for both observed and unobserved determinants of the two processes. Moreover, the model accommodates the potential existence of spillover effects from past poverty to current informality status and from past informality to current poverty status. These dynamic spillover effects might be crucial determinants of the persistence of both poverty and informality that have not been previously studied in the literature. Our results from Argentina show that poverty and informal employment are highly persistent processes at the individual level. For example, the probability of being poor in the current year is about 0.25 for a male household head who was both working in the formal sector and not poor in the previous year. However, this same probability increases at about 0.30 if, other things equal, this same person was poor in the previous year. Similarly, the probability of working in the informal sector in the current year are about 0.43 if a male head was nonpoor and not employed in the informal sector in the previous year, but increase at about 0.50 if, other

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things equal, he was previously employed in the informal sector. Similar persistence effects were also found in the case of female heads. Moreover, statistically significant and positive spillover effects are found running both from past poverty to current informal employment and from past informality to current poverty status, corroborating the view that the two processes are also shaped by interrelated dynamics in segmented labor markets. For example, the probability of being poor in the current year increases at 0.33 if a male head was both poor and working in the informal sector in the previous year, Similarly, the probability of working in the informal sector increases by 1 percentage point (almost 5 percentage points in the case of a female head) if the person was simultaneously poor and working in the informal sector in the previous year. Thus, our results seem to support the view of jobs in the informal sector as a demand-led and second-choice type of employment to which household heads turn as they find it difficult to cover minimum household needs, as in the case of poor households. The involuntary nature of informality should indeed be worrisome for the policy makers; not only is the status of informality a second-choice type of employment, but it is also highly persistent (and, therefore, it is hardly identifiable as ‘‘port of entry’’ to formal employment) and it has, on average, unpleasant poverty implications. These concerns need to be addressed in future labor negotiations, particularly with the share of nonregistered salaried workers on the rise. Better access to capital, better labor-market institutions, efforts in ‘‘formalizing’’ nonregistered salaried workers, policies to enhance firm productivity and policies aimed at improving human capital, and individual productivity (e.g., training programs) could expand workers’ opportunities for advancement, breaking the informality trap and leading to an improvement in families’ well-being. Well-designed income transfer schemes for poorer households are instead an important measure to help households move out of poverty (breaking the poverty trap). Breaking the informality and the poverty traps may not be enough if some individual characteristics strongly increase the individual probabilities to reenter in the negative status and, therefore, being trapped in it once again. Our results show that observed and unobserved heterogeneity matter in determining the probability of being poor and/or working in the informal sector. Therefore, the identification of the groups having high risks of being poor and/or of working in the informal sector naturally provides strategic information to design public policies oriented to alleviate poverty and improving working opportunities.

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NOTES 1. In 1994 it was passed as a reduction in employer contributions for all workers of about 40% on average. 2. In 1995 Law 24,465 included for the first time within the Employment Contract Law the trial period. All new contracts will be under a trial period during the first three months, enabling collective labor agreement to extend the period to six months. Employer contributions to social security were eliminated during this period, although not those related to the health system. The impact of this reform was unambiguous and strong. There is agreement that since 1995, a vast majority of new recruitments in the formal sector of the economy were under a trial period. 3. Informality also includes the beneficiaries of employment plans. These plans were implemented by the government in the second half of the 1990s and may be identified in the data since 2000. 4. We follow INDEC’s methodology for identifying poverty. It consists on computing the number of equivalent adults for each household and then computing a monetary poverty line also for each family. This approach accounts for households’ composition according to sex and age. Total household income is compared to that line and poor households become those with incomes below that value. 5. Arulampalam and Stewart (2007) report that, in their simulation experiments, the two estimators provided similar results once correlated random effects were allowed using the Mundlak approach, which entails the inclusion of longitudinal averages of the explanatory variables x i . As shown by our Eq. (4), our estimates always include the Mundlak terms x i . 6. 6 Moreover, in the conditional MLE we can ignore any stratification that is a function of the initial level of deprivation and of the time-constant explanatory variables; thus, using sampling weights would lead to an efficiency loss. 7. As a simple robustness check, we estimated bivariate probit models on both the balanced sample (consisting of all individuals who are present in each of the four waves of data) and the unbalanced sample (individuals present in at least two waves) and obtained very similar results for the coefficients of the main variables of interest (namely, lagged poverty and lagged informality). These sets of estimates are available from the authors upon request. 8. We observe no variability over the period of study (i.e., marital status) or very limited variability, so we decide to treat these variables as time constant to simplify the specification since the estimation of our model is already computationally demanding. 9. The identification strategy relies upon the observed changes in an individual status of poverty and informal employment, which is convenient to our aims given that the types of exclusion restrictions normally used in the literature (e.g., AmuedoDorantes, 2004) did not hold in our case. 10. The reference group is composed by single low-educated males living in Great Buenos Aires, working in manufacture as white collars in medium-largesized firms. 11. The reference group is composed by single low-educated females living in Great Buenos Aires, working in manufacture as white collars in medium-large-sized firms.

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REFERENCES Aassve, A., Burgess, S., Propper, C., Dickson, M. (2005). Modelling poverty by not modeling poverty: a simultaneous hazard approach to the UK. ISER Working Paper 2005-26. Essex, England. Alessie, R., Hochguertel, S., & van Soest, A. (2004). Ownership of stocks and mutual funds: A panel data analysis. The Review of Economics and Statistics, 86, 783–796. Allison, P. D. (1982). Discrete-time methods for the analysis of event histories. In: S. Leinhardt (Ed.), Sociological methodology (pp. 61–97). San Francisco: Jossey-Bass Publishers. Amuedo-Dorantes, C. (2004). Determinants and poverty implications of informal sector work in Chile. Economic Development and Cultural Change, 52(2), 349–368. Arranz, J. M., & Canto, O. (2008). Measuring the effect of spell recurrence on poverty dynamics, P.T.N.1 5/80, Instituto de Estudios Fiscales, Madrid. Arulampalam, W., & Stewart, M. B. (2007). Simplified implementation of the Heckman Estimator of the dynamic probit model and a comparison with alternative estimators. IZA Discussion Papers 3039. Institute for the Study of Labor (IZA), Bonn, Germany. Bane, M. J., & Ellwood, D. T. (1986). Slipping into and out of poverty: The dynamics of spells. Journal of Human Resources, 21(1), 1–23. Beccaria, L., & Groisman, F. (2008). Informality and poverty in Argentina. Invetigacio´n Econo´mica, LXVII, 266, Me´xico. Biewen, M. (2006). Who are the chronic poor? An econometric analysis of chronic poverty in Germany. Research on Economic Inequality, 13, 31–62. Biewen, M. (2009). Measuring state dependence in individual poverty histories when there is feedback to employment status and household composition. Journal of Applied Econometrics, 24(7), 1095–1116. Capellari, L., & Jenkins, S. P. (2004). Modelling low income transitions. Journal of Applied Econometrics, 19(5), 593–610. Clark, A., & Etile´, F. (2002). Do health changes affect smoking? Evidence from British panel data. Journal of Health Economics, 21(4), 533–562. Devicienti, F. (2001). Poverty persistence in Britain. A multivariate analysis using the BHPS, 1991–1997. Journal of Economics, 9(Suppl.), 1–34. Devicienti, F., & Poggi, A. (In press). Poverty an social exclusion: Two sides of the same coin or dynamically interrelated processes. Applied Economics. DOI:10.1080/ 00036841003670721. Ducan, G. J. (1984). Years of poverty, years of plenty. Ann Arbor, MI: Institute for Social Research. Fields, G. (1975). Rural-urban migration, urban unemployment and underemployment, and job search activity in LDCs. Journal of Development Economics, 2, 165–187. Gasparini, L., & Tornaroli, L. (2007). Labor informality in Latin America and the Caribbean: Patterns and trends from household survey microdata. Working Paper no. 46. CEDLAS, Buenos Aires, Argentina. Groisman, F. (2008). Distributive effects during the expansionary phase in Argentina (2002–2007). Cepal Review, 96, 203–222. Hansen, J., & Wahlberg, R. (2004). Poverty persistence in Sweden. IZA Discussion Paper. Bonn, Germany.

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Heckman, J. J. (1981). The incidental parameters problem and the problem of initial conditions in estimating a discrete time-discrete data stochastic process. In: C. F. Manski & D. McFadden (Eds), Structural analysis of discrete data with econometric applications. Cambridge: MIT Press. Heckman, J., & Sedlacek, G. (1985). Heterogeneity, aggregation and market wage functions: An empirical model of self-selection in the labor market. Journal of Political Economy, 93, 1077–1125. Hussmanns, R. (2005). Measuring the informal economy: From employment in the informal sector to informal employment. Working Paper no. 53. ILO, Geneva. International Labour Organization. (2002). Decent work and the informal economy. Report VI, International Labour Conference, 90th Session, Report IV, Geneva. Jenkins, S. P. (1995). Easy estimation methods for discrete-time duration models. Oxford Bulletin of Economics and Statistics, 57, 129–137. Kalbfleisch, J. D., & Prentice, R. L. (1980). The statistical analysis of failure time data. New York: Wiley. Lillard, L. A., & Willis, R. J. (1978). Dynamic aspects of earnings mobility. Econometrica, 46, 985–1012. Maloney, W. F. (1999). Does informality imply segmentation in urban labor markets? Evidence from sectoral transitions in Mexico. World Bank Economic Review, 13, 275–302. Orme, C. D. (1997). The initial conditions problem and two-step estimation in discrete panel data models. Mimeo, University of Manchester, Manchester, England. Orme, C. D. (2001). Two-step inference in dynamic non-linear panel data models. Mimeo, University of Manchester, Manchester, England. Packard, G. (2007). Do workers in Chile choose informal employment? A dynamic analysis of sector choice. Working Paper no. 4232. World Bank, Washington, DC. Perry, G., Lopez, J. H., Maloney, W., Arias, O., & Serven, L. (2006). Poverty reduction and growth: Virtuous and vicious circles. Washington, DC: World Bank. Perry, G., Maloney, W., Arias, O., Fajnzylber, P., Mason, A., & Saavedra-Chanduvi, J. (2007). Informality: Exit and exclusion in Latin America. Washington, DC: World Bank. Poggi, A. (2007). Does persistence of social exclusion exist in Spain? Journal of Economic Inequality, 5(1), 53–72. Pratap, S., & Quintı´ n, E. (2006). Are labour markets segmented in developing countries? A semiparametric approach. European Economic Review, 50, 507–542. Sosa Escudero, W., & Arias, O. (2008). Trends in informality in Argentina: A cohorts/PVAR approach. Mimeo, World Bank, Washington, DC. Stevens, A. H. (1999). Climbing out of poverty, falling back in (measuring the persistence of poverty over multiple spells). Journal of Human Resources, 34, 557–588. Stewart, M. B. (2007). The interrelated dynamics of unemployment and low-wage employment. Journal of Applied Econometrics, 22, 511–531. Wooldridge, J. M. (2005). Simple solutions to the initial conditions problem for dynamic, nonlinear panel data models with unobserved heterogeneity. Journal of Applied Econometrics, 20, 39–54.

CHAPTER 5 INEQUALITY EVOLUTION IN BRAZIL: THE ROLE OF CASH TRANSFER PROGRAMS AND OTHER INCOME SOURCES Luiz Guilherme Scorzafave and E´rica Marina Carvalho de Lima ABSTRACT This chapter provides a detailed analysis of the recent evolution (1993–2007) of Brazilian income inequality. Particularly, we assess the contribution of different income sources to inequality, using three different decomposition techniques: Shorrocks (1982), Lerman and Yitzhaki (1985), and Gini decomposition. We exploit a recent dataset (PNAD, 2004) that allows the identification of different governmental transfer programs (Bolsa-Famı´ lia, PETI, and BPC) and their impacts into inequality. While informal labor income and self-employment income reduces inequality in almost every measure, the opposite is true for public sector wages. Private formal labor income is becoming less important in explaining Brazilian inequality over time, but its behavior is still important, as it represents more than 40% of the total income. We find that social transfer programs have a limited, but positive impact on equality.

Studies in Applied Welfare Analysis: Papers from the Third ECINEQ Meeting Research on Economic Inequality, Volume 18, 107–129 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2010)0000018008

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On the other hand, dynamics of pensions attenuate the recent path of decreasing inequality in Brazil.

1. INTRODUCTION Brazil is one of the most unequal countries in the world, occupying the 119th position among 127 countries in 1998. However, since 2001, there has been a slight drop in inequality in Brazil and an even stronger fall in poverty. Multiple causes may be responsible for this recent movement. For example, in recent years, Brazilian government is increasingly spending budget resources on conditional cash transfer (CCT) programs in order to alleviate poverty and inequality. But labor market dynamics may also play an important role, as more than 60% of household income comes from the labor market. Different policies can have opposite effects on inequality. CCT programs contribute to reducing inequality, while public sector wages and the retirement system in Brazil do the opposite (Ferreira & Souza, 2004; Hoffmann, 2003). In order to improve our knowledge about the sources of recent trends in inequality, this chapter will decompose different inequality measures according to income sources (formal private labor income, informal labor income, self-employed income, house rents, CCT programs, retirement, pensions, public sector wages) using three different techniques. As far as we know, this is the most detailed inequality source decomposition implemented on Brazilian data. The main contribution of the chapter is to identify which factors are contributing to a decrease (or increase) in inequality. Two aspects of social policy make Brazil an interesting case study of changes in inequality. First, since 2001, there has been a continuous growth in CCTs in Brazil and by 2008 more than 11 million of households benefited from Bolsa-Famı´lia (one of the largest CCT programs in the world). So, it is important to verify if one of the aims of the program – reducing inequality – is being reached. Second, Brazil has an ageing population, increasing the weight of retirement and pensions in the public budget, which has important implications for inequality. The chapter is divided into five sections. In Section 2, we present a short bibliographic review concerning income inequality decomposition in Brazil. Section 3 discusses the decomposition methodologies and data. The Section 4 presents the results, and the final section provides concluding remarks.

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2. LITERATURE REVIEW: WHAT INCOME SOURCE EXPLAINS BRAZILIAN INEQUALITY? The literature on Brazilian inequality is vast and has grown in the past years. Our focus is on papers that specifically decompose inequality according to income sources. Hoffmann (2003) studies the contribution of different income sources to per capita household income inequality in 1999, decomposing the Gini index. The author concludes that pension and retirement income both contribute to increasing Brazilian inequality, especially in metropolitan areas. Ferreira and Souza (2004) adopt the same approach as Hoffmann (2003) and conclude that for specific Brazilian regions (Parana´ State) the contribution of pensions and retirement to inequality is not significant, while the result is opposite for Brazil as a whole. Adopting a different division of income sources, Hoffmann (2004) confirms the importance of retirement and pension income to inequality. Soares (2006) decomposes the inequality in Brazil between 2001 and 2004 and concludes that three-fourth of the recent drop in inequality rates is due to the behavior of labor market, as the labor income becomes less concentrated. Social programs as Bolsa-Famı´lia also play an important role in this process. Barros, Carvalho, Franco, and Mendonc- a (2006) studies the role of nonlabor income, especially of two programs – Bolsa-Famı´lia and Benefı´cio de Prestac- a˜o Continuada (BPC) – and concludes that they are responsible for about half of the recent inequality fall in Brazil. Soares, Soares, Medeiros, and Oso´rio (2006) decomposes Gini index and concludes that between 1995 and 2004, BPC is responsible for 7% of the reduction in inequality. Our chapter is innovative in many ways in Brazilian study of inequality. First, we implement a very detailed decomposition of income sources, incorporating another important social program in Brazil: Programa de Erradicac- a˜o do Trabalho Infantil (PETI) whose aim is to eliminate child labor. Second, we divide labor income in five components: formal labor income, informal labor income, self-employed income, public sector wages, and military wages. This is relevant in the Brazilian case, as there is anecdotic evidence that inequality is larger in public sector than among private sector workers. Third, we assess inequality decomposition using generalized entropy measure with index1. As this measure is very sensitive to changes in the lower tail of income distribution, it is particularly useful to capture the effect of CCT in Brazilian case. Finally, we implement the Lerman and Yitzhaki (1985) methodology that, as far as we know, has not been yet implemented in the Brazilian case.

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LUIZ GUILHERME SCORZAFAVE AND E´RICA MARINA CARVALHO DE LIMA

3. DATA AND METHODOLOGY We measure per capita household income inequality that neglects any income disparity inside family. We also ignore scale economies in household consumption, giving the same weight to all family members, although previous paper (Castro & Scorzafave, 2007) shows no significant differences in evolution of inequality when scale economies are considered in Brazilian case.

3.1. Cash Transfer Programs in Brazil: Bolsa-Famı´ lia, PETI, and BPC We will quickly describe the main characteristics of three important social programs in Brazil. Operating since September 2004 in Brazil, Bolsa-Famı´lia consolidates already existing social programs (Bolsa Escola, Bolsa Alimentaca˜o, Auxı´lio Ga´s, and Carta˜o Alimentac- a˜o). The program has the following design: very poor families, with household per capita income up to R$ 50.00 per month in 2004 (about US$ 20.00) received R$ 50.00 per month. The families with per capita income between R$ 50.00 and R$ 100.00 per month received R$ 15.00 per child, up to three children. So, the transfers varied between R$ 15.00 and R$ 95.00 in 2004. Another important social program in Brazil is BPC. It is a minimum wage benefit (R$ 260.00, in 2004) received by people aged 65 years or more and disabled people with per capita family income of one-fourth of the minimum wage. BPC began in 1996 and cannot be (officially) received together with other social programs, such as Bolsa-Famı´lia. Finally, PETI aims to eliminate child labor in Brazil. This program covers children aged between 7 and 15 years with per capita income below half the minimum wage. In 2004, the benefit varies between R$ 20.00 and R$ 40.00 per child, depending on the city size. In 2006, PETI was incorporated into the Bolsa-Famı´lia program.

3.2. Data The data used in this chapter are from Pesquisa Nacional por Amostra de Domicı´lios (PNAD),1 covering the following years: 1993, 1995, 1997, 1999, 2001, 2002, 2003, 2004, 2005, 2006, and 2007. Since 2004, PNAD covers the rural area of the north region. To maintain comparability over time, we exclude data from this region in 2004, 2005, 2006, and 2007. Although

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The Role of Cash Transfer Programs and Other Income Sources

covering 11 surveys, we study carefully the 2004 data, because this year brings supplementary information concerning cash transfer programs. PNAD reports different income sources for each household. The variable ‘‘other incomes,’’ is a residual one, encompassing government transfers. For 2004, we can decompose this variable into four parts (Bolsa-Famı´lia, PETI, BPC, and others). The variable Bolsa-Famı´lia was created for everyone that reported receiving any kind of governmental transfer other than PETI and BPC, including programs such as Bolsa Escola, Auxı´lio-Ga´s, and Carta˜o Alimentac- a˜o. In these cases, we keep the declared values. For those families that declared receiving Bolsa-Famı´lia, but do not declare the received value, we have imputed the program values, according to income and number of children of the household.2 Labor income was disaggregated as there is evidence that public sector wages and military wages have a distinct distribution when compared with private sector wages. In fact, Belluzzo, Anuatti, and Pazello (2005) show that there is substantial wage differential between public and private workers in Brazil. Beyond this, the informal sector in Brazil is very important, encompassing more than 40% of Brazilian workers. So, it is also necessary to decompose the private labor wage into three distinct components: private formal labor income, for the private sector workers who are employees and were covered by the Brazilian labor regulations; informal labor income, for those employees not covered by regulations; and self-employed income.

3.3. Methodology In this section, we present three inequality decomposition techniques by income sources applied to Brazilian data: Shorrocks (1982), Gini decomposition, and Lerman and Yitzhaki (1985). 3.3.1. Shorrocks (1982) Shorrocks (1982) is one of the most important contributions in the field of inequality decomposition by income sources. He notes, ‘‘alternative decompositions are available because the functional representation used by any inequality index is not uniquely determined’’ (1982, p. 208). He shows that the contribution of any factor (as proportion of total inequality) can take any value depending on the chosen method. To solve this problem, the author assumes some restrictions. Following Shorrocks (1982), let Y ki be individual income P ði ¼ 1; . . . ; nÞ of source kðk ¼ 1; . . . ; KÞ and let Y ¼ ðY1 ; . . . ; Yn Þ ¼ Yk k

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LUIZ GUILHERME SCORZAFAVE AND E´RICA MARINA CARVALHO DE LIMA

be the distribution of total income, whose variance is: X XX s2 ðYÞ ¼ s2 ðYk Þ þ rjk sðYj ÞsðYk Þ k

jak

(1)

k

that rjk is the correlation coefficient between Yj and Yk. Assuming that the different kinds of incomes are not correlated: X s2 ðYk Þ (2) s2 ðYÞ ¼ k

Shorrocks (1982) assumes that I (Y ), a generic inequality measure, is continuous and symmetric and that I (Y ) ¼ 0 if and only if Y ¼ me, where e ¼ (1,1, y ,1). If K disjoint and exhaustive income sources could be identified, the contribution of factor k to total inequality can be represented by Sk(Y1, y , YK;K ), that is continuous in Yk. He also assumes symmetry of factors, that contribution of factor k should be independent of the disaggregation level of total income. He also assumes consistency: X X S k ðY1 ; . . . ; YK ; KÞ ¼ SðYk ; YÞ ¼ IðYÞ (3) k

Finally, he assumes that Sðmk e; YÞ ¼ 0 for every mk and two factor symmetry: SðY1 ; Y1 þ Y1 PÞ ¼ SðY1 P; Y1 þ Y1 PÞ, for every permutation matrices P. Assuming these properties, sk is the relative contribution of factor k to income inequality: sk ðIÞ ¼

SðYk ; YÞ covðYk ; YÞ for every Yame ¼ IðYÞ s2 ðYÞ

(4)

or sk ¼

covðYk ; YÞ s2 ðYÞ

(40 )

3.3.2. Gini Decomposition Following Hoffmann (2004), let Y0 ¼ ðY 01 ; Y 02 ; . . . ; Y 0n Þ denotes the income distribution of a population with n families and Y K ¼ ðY K1 ; Y K2 ; . . . ; Y Kn Þ be the distribution of source K income. So, if Y K1  Y K2  . . .  Y Kn , the Gini coefficient can be written as: X ai ðY0 ÞY 0i (5) G¼ i

The Role of Cash Transfer Programs and Other Income Sources

113

so that ai ðY0 Þ ¼ ai ðY0 ; GÞ ¼P2=ðn2 mÞði  ðn þ 1Þ=2Þ is the weight associated with Y0. Changing Y0i for Yki, we obtain:  X X X X 2  nþ1 Sk i  S k G k ¼ S k Rk Gk (6) Y ki ¼ G¼ 2 n mk 2 i k k k where Rk is the ‘‘Gini correlation’’ between source k and total income, with 0  R  1; Gk is the Gini concerning the source k and Sk represents the share of source k in total income. We can also define the concentration ratio of source k as: Ck ¼ Rk Gk

(7)

We can define the contribution of factor k to income inequality as: sk ¼

S k Rk Gk G

(8)

If C k 4G, the income source k contributes to raise inequality. 3.3.3. Lerman and Yitzhaki (1985) Following Lerman and Yitzhaki (1985), we can assess the impact on to inequality of a variation bYk in source k, where b approaches to 1. According to Eq. (6), we can write: @G ¼ S k ðRk Gk  GÞ @bk

(9)

This approach is used to examine the effect of marginal changes in each income source on total inequality. For instance, we can assess the impact of a marginal increase in the benefit paid by Bolsa-Famı´lia on inequality. This is important because in practice, we show modest changes in each source in the short run and it is interesting to have a method that permits to analyze the importance of these changes.

4. RESULTS Our results (Fig. 1) confirm previous findings that point to a consistent decrease in Brazilian inequality, especially since 2001. This fact is reflected not only in Gini, but also in generalized entropy measures. It is interesting to note that GE(1), which is very sensitive to income variations in the lower tail of distribution, shows two periods of falling inequality. The first one is between 1993 and 1995, probably because the

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LUIZ GUILHERME SCORZAFAVE AND E´RICA MARINA CARVALHO DE LIMA

Fig. 1. Income Inequality Measures: 1993–2007.

poor benefit from Plano Real, the successful inflation stabilization plan in 1994. The second moment is 2004; in the next sections we will investigate this year carefully. The other indexes also show a consistent fall in Brazilian household per capita income inequality. For example, Theil-T decreases almost 18% between 1993 and 2007 and Gini falls 7.5% over the same period of time.

The Role of Cash Transfer Programs and Other Income Sources

115

4.1. Income Inequality Decompositions: 1993–2007 In this section, we show the results of decomposition techniques. Fig. 2 illustrates Shorrocks (1982) decomposition for Brazilian per capita household income between 1993 and 2007. Private formal labor income (hereafter private labor income) is the main factor contributing to inequality, as more than 45% of inequality comes from this factor. However, its contribution shows a decreasing path over time. Retirement payments (9–15%) and public sector wages (8–14%) also contributes importantly to inequality. These two factors, together with pension payments are becoming more important over time in explaining inequality in Brazil. It worth emphasizes the role of selfemployed income and informal labor income. They do not present a clear trend over time, although their contribution to inequality is also important (around 12 and 5%, respectively). Military wages and donations represent less than 1% of inequality and are not reported in Fig. 2. The Gini decomposition confirms results of the Shorrocks decomposition (Fig. 3). Once again, private labor income contributes to near 40% of inequality, but its importance to inequality is decreasing over time. The path for retirement and pension payments are different in this case. While the contribution of pensions exhibits a growth path, retirement income is constant since 1999. It is interesting that ‘‘other incomes’’ shows an increasing importance after 2003, probably due to cash transfer programs of Brazilian government. Gini decompositions highlight the factors that most contribute to inequality. If the concentration ratio of factor k is higher than Gini, this factor contributes to increase inequality. Results show that formal private labor income is contributing to increase inequality (in Table A3, C k 4G). The same is true for public sector wages. On the other hand, the other labor components of income (informal and self-employment income) act in the opposite way. The role of pensions and retirement are interesting. Depending on the considered year, they contribute to increase or to decrease inequality. In fact, their concentration ratios are very close to the Gini index of the whole income distribution. We use the Lerman and Yitzhaki (1985) decomposition to assess the impact of a marginal increase of each income source on inequality (Fig. 4). Inequality remains practically the same when the income sources changes marginally. For example, if each formal private worker receives a 1% higher wage in 1995, inequality would rise only 0.04%. Although very small,

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LUIZ GUILHERME SCORZAFAVE AND E´RICA MARINA CARVALHO DE LIMA

Fig. 2.

Shorrocks Decomposition by Income Sources: 1993–2007.

117

The Role of Cash Transfer Programs and Other Income Sources Retirement 18,0

42,0

16,0

40,0

14,0

38,0

12,0

%

%

Formal Private Labour Income 44,0

36,0

10,0

34,0

8,0

32,0

6,0 93

95

97

99

01

02

03

04

05

06

93

07

95

97

Informal Labour Income

99

01

02

03

04

05

06

07

Self-Employed Income

12,0

20,0 18,0

10,0

16,0 14,0

8,0 %

%

12,0 6,0

10,0 8,0

4,0

6,0 4,0

2,0

2,0

Public Sector Wages

10,8

07

05

06

03

04

House Rent

3,0

10,6

02

01

99

95

97

93

07

05

06

04

03

02

01

99

97

95

0,0 93

0,0

2,5

10,4 2,0 10,0

%

%

10,2

9,8

1,5 1,0

9,6 0,5 9,4 0,0

9,2 93

95

97

99

01

02

03

04

93

05

Pensions

8,0

97

99

01

02

03

04

05

06

07

04

05

06

07

Other Incomes

3,5

7,0

3,0

6,0

2,5

5,0

2,0

4,0

%

%

95

1,5

3,0

1,0

2,0

0,5

1,0 0,0

0,0 93

95

97

99

01

02

03

Fig. 3.

04

05

06

07

93

95

97

99

01

Gini Decomposition: 1993–2007.

02

03

LUIZ GUILHERME SCORZAFAVE AND E´RICA MARINA CARVALHO DE LIMA

118

Private Labour Income

Retirement

0,045

0,012

0,040

0,010

0,035

0,008

0,030

0,006

0,025 0,004

0,020 0,002

0,015 0,010

0,000

0,005

-0,002

0,000

-0,004

93

95

97

99

01

02

03

04

05

06

07

93

95

97

99

01

02

03

04

05

06

07

Self-Employed Income

Informal Labour Income -0,035

0,000

-0,036

-0,005

-0,037

-0,010 -0,038

-0,015

07

06

04

05

03

01

02

97

99

93

06

07

05

03

04

01

02

97

-0,025 99

-0,041 93

-0,020

95

-0,040

95

-0,039

House Rent

Public Sector Wages 0,009

0,045

0,035

0,007

0,030

0,006

0,025

0,005 %

0,040

0,008

0,020

0,004

0,015

0,003

0,010

0,002

0,005

0,001

0,000

0,000 93

95

97

99

01

02

03

04

05

06

07

Pensions

0,001

93

95

97

99

01

02

03

04

05

06

07

05

06

07

Other Incomes

0,015 0,010 0,005

-0,001 0,000 -0,005

-0,003

-0,010 -0,015 -0,020

-0,005 93

95

97

99

01

Fig. 4.

02

03

04

05

06

07

93

95

97

99

01

02

03

Lerman and Yitzhaki (1985) Decomposition.

04

The Role of Cash Transfer Programs and Other Income Sources

119

bootstrap standard deviations show that they are statistically significant, in general, with few exceptions. This methodology gives the direction of each factor contribution to inequality. Marginal changes in formal labor income are becoming less important in explaining inequality over time (in 2007 the impact was only one-third of the 1993 one). Public sector wages acts in an opposite way, with positive and growing impact on inequality. Informal labor and selfemployment incomes both contribute to reduce inequality, and pensions and donations both contribute to decreasing inequality. Finally, ‘‘other incomes’’ also changes its behavior over time; as additional cash transfers are phased-in this source now contributes to a decrease in inequality.

4.2. Cash Transfer Programs and Inequality: A Detailed Analysis for 2004 As mentioned before, PNAD 2004 provides a supplement that permits us to assess the impact of different governmental programs on inequality. We investigate which programs are most important in reducing (or increasing) inequality in Brazil. 4.2.1. Descriptive Statistics Table 1 provides some descriptive statistics concerning the different income sources. Formal workers, military, and public sector workers have higher mean wages than other workers. It is also interesting to note that about Table 1. Income source Formal labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Other sources Military wages Public sector wages Bolsa-Famı´lia PETI BPC

Descriptive Statistics.

Observations

Mean

SD

Minimum

Maximum

22,608,752 16,006,633 15,624,343 1,678,045 7,441,525 12,265,415 2,040,912 2,315,109 187,230 4,724,228 6,828,595 348,843 718,159

415.57 158.64 231.62 159.26 237.15 334.94 252.42 108.98 334.83 484.54 10.84 13.02 97.42

887.24 324.79 487.52 334.00 558.90 620.65 561.11 508.40 431.65 795.85 9.99 6.08 67.77

0.167 0.33 0.125 1 1.2 8.89 2.5 0.14 14 6.67 0.14 2.08 20

61,250 13,100 20,000 5,000 19,000 24,288 10,500 15,000 3,500 16,000 130 32 260

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LUIZ GUILHERME SCORZAFAVE AND E´RICA MARINA CARVALHO DE LIMA

Table 2.

Per Capita Household Income Inequality: 2004.

Inequality Measures Gini Theil-T–GE(1) Theil-L–GE(0) GE(1) GE(2)

Without Transfers

Observed

0.584 0.703 0.640 1.273 1.753

0.580 0.696 0.624 1.206 1.743

Obs: GE, generalized entropy.

7 million households received Bolsa-Famı´lia and 1 million households received PETI or BPC in 2004. Finally, BPC is the most generous program, paying the highest per capita benefit (R$ 97.42 per month). In a simple procedure to assess the impact of governmental programs on inequality, we compared inequality indicators using two concepts of income: one that excludes governmental transfers from per capita household income and another that includes them. Examining Table 2 we find that the Gini is 0.6% higher if we exclude income transfer. Furthermore, the results are similar for Theil-T (1%) and for GE(2) (0.6%). GE(1) has a different behavior (5.5%); yet, this is to be expected as this indicator gives more weight to income variations in the bottom part of income distribution (where the beneficiaries of cash transfer programs are concentrated). Although the amount of money in this program is small relative to other income sources, it does appear to contribute to reducing inequality. 4.2.2. Decomposition Results Table 3 shows the decomposition results. According to the Shorrocks decomposition, all social programs have a small impact on inequality because the total amount of resources is a very small fraction of total income of society. It is interesting to note that retirement (14.54%), public sector wages (9.78%), and self-employment (10.12%) have important impacts on inequality. Next, we present results concerning Gini decomposition and Lerman and Yitzhaki (1985) that permits to focus on the effects of social programs on inequality. According to the Gini decomposition (Table 4), informal labor income, self-employment income, donations, pensions, and, especially, cash transfer programs (Bolsa-Famı´lia, PETI, and BPC) tend to decrease income inequality. On the other hand, formal labor income, retirement, house rents, and public sector wages increases income inequality.

121

The Role of Cash Transfer Programs and Other Income Sources

Table 3.

Inequality Decomposition by Income Sources: Shorrocks (1982)  2004.

Source

Contribution (%)

Formal labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public Sector Wages Bolsa-Famı´lia PETI BPC Other Income Total

Table 4.

47.30 4.96 10.13 0.49 6.91 14.54 3.17 0.16 9.78 0.06 0.01 0.04 2.66 100.0

Inequality Decomposition by Income Sources: Gini – 2004.

Source Formal labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Bolsa-Famı´lia PETI BPC Other income

Sk

Gk

Rk

Ck

0.365 0.105 0.143 0.011 0.068 0.162 0.020 0.003 0.106 0.003 0.000 0.003 0.010

0.815 0.860 0.882 0.989 0.942 0.901 0.985 0.998 0.958 0.920 0.994 0.990 0.991

0.759 0.412 0.569 0.513 0.606 0.682 0.782 0.696 0.783 0.649 0.604 0.077 0.676

0.618 0.354 0.502 0.507 0.571 0.615 0.770 0.695 0.750 0.598 0.601 0.076 0.670

Obs: Total Income Gini ¼ 0.580.

Table 5 provides the Lerman–Yitzhaki decomposition results. Not only do the cash transfer programs contribute to decrease inequality, but so does informal labor income, self-employed income, donations, and pensions. However, the magnitude of impacts is modest. For example, a 1% increase

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LUIZ GUILHERME SCORZAFAVE AND E´RICA MARINA CARVALHO DE LIMA

Table 5.

Inequality Decomposition by Income Sources: Lerman and Yitzhaki (1985) – 2004.

Source

Change (%)

Formal labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Bolsa-Famı´lia PETI BPC Other income

0.0237 0.0407 0.0193 0.0014 0.0011 0.0096 0.0064 0.0006 0.0311 0.0065 0.0004 0.0036 0.0015

in informal labor income decreases the Gini by 0.041%. The impact of governmental programs is somewhat smaller.

5. CONCLUDING REMARKS We assess the effect of different income sources on Brazilian income inequality. Different income sources play very different roles in the evolution of income inequality. While informal labor income and self-employment income sectors reduce inequality in almost every measure, the reverse is true for public sector wages. Private formal labor income is becoming less important in explaining Brazilian inequality over time, but its behavior is still important, as it represents more than 40% of total income. It is interesting to note that the concentration ratio of this income source drops 13% between 1993 and 2007. If this path continues, private labor income will become a very important factor inducing the continuity of inequality fall in Brazil. On the other hand, although governmental programs such as Bolsa-Familia exhibit only a limited effect in inequality, we find evidence that these policies benefit the bottom part of income distribution. The results for retirement and pensions are not conclusive, confirming the previous literature ambiguities. Concerning pensions, our result is stronger in the sense that this income source seems to have an increasing contribution to Brazilian inequality. With the ageing of population, retirement and

The Role of Cash Transfer Programs and Other Income Sources

123

pensions are becoming important sources of family income. This implies that the government should worry about the inequality of these two kinds of income. The same recommendation is valid for public sector wages that is also increasingly contributing to inequality. In order to keep on the path of reducing inequality in Brazil, some practices should be implemented. First, the government should improve the efficiency of CCT programs, particularly its focus on the poorest citizens. Second, investments in the public education system certainly would also contribute to a path of decreasing labor income inequality.

NOTES 1. PNAD is a Brazilian household survey covering many aspects of households’ members, such as employment, migration, schooling, etc. 2. We also implement the decompositions without this imputation. The results are in the appendix, but there is not substantial difference between the imputed one.

REFERENCES Barros, R. P., Carvalho, M., Franco, S. O., & Mendonc- a, R. (2006). Uma ana´lise das principais causas da queda recente na desigualdade de renda brasileira. Econoˆmica. Rio de Janeiro, 8(1), 117–147. Belluzzo, W., Anuatti, F., & Pazello, E. T. (2005). Distribuic- a˜o de sala´rios e o diferencial pu´blico-privado no Brasil. Revista Brasileira de Economia, 59, 511–533. Castro, S., & Scorzafave, L. (2007). Ricos? Pobres? Uma Ana´lise da Polarizac- a˜o de Renda no Caso Brasileiro. Pesquisa e Planejamento Econoˆmico, 37(2), 283–297. Ferreira, C. R., & Souza, S. C. I. (2004) Prevideˆncia social e desigualdade: A participac- a˜o das aposentadorias e penso˜es na distribuic- a˜o de renda do Brasil – 1981 A 2001 – XXXII Encontro Nacional de Economia – ANPEC. Joa˜o Pessoa. Hoffmann, R. (2003). Inequality in Brazil: The contribution of pensions. Revista Brasileira de Economia, Rio de Janeiro, 57, 755–773. Hoffmann, R. (2004). Decomposition of Mehran and Piesch inequality measures by factor components and their application the distribution of per capita household income in Brazil. Brazilian Review of Econometrics, Rio de Janeiro, 24(1), 149–171. Lerman, R. I., & Yitzhaki, S. (1985). Income inequality effects by income source: A new approach and applications to the United States. The Review of Economics and Statistics, 67(1), 151–156. Shorrocks, A. F. (1982). Inequality by factor components. Econometrica, 50(1), 193–212. Soares, S. S. D. (2006). Distribuic- a˜o de renda no Brasil de 1976 a 2004 com eˆnfase no perı´ odo de 2001 a 2004. IPEA. Brası´ lia. Soares, F. V., Soares, S. S., Medeiros, M., & Oso´rio, R. G. (2006) Programas de transfereˆncia de renda no Brasil: Impactos sobre a desigualdade, in: IPEA (2006). Sobre a recente queda da desigualdade de renda no Brasil. Brası´ lia: IPEA, ago.2006. (Nota Te´cnica).

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APPENDIX Table A1.

Inequality Measures.

1993 1995 1997 1999 2001 2002 2003 2004 2005 2006 2007 Gini Theil-T–GE(1) Theil-L–GE(0) GE(1) GE(2)

0.614 0.807 0.722 3.036 2.535

0.613 0.781 0.707 1.497 1.969

0.612 0.781 0.707 1.483 2.140

0.607 0.759 0.691 1.413 1.856

0.607 0.768 0.699 1.638 1.913

0.599 0.739 0.681 1.746 1.769

0.591 0.715 0.666 2.173 1.701

0.580 0.696 0.624 1.206 1.743

0.584 0.704 0.647 1.743 1.799

0.578 0.690 0.638 1.733 1.763

0.568 0.662 0.614 1.745 1.568

54.489 4.573 11.089 0.227 2.282 9.529 1.576 0.289 7.363 8.583

43.19 8.68 16.42 0.70 4.46 13.18 1.27 0.70 9.79 3.28

Formal labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Other Income Bolsa-Famı´lia PETI BPC

1993

Formal labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Other income Bolsa-Famı´lia PETI BPC

Source

39.83 9.29 17.78 0.97 4.71 13.28 2.67 0.41 10.39 1.130

45.363 6.554 15.098 0.342 4.168 10.295 4.952 0.160 10.009 3.059

1995

40.05 9.84 17.25 0.94 5.00 13.54 2.69 0.42 9.78 0.98

51.928 4.335 12.136 0.405 3.650 10.086 4.627 0.280 7.904 4.649

1997

37.12 10.02 16.19 1.12 5.60 16.05 2.55 0.85 9.68 0.97

46.984 5.362 10.783 0.535 4.427 15.149 4.738 0.757 9.029 2.238

36.88 10.96 15.40 0.97 6.32 15.99 2.32 0.41 10.15 1.08

44.767 6.686 11.120 0.426 5.582 14.358 3.863 0.419 8.662 4.118

2001 49.099 5.772 10.703 0.488 5.660 14.299 3.327 0.191 9.160 1.301

36.86 10.63 14.90 1.10 6.52 16.87 1.92 0.30 10.39 1.09

Gini (%) 37.02 11.18 14.83 1.15 6.19 15.91 2.23 0.29 10.62 1.10

2003

45.874 6.564 10.714 0.651 5.101 14.463 3.374 0.269 11.036 1.955

2002

Shorrocks (1982)

Decomposition Results.

1999

Table A2.

36.54 10.46 14.31 1.12 6.79 16.24 1.96 0.29 10.63 1.66 0.32 0.02 0.32

47.289 4.958 10.128 0.486 6.914 14.539 3.173 0.162 9.781 2.563 0.064 0.004 0.038

2004

37.53 10.76 13.99 1.10 6.70 16.33 2.03 0.27 10.38 1.78

49.150 6.179 9.127 0.572 4.445 13.325 4.117 0.127 10.163 2.795

2005

37.09 10.42 13.89 1.11 6.66 16.05 1.98 0.31 11.36 2.22

47.900 5.036 10.847 0.959 4.360 12.521 3.913 0.139 11.684 2.641

2006

36.59 10.40 15.05 0.87 6.86 16.05 1.62 0.27 11.50 1.69

44.418 5.255 12.509 0.474 5.221 14.226 2.801 0.158 13.547 1.391

2007

The Role of Cash Transfer Programs and Other Income Sources 125

126

LUIZ GUILHERME SCORZAFAVE AND E´RICA MARINA CARVALHO DE LIMA

Table A3. Source

Gini Decomposition.

Sk

Gk

Rk

Ck

0.823 0.871 0.879 0.992 0.957 0.913 0.988 0.997 0.959 0.986

0.813 0.385 0.600 0.488 0.583 0.653 0.775 0.731 0.775 0.815

0.670 0.335 0.527 0.484 0.558 0.596 0.766 0.729 0.744 0.804

0.824 0.867 0.872 0.990 0.955 0.916 0.984 0.997 0.957 0.991

0.800 0.406 0.617 0.524 0.594 0.668 0.821 0.708 0.785 0.772

0.660 0.352 0.538 0.519 0.567 0.613 0.808 0.706 0.751 0.765

0.825 0.867 0.878 0.990 0.953 0.909 0.985 0.997 0.958 0.993

0.802 0.426 0.622 0.496 0.599 0.658 0.816 0.734 0.776 0.775

0.662 0.369 0.546 0.491 0.571 0.598 0.804 0.732 0.743 0.770

0.830 0.868 0.876 0.988 0.950 0.908 0.985

0.785 0.426 0.596 0.466 0.598 0.697 0.811

0.651 0.370 0.522 0.461 0.569 0.633 0.799

1993 Private labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Other income

0.431 0.086 0.164 0.007 0.044 0.131 0.012 0.007 0.097 0.032

Private labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Other income

0.398 0.093 0.178 0.010 0.047 0.133 0.027 0.004 0.104 0.011

Private labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Other income

0.400 0.098 0.172 0.009 0.050 0.135 0.026 0.003 0.097 0.009

Private labor income Informal labor income Self-employed income Donations Pensions Retirement House rent

0.371 0.100 0.162 0.011 0.056 0.161 0.026

1995

1997

1999

The Role of Cash Transfer Programs and Other Income Sources

Table A3. Source

127

(Continued )

Sk

Gk

Rk

Ck

Military wages Public sector wages Other income

0.009 0.097 0.010

0.997 0.989 0.991

0.751 0.777 0.661

0.655 0.749 0.768

Private labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Other income

0.368 0.109 0.154 0.009 0.063 0.159 0.023 0.004 0.101 0.010

0.828 0.864 0.882 0.988 0.949 0.904 0.985 0.998 0.959 0.988

0.779 0.456 0.600 0.448 0.630 0.688 0.808 0.760 0.784 0.600

0.645 0.394 0.529 0.443 0.598 0.622 0.797 0.759 0.752 0.593

Private labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Other income

0.370 0.111 0.148 0.011 0.061 0.159 0.022 0.002 0.106 0.011

0.824 0.864 0.879 0.987 0.944 0.900 0.985 0.998 0.959 0.979

0.777 0.462 0.578 0.483 0.599 0.679 0.804 0.717 0.794 0.594

0.640 0.399 0.509 0.477 0.566 0.612 0.792 0.716 0.762 0.582

Private labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Other income

0.368 0.106 0.149 0.011 0.065 0.168 0.019 0.003 0.103 0.010

0.821 0.865 0.880 0.988 0.944 0.895 0.985 0.998 0.957 0.972

0.768 0.435 0.579 0.471 0.599 0.681 0.787 0.730 0.778 0.390

0.631 0.376 0.509 0.466 0.566 0.609 0.776 0.728 0.744 0.379

Private labor income Informal labor income Self-employed income Donations

0.365 0.105 0.143 0.011

0.815 0.860 0.882 0.989

0.759 0.412 0.569 0.513

0.618 0.354 0.502 0.507

2001

2002

2003

2004

128

LUIZ GUILHERME SCORZAFAVE AND E´RICA MARINA CARVALHO DE LIMA

Table A3. Source

(Continued )

Sk

Gk

Rk

Ck

Pensions Retirement House rent Military wages Public sector wages Bolsa-Famı´lia PETI BPC Other income

0.068 0.162 0.020 0.003 0.106 0.003 0.000 0.003 0.010

0.942 0.901 0.985 0.998 0.958 0.920 0.994 0.990 0.991

0.606 0.682 0.782 0.696 0.783 0.649 0.604 0.077 0.676

0.571 0.615 0.770 0.695 0.750 0.598 0.601 0.076 0.670

Private labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Other income

0.375 0.107 0.139 0.011 0.067 0.163 0.020 0.002 0.103 0.017

0.813 0.862 0.878 0.988 0.938 0.897 0.985 0.998 0.959 0.958

0.766 0.433 0.554 0.476 0.591 0.682 0.792 0.689 0.785 0.239

0.623 0.373 0.486 0.470 0.554 0.612 0.780 0.688 0.753 0.229

Private labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Other income

0.370 0.104 0.138 0.011 0.066 0.160 0.019 0.003 0.113 0.022

0.808 0.860 0.879 0.98 0.938 0.895 0.98 0.998 0.958 0.935

0.755 0.413 0.555 0.494 0.590 0.672 0.791 0.705 0.800 0.147

0.610 0.355 0.488 0.488 0.554 0.602 0.779 0.704 0.766 0.137

Private labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Other income

0.365 0.104 0.150 0.008 0.068 0.160 0.016 0.002 0.115 0.017

0.800 0.861 0.880 0.990 0.935 0.894 0.987 0.998 0.956 0.948

0.738 0.407 0.582 0.476 0.569 0.659 0.777 0.725 0.795 0.079

0.591 0.351 0.512 0.471 0.532 0.590 0.767 0.724 0.761 0.074

2005

2006

2007

The Role of Cash Transfer Programs and Other Income Sources

Table A4.

129

Decompositions without Imputing Bolsa-Famı´lia Values: 2004.

Source

Shorrocks

Gini

Lerman

0.365 0.104 0.150 0.008 0.068 0.160 0.016 0.002 0.115 0.003 0.000 0.003 0.0169

0.618 0.354 0.502 0.507 0.571 0.615 0.770 0.695 0.750 0.598 0.602 0.077 0.670

0.738 0.407 0.582 0.476 0.569 0.659 0.777 0.725 0.795 0.651 0.608 0.077 0.079

Private labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Bolsa-Famı´lia PETI BPC Other income Source Private labor income Informal labor income Self-employed income Donations Pensions Retirement House rent Military wages Public sector wages Bolsa-Famı´lia PETI BPC Other income

Sk

Gk

Rk

Ck

0.365 0.105 0.143 0.011 0.068 0.162 0.020 0.003 0.106 0.003 0.000 0.003 0.010

0.815 0.860 0.882 0.989 0.942 0.901 0.985 0.998 0.958 0.920 0.994 0.990 0.991

0.759 0.412 0.569 0.513 0.606 0.682 0.782 0.696 0.783 0.650 0.605 0.078 0.676

0.618 0.354 0.502 0.507 0.571 0.615 0.770 0.695 0.750 0.598 0.602 0.077 0.670

CHAPTER 6 INEQUALITY REDUCING TAXATION RECONSIDERED Udo Ebert ABSTRACT The chapter investigates inequality reducing taxation for various inequality views. Using the general definition of an inequality concept (Ebert, 2004), corresponding definitions of Lorenz dominance, inequality reduction, and measures of tax progression are provided. The framework allows us to simplify and clarify the different approaches found in the literature, to extend this analysis, and to present brief and transparent proofs.

1. INTRODUCTION Nowadays one of the objectives of income taxation is inequality reduction. Public economists seem to agree that Lorenz dominance is the appropriate criterion for ranking income distributions. If the distribution of posttax income dominates the distribution of pretax income, inequality is reduced. When this is the case for any income distribution, the tax function is progressive. Progression can also be determined by the properties of the tax schedule itself: The average tax rate has to be increasing, or, equivalently, residual progression – a measure of progression – has to be less than unity for all levels of income. Studies in Applied Welfare Analysis: Papers from the Third ECINEQ Meeting Research on Economic Inequality, Volume 18, 131–152 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2010)0000018009

131

132

UDO EBERT

All this is well known and well established. Furthermore, the Lorenz criterion can be justified convincingly. When net incomes dominate gross incomes, the posttax income distribution is also unanimously preferred to the pretax income distribution by all inequality measures possessing few important properties. The measures have to be relative (equal proportional changes of all incomes do not change inequality) and symmetric (the individuals’ identity does not play a role). Moreover, they have to satisfy the principle of progressive transfers (a rank-preserving redistribution of income from a richer to a poorer individual reduces inequality). These properties seem to be indispensable for inequality measurement. On the other hand, these characterizations are based on the relative inequality view. Here inequality is not altered if all incomes are changed in the same proportion. But this concept of inequality is not the only one considered by economists. Kolm (1976) has already proposed the concept of absolute inequality (inequality is not changed if incomes are changed by the same amount); Bossert and Pfingsten (1990) have introduced intermediate inequality (which can be interpreted as a compromise between relative and absolute inequality). This concept is also mentioned and, respectively, investigated in Ebert and Moyes (2000, 2002) and Chakravarty (2009). A number of other concepts have been proposed recently (see, e.g., Pfingsten & Seidl, 1997; Zoli, 1998; del Rio & Ruiz-Castillo, 2000; Ebert, 2004; Yoshida, 2005). Furthermore, empirical studies demonstrate that the relative inequality view is not unanimously accepted (see, e.g., Amiel & Cowell, 1992). On the contrary, further attitudes toward inequality can be observed. Thus, the question arises whether inequality reducing taxation can be defined with respect to further concepts and how it can be characterized. The present chapter deals with this problem. In order to not restrict the analysis too much a priori, we start by using the general definition of an inequality concept provided in Ebert (2004). Then, in a first step, we confine ourselves to the subclass of coherent inequality concepts, namely to those for which we are able to describe the operations leaving inequality unchanged by linear transformations. We also need a corresponding concept of Lorenz dominance. It is required for checking the reduction of inequality. The dominance concept is proposed in the second step. Given this device, finally, the set of inequality reducing tax functions is described and corresponding concepts and measures of progression are defined and investigated. The chapter is organized as follows: The notation is defined in Section 2. Section 3 provides a general definition of an inequality concept (originally proposed in Ebert, 2004). Its basic idea is to characterize the equivalence

Inequality Reducing Taxation Reconsidered

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relation of having equal inequality by a one-parameter family of transformations that leave inequality unchanged. It then turns out that every coherent concept can be uniquely described by its domain and a characteristic function. The characteristic function maps the domain of incomes onto the set of strictly positive incomes (the domain of the relative inequality concept), and the characteristic function defines an isomorphism between both concepts. It therefore allows us to describe the original concept by means of the concept of relative inequality. As a consequence, this one-to-one relationship can be exploited later on; we borrow many definitions and ideas from the relative inequality concept. Furthermore, the proof of many results makes use of the isomorphism. The class of coherent concepts then consists of the relative, absolute, intermediate, and referencepoint inequality views. In Section 4, Lorenz dominance is investigated that forms an important ingredient when inequality reducing taxation is to be considered. Then it is required that a progressive tax schedule decreases inequality in every case, that is, for every income distribution. Given the characteristic function of a coherent inequality concept, it is obvious how to define the corresponding Lorenz curve. All incomes are transformed by means of the characteristic function and then the usual Lorenz curve is used. Following the discussion of Lorenz dominance, it is clear how inequality reducing taxation has to be introduced. In Section 5, the corresponding definition is given. We call a tax schedule progressive if and only if it is inequality reducing. For a given inequality concept, a tax schedule reduces inequality if and only if the corresponding tax schedule, which can be derived by means of the characteristic function, is progressive in the standard sense. Analogous results are presented in Section 6 for measures of tax progression. A measure of residual income progression can be defined by transforming income by means of the characteristic function and employing the standard measure. Finally, Section 7 concludes. The chapter makes several contributions to the literature: First, it leads to a clarification of ideas and definitions. Using the close relationship between the coherent inequality concepts considered and the relative inequality view, we are able to present simple definitions of Lorenz dominance, progression, and measures of progression. These definitions are obvious – given the above isomorphism – but often replace and always simplify the definitions proposed in the literature for the intermediate inequality view. Above that, the isomorphism facilitates the application of the nonstandard inequality concepts to practical and empirical analysis. Up to now, the standard

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approach is to use the relative inequality view. Second, the basic results in this area can now be derived directly by making use of those for relative inequality. Thus, a number of proofs that can be found in the literature now reduce to few lines. Third, some results already proven in the past are here collected and presented by a unifying approach. Fourth, all definitions and results for the concepts of reference-point inequality have not been discussed up to now.

2. NOTATION We consider a population consisting of n  3 individuals. They are supposed to be identical with respect to all attributes but possibly income. Let Od be the set of feasible incomes. We confine the analysis to Od :¼ ðd; 1Þ for d 2 R and O1 :¼ R for d ¼ 1. Incomes are either bounded from below by income d or may be arbitrary. Sometimes also negative incomes are permitted. Income can be negative in a given period (as long as an individual is able to survive by getting credit or by using savings). The income d can be interpreted as reference or minimum income. Individual i’s income is denoted by X i 2 Od ; i ¼ 1; . . . ; n. An income distribution is represented by a vector X ¼ ðX 1 ; . . . ; X n Þ 2 Ond . The vector of X in X ð Þ ¼ ðX ð1Þ ; . . . ; X ðnÞ Þ is generated by permuting the components P such a way that incomes are nondecreasing. mðXÞ ¼ ð1=nÞ ni¼1 X i is the average income of X and 1 a vector containing n ones, 1 ¼ ð1; . . . ; 1Þ. F ðOd Þ denotes the set of all continuous and nondecreasing functions F:Od-Od. For every function F 2 F ðOd Þ, we define a (corresponding) transformation F: Ond ! Ond by FðXÞ :¼ ðFðX 1 Þ; . . . ; FðX n ÞÞ. It is individualistic [i.e., FðX i Þ depends only on individual i’s income] and symmetric (all individuals are treated identically). Sometimes we confine ourselves to the subset F c ðOd Þ F ðOd Þ containing all functions, which are once continuously differentiable. T ðOd Þ F ðOd Þ denotes the set of all functions being strictly increasing and one-to-one. For F 2 F c ðOd Þ, we define the elasticity of FðXÞ with respect to X by ZðF; XÞ :¼ ðdF=FÞ=ðdX=XÞ ¼ ðdF=dXÞðX=FðXÞÞ.

3. INEQUALITY CONCEPTS Whenever we are talking about ‘‘inequality reducing taxation,’’ the meaning of the term ‘‘inequality’’ has to be made precise. This section is based on

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135

Ebert (2004) and provides a general definition of an inequality concept that represents an inequality view.

3.1. Definition The standard example of an inequality concept is given by the relative inequality view. Here we assume that incomes are strictly positive, that is, O0 ¼ Rþþ . Furthermore, according to this view, equiproportional changes of all incomes leave inequality unchanged. In other words, every income distribution X 2 Rnþþ and its transform Y ¼ lX :¼ ðlX 1 ; . . . ; lX n Þ (a priori) possess the same degree of inequality for l40; that is, in this case, both income distributions having different means are related as far as inequality is concerned. This relationship defines a (mathematical) relation on the set of feasible income distributions: X rel Y : 3There is l40 such that Y ¼ lX It is easily seen that rel is an equivalence relation, that is, X rel Y and Y rel Z imply that X rel Z (transitivity) and that Y rel X (symmetry) for all X; Y; Z 2 Rnþþ . Moreover, we have X rel X (reflexivity) for all X 2 Rnþþ . More formally, the relative inequality view can be characterized by the equivalence relation rel that in turn is based on a set of transformations or, more simply, on a set of functions defining the admissible transformations. Let S l : Rþþ ! Rþþ be given by S l ðXÞ ¼ lX for X 2 Rþþ ; l40. Then we introduce the set of functions T rel :¼ fS l jl 2 Rþþ g. Since the individuals considered can differ only in income, we treat them identically. S l defines the transformation S l ðXÞ :¼ ðSl ðX 1 Þ; . . . ; S l ðX n ÞÞ for all X 2 Rnþþ ; S l 2 T rel . Then the relative inequality view is uniquely described by the relation rel : X rel Y3There is Sl 2 T rel such that Y ¼ S l ðXÞ that is, X and Y possess the same degree of inequality according to this view if Y can be generated from X by means of a(n admissible) transformation S l , where S l 2 T rel . This idea of characterizing an inequality view can be generalized to any domain Od by an appropriate definition of an equivalence relation based on a (one-parameter) family of functions T . We therefore introduce

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Definition 1. Given a domain Od and a set T ¼ fT l jl 2 Rþþ g T ðOd Þ, we call a relation dT defined on Ond an inequality concept1 if (a) for all X; Y 2 Ond : X dT Y3 There is T l 2 T such that Y ¼ T l ðXÞ (b) (i) T 1 ðXÞ ¼ X for all X 2 Od . 1 (ii) T 1 l ¼ T 1=l for all l40, where T l denotes the inverse of T l . (iii) T l  T k ¼ T lk for all l; k 2 Rþþ , where the operation (composition)  is defined by T l  T k ðXÞ ¼ T l ðT k ðXÞÞ for all X 2 Od and l; k 2 Rþþ . [T is a(n algebraic) group with the group operation  (see Lang, 1968)]. (c) T l ðXÞ is continuous and strictly increasing in l 2 Rþþ for all X 2 Od . Thus, an inequality view is described by a binary relation dT satisfying Definition 1. It is completely determined by the domain Od and the set of admissible transformations characterized by T . Indifference between X and Y according to dT means that these distributions have the same degree of inequality. Given the properties of T the relation dT is an equivalence relation: b(iii) implies transitivity, b(ii) symmetry, and reflexivity is an implication of b(i). The set T is a one-parameter family. Parameters are restricted to be strictly positive. Condition (c) is a regularity condition. Monotonicity in l guarantees that equivalence classes fT l ðXÞjl 2 Rþþ g are ‘‘thin’’; that is, for every income distribution, there is always an arbitrarily close distribution being less or more unequal. It is obvious that rel is the inequality concept representing the relative inequality view. Another concept abs can be defined by setting Od ¼ O1 ¼ R and T 1 :¼ fT l : R ! RjT l ðXÞ ¼ X þ ln l for l 2 Rþþ g. It corresponds to the absolute inequality view. Inequality is not changed by equal additions to all incomes. The functions involved may also be nonlinear: Consider, for example, the inequality concept exp given by Od ¼ O1 and T exp ¼ fT l : O1 ! O1 jT l ðXÞ ¼ X l for l 2 Rþþ g. Further examples are discussed below.

3.2. Properties Although the idea of defining inequality concepts is a general one and the variety of concepts is great, inequality concepts can be described simply. We have

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Inequality Reducing Taxation Reconsidered

Theorem 1. (Ebert, 2004) Let dT be an inequality concept. Then there exists a continuous, strictly increasing, and surjective function G : Od ! Rþþ such that T ¼ fT l jT l ðXÞ ¼ G1  S l  GðXÞ for all X 2 Od ;

Sl 2 T rel g

given an appropriate parameterization of the functions Tl. The function G is unique up to a nonzero multiplicative factor. The set of inequality concepts can be completely characterized. The admissible transformations T l possess a simple and clear structure (when written down appropriately). Therefore, an inequality concept dT can also be described by its domain Od and the characteristic function G : Od ! Rþþ which is unique up to a multiplicative constant and which connects the domain Od with O0 ¼ Rþþ . The relative inequality concept rel is given by O0 ¼ Rþþ , and GðXÞ ¼ X, the absolute concept abs by O1 ¼ R, and GðXÞ ¼ eX , and the nonlinear concept exp by O1 , and GðXÞ ¼ ln X. The function G allows the definition of a one-toone correspondence between T and T rel and is consistent with the composition 3: T l  T k ¼ ðG1  S l  GÞ  ðG1  S k  GÞ ¼ G1  Sl  Sk  G ¼ G1  S lk  G ¼ T lk Furthermore, the equivalence relations are linked in a simple manner: X dT Y3There is T l 2 T such that Y ¼ T l ðXÞ 3There is S l 2 T rel such that Y ¼ G 1  S l  GðXÞ 3There is S l 2 T rel such that GðYÞ ¼ S l  GðXÞ 3GðXÞ rel GðYÞ X and Y possess the same degree of inequality according to dT if and only if the transformed income distributions GðXÞ and GðYÞ are equivalent according to the relative inequality view. This relationship is used below intensively. Theorem 1 says that all inequality concepts are essentially isomorphic to the relative inequality concept. The relationship is characterized by the characteristic function G, that is, we obtain Corollary 2. Let dT be an inequality concept. Then X dT Y3GðXÞ rel GðYÞ

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In this chapter, we confine ourselves to inequality concepts for which the admissible transformations are linear.

3.3. Linear Concepts In order to characterize linear concepts, we introduce progressive transfers and the property of transfer consistency: Definition 2. An income distribution X 0 2 Ond is derived from X 2 Ond by a progressive transfer if there are i and j, iaj; 1  i  n; 1  j  n, and 40 such that X i oX 0i ¼ X i þ   X j   ¼ X 0j oX j and X 0k ¼ X k for k ¼ 1; . . . ; n;

kai;

kaj

Progressive transfers redistribute a small amount of income from a richer to a poorer individual. Definition 3. An inequality concept dT is called transfer-consistent if it satisfies the following condition for all X; X 0 2 Ond , and T l 2 T : If X 0 is derived from X by a progressive transfer then Y 0 ¼ T l ðXÞ can be derived by a(n appropriate) transfer from Y ¼ T l ðXÞ. In this case, the sequence of the transformation T l and the redistribution of income does not play a role. We then get Theorem 3. (Ebert, 2004) Let the inequality concept dT be given. Then dT is transfer-consistent if and only if T d ¼ fT dl jl 2 Rþþ g where T dl ðXÞ ¼ lðX  dÞ þ d

for d 2 R and

T dl ðXÞ

for d ¼ 1

¼ X þ ln l

for all X 2 Od and l 2 Rþþ . Thus, the admissible transformations have to be linear for transferconsistent inequality concepts. These concepts are called coherent. Now some comments can be made. (1) If we have T l ðXÞ ¼ lðX  dÞ þ d for d 2 R, the characteristic function G is also linear: GðXÞ ¼ aðX  dÞ. Obviously, there exists exactly one coherent inequality concept defined on every Od . It is

Inequality Reducing Taxation Reconsidered

139

denoted by

d dT d , where T d ¼ fT dl jT dl ðXÞ ¼ lðX  dÞþ d; l 2 Rþþ g. Then T l ðXÞ ¼ lðX  d1Þ þ d1 for all X 2 Ond , that is, d1 is the point in the ‘‘lower left corner’’ of Ond , just outside the domain. T l translates X at first, then applies a relative transformation (equiproportional scaling), and then translates back. The distribution d1 represents a reference point. We denote the characteristic function2 simply by DðXÞ :¼ X  d for X 2 Od and D1 ðXÞ :¼ X þ d for X 2 O0 ¼ Rþþ The functions T dl are indexed explicitly by d if necessary. For d ¼ 1, we obtain T l ðXÞ ¼ X þ ln l and define DðXÞ :¼ expðXÞ. (2) If do0, we obtain Bossert and Pfingsten’s (1990) intermediate inequality concepts. According to their notation, the domain is equal to Od ¼ f½ð1  yÞ=y; 1g where y 2 ð0; 1 and d ¼ ½ð1  yÞ=y. They consider the transformations T yk ðXÞ ¼ X þ k½yX þ ð1  yÞ1 for k4  1=y. The concept is identical to rel for y ¼ 1 and to abs for y ¼ 0. If y 2 ð0; 1Þ, we obtain d for d ¼ ð1  yÞ=y. It turns out that T yk coincides with T dl 2 T d if l ¼ 1 þ yk. (3) For d40, we get a class of inequality concepts for which the reference point d140 is relevant. d can be interpreted as some basic income that is necessary for surviving. Therefore, only the surplus income X  d is taken into account (cf. Ebert, 2004). In the following sections, we restrict the analysis to the coherent inequality concepts d for d 2 R [ f1g, that is, to the relative, absolute, intermediate, and reference-point inequality view. The definitions presented and the results derived will be related to the literature if possible. Furthermore, if not mentioned explicitly, the citations refer to intermediate inequality. To the best knowledge of the author, there are no papers – apart from Ebert (2004) – dealing with reference-point inequality. Therefore, the corresponding definitions and results are new.

4. LORENZ DOMINANCE Having clarified the concept of inequality we will use, we have to deal with a dominance criterion in the next step. Therefore, we now introduce the corresponding notions of Lorenz dominance for d 2 R [ f1g. Since the characteristic function D allows us to define an isomorphism from ðT d ; Þ to ðT rel ; Þ, the following definitions seem to be obvious in view of Corollary 2.

140

UDO EBERT

It will turn out that they are appropriate. At first, we define a d-Lorenz curve by LCd ð0; XÞ :¼ 0,   i DðX ðkÞ Þ i 1X LCd ; X :¼ n n k¼1 DðmðXÞÞ LCd ðp; XÞ is linear on ½ði=nÞ; ði þ 1Þ=n for i ¼ 1; . . . ; n  1, X 2 Ond , and d 2 R [ f1g. For d ¼ 0, this definition coincides with the usual one; we get the ordinary Lorenz curve LC :¼ LC0 . Whenever d 2 R incomes are translated, that is X ! DðXÞ ¼ X  d and the ordinary concept of Lorenz curve is computed for the vector of translated incomes DðXÞ; that is, we obtain LCd ðp; XÞ ¼ LCðp; DðXÞÞ for X 2 Ond and p 2 ½0; 1 For do0, this definition has already been proposed in a remark by Besley and Preston (1988, p. 162). But in the literature (cf., e.g., Moyes, 1992), a different concept has been used for d  0, namely    i  X ðkÞ  mðXÞ i 1X þy LCy ; X ¼ n n k¼1 ymðXÞ þ ð1  yÞ for X 2 Ond and d ¼ 

1y y

where y 2 ð0; 1 and LCy



i ;X n

 ¼

i 1X ½X ðkÞ  mðXÞ n k¼1

for X 2 Ond and y ¼ 0 It is also identical to LC for y ¼ 1 or d ¼ 0, but differs from LCd for y 2 ð0; 1Þ and do0. But there is a definite relationship between both concepts:     i 1 LCd ðði=nÞ; XÞ y i d i ; X ¼ ð1  dÞLC ;X þ ¼ (1) LC n n n1  d y þ ði=nÞy In other words, LCy is a transform of LCd . The concept of LCd is much simpler than the corresponding LCy since it directly reveals the underlying idea. On the other hand, the transform LCy is attractive if the concept of absolute inequality is investigated. The curves LCy converge pointwise to the

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absolute Lorenz curve as defined by Moyes (1987) if y tends to zero. This is no longer true for the curves LCd . They do not converge to LC1 for d ! 1 since there is no direct functional relationship between LC1 ði=n; XÞ and Moyes’ P Lorenz curve for absolute inequality. We have LC1 ði=n; XÞ ¼ ð1=nÞ ik¼1 exp½X ðkÞ  mðXÞ. Now we are in the position to introduce Definition 4. Let X; Y 2 Ond . Then X d-Lorenz dominates Y, that is, XhdL Y, if and only if     d i d i ; X  LC ;Y for i ¼ 1; . . . ; n  1 LC n n The symmetric part of this relation is denoted by dL , its asymmetric one by dL . For d ¼ 0, hdL is abbreviated by hL . It should be clear that further concepts like generalized Lorenz dominance or ratio dominance (see, e.g., Moyes, 1992) can be extended in the same way. As they are not discussed below, here no definition is presented. Furthermore, the relationship (1) demonstrates that for d ¼ 0, d-Lorenz dominance can equivalently be defined by means of the Lorenz curves LCy . Moreover, Lorenz dominance for d ¼ 1 is equivalent to the dominance criterion for absolute inequality suggested in Moyes (1987). Given this definition, we can establish a number of results. At first, we investigate whether the definition of d-Lorenz dominance is well chosen. We establish3 Proposition 4. Let d 2 R [ f1g. Then (a) For all X; Y 2 Ond : XhdL Y3DðXÞhL DðYÞ (b) For all X; Y 2 Ond : X d Y3X dL Y Part (a) of Proposition 4 reflects the definition of the d-Lorenz curve and of d-Lorenz dominance. It demonstrates again the simple relationship between d and rel . A minimal requirement for the ordering hdL is its compatibility with the inequality concept d and the idea that two income distributions possess the same degree of inequality if one is a transform by T dl of the other one. Thus, we expect that two income distributions having the same degree of inequality should be equivalent as far as d-Lorenz dominance is concerned. Part (b) shows even more: The criterion of d-Lorenz dominance is very sensitive. Whenever two income distributions are not equivalent, as far as

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inequality is concerned, they cannot be equivalent with respect to the Lorenz criterion. Next we consider some implications of the definition of d-Lorenz dominance. They will be used below when taxation is examined. We obtain: Proposition 5. (a) Let H : Od ! Od . Then X dL HðXÞ for all X 2 Ond 3H 2 T d . (b) Let H i : Od ! Od and let H i be nonconstant for i ¼ 1; 2. Then H 1 ðXÞ dL H 2 ðXÞ for all X 2 Ond . 3There is T dl 2 T d such that H 1 ðXÞ ¼ T dl ðH 2 ðXÞÞ for X 2 Od . (c) Let H : Od ! Od . Then [For all X; Y 2 Ond : XhdL Y ) HðXÞhdL HðYÞ3H 2 T d or H is constant. Parts (a) and (b) consider the relationship between dL and symmetric transformations. Part (a) is related to part (b) of Proposition 4. Equivalence with respect to d-Lorenz dominance requires that the transformation is admissible. Part (b) derives an analogous relationship for two transformations. The last part (c) is important for the discussion of tax schedules. d-Lorenz dominance is only preserved if the transformation applied is admissible ðH 2 T d Þ or always maps an income vector into the same distribution. It is a result already proven in Theorem 3.3 of Moyes (1992) and Proposition 4.6 of Ebert and Moyes (2002) for d  0. But it holds for all d 2 R [ f1g. The analysis of this section can be summarized simply. The relation hdL , that is, d-Lorenz dominance, possesses the same properties as hL when interpreted appropriately.

5. TAXATION Next we consider the taxation of income. Given an inequality concept d for d 2 R [ f1g, we examine a net income function F d 2 F ðOd Þ. It assigns to any pretax income X 2 Od the posttax income F d ðXÞ. Since F d ðXÞ 2 Od , negative net incomes are admitted if do0. Monotonicity implies that there are no rank reversals, that is, X  X 0 ) F d ðXÞ  F d ðX 0 Þ for all X; X 0 2 Od . This condition can be interpreted as the property of incentive preservation. For F d , the corresponding tax liability can be derived

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by defining td ðXÞ ¼ X  F d ðXÞ for X 2 Od . Since there is a duality between F d and td (cf. Moyes & Shorrocks, 1998), it is sufficient to investigate net income functions. Corresponding results for tax schedules can then be derived by duality. The sets of net income functions for different d 2 R [ f1g are isomorphic, that is, they are essentially ‘‘identical’’ with F ðO0 Þ since we have F d 2 F ðOd Þ3D  F d  D1 2 F ðO0 Þ and F 0 2 F ðO0 Þ3D1  F 0  D 2 F ðOd Þ There is also a one-to-one correspondence between the respective tax functions for d 2 R: td ðXÞ ¼ t0 ½DðXÞ3F d ¼ D1  F 0  D For the evaluation of taxation, we introduce the usual concepts (cf., e.g., Pfingsten, 1986): Definition 5. F d 2 F ðOd Þ is called d-inequality preserving [reducing] if and only if F d ðXÞ dL X½F d ðXÞhdL X for all X 2 Ond A net income function is d-inequality preserving (reducing) if the distribution of net incomes possesses the same (less) inequality than the distribution of gross incomes (for all income distributions). The implications of taxation are evaluated by means of the criterion of d-Lorenz dominance. Now we are able to establish Proposition 6. Let F d ; F d1 ; F d2 2 F ðOd Þ, and d 2 R [ f1g. Then (a) F d is d-inequality preserving 3F d 2 T d . (b) F d is d-inequality reducing 3DðF d ðXÞÞ=DðXÞ is nonincreasing in X 2 Od . (c) Let F d1 and F d2 be not constant. F d1 ðXÞ dL F d2 ðXÞ for all X 2 Ond 3 There is T dl 2 T d such that F d1 ¼ T dl  F d2 . Part (a) is not surprising given Proposition 5(a). If F d is d-inequality preserving there exists l40 such that F d ðXÞ ¼ lðX  dÞ þ d. Then the corresponding tax function has a particular simple form td ðXÞ ¼ ð1  lÞDðXÞ. Our approach simplifies Pfingsten’s (1986) Theorem 4.2 considerably. But here the factor 1  l can be positive or negative in contrast to Pfingsten’s result. This is a consequence of the way the domains are chosen. Pfingsten considers only strictly positive pre- and post-tax incomes.

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Part (b) demonstrates for d 2 R that d-inequality reduction is equivalent to the fact that the average retention rate – measured for the ‘‘appropriately translated net and gross income’’ DðF d ðXÞÞ and DðXÞ ¼ ðX  dÞ – is nonincreasing in income. This condition is reasonable since d represents the basic income, which is the point of reference. For d ¼ 0, a net income function F d reduces inequality if and only if F d is progressive (in the standard sense) (cf. Jakobsson, 1976; Eichhorn, Funke, & Richter, 1984); for d ¼ 1, the condition requires that F d ðXÞ  X is nonincreasing. The condition for inequality reduction presented here is much simpler than the definition in Pfingsten (1988) for do0. It requires F d ðXÞ  X to be nonincreasing in X yX  1 þ y It is easy to see that this condition is equivalent to part (b):   F d ðXÞ  X 1 DðF d ðXÞÞ ¼ 1 yX  1 þ y y DðXÞ 1y and y 2 ð0; 1 for d ¼  y

ð2Þ

Therefore, the following definition suggests itself: Definition 6. F d 2 F ðOd Þ is called d-progressive if and only if DðF d ðXÞÞ=DðXÞ is nonincreasing in X 2 Od . Accordingly, we define F^ ðOd Þ :¼ fF d 2 F ðOd ÞjDðF d ðXÞÞ=DðXÞ is nonincreasing in X 2 Od g. The set F^ ðOd Þ consists of all d-progressive net income functions. Furthermore, we have a direct implication: F d 2 F ðOd Þ is d-progressive 3D  F d  D1 2 F ðO0 Þ is 0-progressive F 0 2 F ðO0 Þ is 0-progressive 3D1  F 0  D 2 F ðOd Þ is d-progressive. Part (c) of Proposition 6 demonstrates that two net income functions are equivalent as far as inequality reduction is concerned if and only if they satisfy a simple relationship. One must be a T dl -transform of the other. Again part (c) is more general than a corresponding result by Pfingsten’s (1986) Theorem 3.4 for d  0. Moreover, in our framework, the connection between the corresponding tax functions is more transparent. It is described by td1 ðXÞ ¼ ð1  lÞDðXÞ þ ltd2 ðXÞ for X 2 Od for d 2 R. The factor ð1  lÞ can again be negative or positive. td1 is a combination of a proportional tax/transfer and the tax schedule td2 . Summing up, the characterization of a d-inequality reducing net income function F d is straightforward. Since d is the basic income, the translated

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(and corresponding) net income function that assigns DðF d ðXÞÞ to DðXÞ has to be inequality reducing and therefore progressive in the standard sense. Moreover, the usual results can be extended directly.

6. MEASURES OF TAX PROGRESSION Up to now it has been examined under what condition a net income function is d-progressive. The degree of progression for a given income tax has not been investigated. In the following text, measures of progression are introduced for the inequality concepts d where d 2 R [ f1g. For the rest of the chapter, we assume that all net income functions considered are once continuously differentiable, that is, we confine ourselves to F c ðOd Þ and to F^ c ðOd Þ :¼ F^ ðOd Þ \ F c ðOd Þ since we want to employ elasticities. We propose a measure of tax progression. Definition 7. Let F d 2 F c ðOd Þ and d 2 R [ f1g. The elasticity Zd ðF d ; XÞ :¼ ZðD  F d ; DðXÞÞ is called d-residual progression. The way d-residual progression is defined is suggested by earlier definitions. Since F d ¼ D1  F 0  D for F d 2 F c ðOd Þ where F 0 2 F c ðO0 Þ, we obtain Zd ðF d ; XÞ ¼ ZðD  ðD1  F 0  DÞ; DðXÞÞ ¼ Z0 ðF 0 ; YÞjY¼DðXÞ In other words, d-residual progression is measured by the corresponding standard residual progression for the translated net income function and income. The definition of Zd ðF d ; XÞ is simpler than the measure of progression introduced by Pfingsten (1987). He proposed for y 2 ð0; 1Þ and d ¼ ð1  yÞ=yo0 0

0

yX½td ðXÞ  td ðXÞ=X þ ð1  yÞtd ðXÞ P ðt ; XÞ ¼ yX½1  td ðXÞ=X þ ð1  yÞ y

d

when td corresponds to F d . It turns out that Py ðtd ; XÞ ¼ 1  Zd ðF d ; XÞ

(3)

(See the appendix; cf. also Besley and Preston’s (1988) remark on this point and Lambert’s (2001) presentation.)

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Now it is not surprising that the following results hold: Proposition 7. Let F d ; F d1 ; F d2 2 F c ðOd Þ, and d 2 R [ f1g. (a) F d 2 F^ c ðOd Þ3Zd ðF d ; XÞ  1 for all X 2 Od . (b) Zd ðF d ; XÞ ¼  for all X 2 Od 3There is a40 such that F d ðXÞ ¼ D1 ðaDðXÞ Þ for all X 2 Od . d (c) F 1 ðXÞhdL F d2 ðXÞ for all X 2 Ond 3Zd ðF d1 ; XÞ  Zd ðF d2 ; XÞ for all X 2 Od . Part (a) demonstrates that the definition of the d-residual progression Zd is appropriate. F d is d-progressive if and only if Zd ðF d ; XÞ is less than or equal to unity for all incomes. Part (b) considers net income functions possessing constant d-residual progression. For d 2 R, they are given by F d ðXÞ ¼ aðX  dÞ þ d for an a40. For d ¼ 1, we obtain F 1 ðXÞ ¼ g þ X. The result is related to Theorem 3.2 in Moyes (1992). Finally, part (c) is a generalization of a result derived by Jakobsson (1976) for d ¼ 0. A net income function is more d-inequality reducing than another one if and only if its d-residual progression is not greater than the d-residual progression of the other one (cf. also Theorem 2.2 in Moyes, 1992). In summary, we find that the criterion of d-residual progression defined above is a reasonable analogue to the standard residual progression.

7. CONCLUSION The chapter investigates d-Lorenz dominance and d-inequality reduction for the coherent inequality concepts d where d 2 R [ f1g. The characteristics of these concepts are again summarized in Table 1. It has turned out that these inequality concepts are essentially equivalent to the relative inequality concept. The only difference is that the reference point d is different from 0 (whenever da0). The definitions introduced and proposed Table 1.

d ¼0 d_0 d ¼ 1

Summary of Characteristics of the Coherent Inequality Concepts. Od

DðXÞ

T l ðXÞ

Rþþ ðd; 1Þ R

X Xd expðXÞ

lX lðX  dÞ þ d X þ ln l

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are identical with the usual ones – apart from the fact that the reference point differs. Therefore, it is easy to describe the set of d-inequality reducing or d-progressive tax functions. There is a one-to-one mapping onto the respective set of relative inequality measures and, respectively, the set of (in the standard sense) progressive tax functions. Given these relationships, it is not surprising that the results we obtain in Propositions 5–7 are generalizations of the results we know from the usual framework. Nevertheless, the chapter clarifies these relationships and presents simple and transparent definitions and brief and elegant proofs of the (general) results. Furthermore, an investigation of reference-point inequality has not yet been performed before. Finally, a number of results for the intermediate inequality view, which can be found in the literature, are collected and extended, and presented in a unifying framework. Moreover, the results of this chapter demonstrate that the nonstandard inequality concepts considered are closely related and isomorphic to the relative inequality view. Therefore, there should be no difficulties to apply them in practice. The examination has proven that these concepts are not all exotic and also present a feasible and consistent framework for empirical work. The analysis has been performed under the assumption that the individuals considered differ only with respect to income and are identical otherwise. In principle, the framework can be extended. Then things are more complicated since the differences in attributes have to be taken into account. One possibility is to use equivalence scales that reflect the type of household. Ebert and Moyes (2003) investigate the implications of reasonable conditions for the definition of Lorenz dominance in a heterogeneous framework. In Ebert and Moyes (2002), welfare and inequality are examined when households are heterogeneous. The corresponding problem of inequality reducing taxation is examined in Ebert and Moyes (2000). Finally, Ebert and Lambert (2004) define measures of progression in the extended model. In all these cases, various concepts of inequality are admitted. It is well known that criteria like Lorenz dominance are incomplete. They do not allow us to compare two income distributions if the corresponding Lorenz curves intersect. In this case, it is helpful to employ summary measures of inequality. In the literature, many inequality measures are considered. An inequality measure is in general only invariant with respect to exactly one type of transformation. Relative measures are not changed if all incomes are changed in the same proportion. Absolute measures are not altered if each income is changed by the same amount. One can similarly derive measures that are invariant with respect to the other coherent inequality concepts. Nevertheless, in some cases, it is possible to define

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inequality measures, which are invariant with respect to various inequality concepts, by a slight variation. For example, the Gini coefficient is a relative measure: P ð1=n2 Þ ni¼1 ðn þ 1  2iÞX ½i  GðXÞ ¼ mðXÞ where X ½  is generated from X by permuting the components such that incomes are nonincreasing. The measure can be interpreted as the relative welfare loss evaluated by means of the Gini welfare function. If we renormalize this welfare loss, we get an analogue to the Gini coefficient by P ð1=n2 Þ ni¼1 ðn þ 1  2iÞX ½i  GðX; dÞ ¼ mðXÞ  d and, respectively,  X n 1 GðX; 1Þ ¼ ðn þ 1  2iÞX ½i  n2 i¼1 for d 2 R [ f1g. Thus, in this situation, the renormalization guarantees that the measure is invariant with respect to the inequality concept d (cf. Ebert, 1997). The chapter has confined itself to an investigation of inequality reducing taxation. But it should be emphasized that everything shown and proven for d-inequality reducing tax functions can also be derived for d-inequality increasing tax functions. Then in the Definition 5, the inequality sign has to be reversed ½F d ðXÞ"dL X, that is, the posttax income distribution has to possess more inequality than the pretax income distribution. Similarly, a net income schedule F d is called d-regressive if and only if DðF d ðXÞÞ=DðXÞ is nondecreasing in income X (Definition 6). In this case, the elasticity representing d-residual progression has to be (weakly) greater than unity. Thus, we observe the same kind of symmetry we know from the standard framework (cf. Lambert, 2001).

NOTES 1. In Ebert (2004), the relation dT is called a path-independent inequality concept. Since in this chapter only this type of concept is considered, the attribute is dropped.

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2. Since the factor a is not relevant, we set a ¼ 1. The notation D should remind the reader of d. 3. The proofs of all propositions and of Eqs. (1)–(3) have been collected in the appendix.

ACKNOWLEDGMENT I thank Patrick Moyes and an anonymous referee for helpful comments and suggestions.

REFERENCES Amiel, Y., & Cowell, F. A. (1992). Measurement of income inequality. Experimental test by questionnaire. Journal of Public Economics, 47, 3–26. Arnold, B. C. (1990). The Lorenz order and the effects of taxation policies. Bulletin of Economic Research, 42, 249–265. Besley, T. J., & Preston, I. P. (1988). Invariance and the axiomatics of income tax progression: A comment. Bulletin of Economic Research, 40, 159–163. Bossert, W., & Pfingsten, A. (1990). Intermediate inequality: Concepts, indices, and welfare implications. Mathematical Social Sciences, 19, 117–134. Chakravarty, S. R. (2009). Deprivation, inequality and welfare. Japanese Economic Review, 60, 172–190. del Rio, C., & Ruiz-Castillo, J. (2000). Intermediate inequality and welfare. Social Choice and Welfare, 17, 223–239. Ebert, U. (1997). Linear inequality concepts and social welfare. DARP Discussion Paper 33. London School of Economics, London. Ebert, U. (2004). Coherent inequality views: Linear invariant measures reconsidered. Mathematical Social Sciences, 47, 1–20. Ebert, U., & Lambert, P. J. (2004). Horizontal equity when equivalence scales are not constant. Public Finance Review, 32, 426–440. Ebert, U., & Moyes, P. (2000). Consistent income tax structures when households are heterogeneous. Journal of Economic Theory, 90, 116–150. Ebert, U., & Moyes, P. (2002). Welfare, inequality and the transformation of incomes. The case of weighted income distributions. In: P. Moyes, C. Seidl, & A. F. Shorrocks (Eds), Inequalities: Theory, measurement and applications. Journal of Economics, Suppl. 9, pp. 9–50. Ebert, U., & Moyes, P. (2003). Equivalence scales reconsidered. Econometrica, 71, 319–343. Eichhorn, W., Funke, H., & Richter, W. F. (1984). Tax progression and inequality of income distribution. Journal of Mathematical Economics, 13, 127–131. Jakobsson, U. (1976). On the measurement of the degree of progression. Journal of Public Economics, 5, 161–168. Kolm, S.-C. (1976). Unequal inequalities I. Journal of Economic Theory, 12, 416–442. Lambert, P. J. (2001). The distribution and redistribution of income (3rd ed.). Manchester: Manchester University Press.

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Lang, S. (1968). Algebra. Reading, MA: Addison-Wesley. Moyes, P. (1987). A new concept of Lorenz domination. Economics Letters, 23, 203–207. Moyes, P. (1992). The through-time redistributive effect of income taxation: The intermediate inequality view. Mathematical Social Sciences, 24, 59–71. Moyes, P., & Shorrocks, A. F. (1998). The impossibility of a progressive tax structure. Journal of Public Economics, 69, 49–65. Pfingsten, A. (1986). Distributionally-neutral tax changes for different inequality concepts. Journal of Public Economics, 30, 385–393. Pfingsten, A. (1987). Axiomatically based local measures of tax progression. Bulletin of Economic Research, 39, 211–223. Pfingsten, A. (1988). Progressive taxation and redistributive taxation: Different labels for the same product? Social Choice and Welfare, 5, 235–246. Pfingsten, A., & Seidl, C. (1997). Ray invariant inequality measures. In: S. Zandvikili (Ed.), Taxation and inequality, research on economic inequality (Vol. 7, pp. 107–129). Oxford: JAI Press. Yoshida, T. (2005). Social welfare rankings of income distributions – a new parametric concept of intermediate inequality. Social Choice and Welfare, 24, 557–574. Zoli, C. (1998). A surplus sharing approach to the measurement of inequality. Mimeo, University of York and University of Pavia, York.

APPENDIX Proof of Eq. (1). Observe that y ¼ 1=ð1  dÞ40. Then X ðkÞ  mðXÞ X ðkÞ  mðXÞ 1 þy¼ þ ymðXÞ þ ð1  yÞ mðXÞ=ð1  dÞ þ 1  1=ð1  dÞ 1  d X ðkÞ  d mðXÞ  d 1 þ  ð1  dÞ ¼ ð1  dÞ mðXÞ  d 1  d mðXÞ  d DðX ðkÞ Þ 1 þ ¼ ð1  dÞ DðmðXÞÞ 1  d & Proof of Proposition 4. (a) Obvious (b) X T d Y3There is T dl 2 T d such that Y ¼ T dl ðXÞ 3There is Sl 2 T rel such that Y ¼ D1  S l  DðXÞ 3There is Sl 2 T rel such that DðYÞ ¼ S l ½DðXÞ 3DðXÞ L DðYÞ & Proof of Proposition 5. (a) See part (b) for H 1 ðXÞ ¼ X and H 2 ðXÞ ¼ HðXÞ

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(b) H 1 ðXÞ dL H 2 ðXÞ for all X 2 Ond 3D½H 1 ðXÞ L D½H 2 ðXÞ for all X 2 Ond 3There is l 2 Rþþ such that DðH 1 ðXÞÞ ¼ S l ðDðH 2 ðXÞÞÞ for all X 2 Ond 3H 1 ðXÞ ¼ D1  S l  DðH 2 ðXÞÞ for all X 2 Ond 3H 1 ðXÞ ¼ T dl ½H 2 ðXÞ for all X 2 Od . (c) ½XhdL Y ) HðXÞhdL HðYÞ for all X; Y 2 Ond  3½DðXÞhL DðYÞ ) DðHðXÞÞhL DðHðYÞÞ for all X; Y 2 Ond  3½DðXÞhL DðYÞ ) D  H  D1 ðDðXÞÞhL D  H  D1 ðDðYÞÞ for all X; Y 2 Ond  3½XhL Y ) ðD  H  D1 ÞðXÞhL ðD  H  D1 ÞðYÞ for all X; Y 2 On0  3There is l 2 Rþþ such that D  H  D1 ¼ S l or H is constant (cf. Theorem 3.1 in Arnold, 1990). 3H ¼ D1  S l  D ¼ T dl 2 T d or H is constant. & Proof of Eq. (2).   F d ðXÞ  X 1 F d ðXÞ  X 1 F d ðXÞ  d X  d ¼ ¼  yX  1 þ y y X  ð1  yÞ=y y X d X d   d 1 DðF ðXÞÞ 1 ¼ y DðXÞ Proof of Proposition 6. (a) See Proposition 5(a). (b) F d ðXÞhdL X for all X 2 Ond 3DðF d ðXÞÞhL DðXÞ for all X 2 Ond 3DðF d ðXÞÞ=DðXÞ is nonincreasing in X (see Jakobsson, 1976). & (c) See Proposition 5(b). Proof of Proposition 7. Let F d 2 F c ðOd Þ: There is F 0 2 F c ðO0 Þ such that F d ¼ D1  F 0  D. Then 0 (a) F d 2 F^ c ðOd Þ3F 0 2 F^ c ðOd Þ3Z0 ðF 0 ; YÞ  1 for all Y 2 O0 . 3Zd ðF d ; XÞ  1 for all X 2 Od . d (b) Z ðF d ; XÞ ¼  for all X 2 Od 3Z0 ðF 0 ; YÞ ¼  for all Y 2 O0 . 3There is a40 such that F 0 ðYÞ ¼ aY  for all Y 2 O0 . 3There is a40 such that F d ðXÞ ¼ D1 ðaDðXÞ Þ for all X 2 Od (See Jakobsson, 1976). (c) Analogous, cf. Jakobsson (1976). &

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Proof of Eq. (3). 0

0

yXðtd ðXÞ  td ðXÞ=XÞ þ ð1  yÞtd ðXÞ P ðt ; XÞ ¼ yXð1  td ðXÞ=XÞ þ ð1  yÞ y

d

0

0

td ðXÞX  td ðXÞ þ ½ð1  yÞ=ytd ðXÞ ¼ X  td ðXÞ þ ð1  yÞ=y 0

¼

X  td ðXÞ  X þ td ðXÞðX  dÞ X  td ðXÞ þ d 0

F d ðXÞ  d þ ðd  XÞ þ td ðXÞðX  dÞ F d ðXÞ  d 0 ð1  td ðXÞÞðX  dÞ ¼1 F d ðXÞ  d DðXÞ 0 ¼ 1  F d ðXÞ DðF d ðXÞÞ DðXÞ 0 ¼ 1  D0 ðF d ðXÞÞF d ðXÞ DðF d ðXÞÞ

¼

¼ 1  ZðF d ; XÞ

CHAPTER 7 COUNTING POVERTY ORDERINGS AND DEPRIVATION CURVES Ma Casilda Lasso de la Vega ABSTRACT Purpose – A counting approach based on the number of deprivations suffered by the poor is quite an appropriate framework to measure multidimensional poverty with ordinal or categorical data. A method to identify the poor and a number of poverty indices have been proposed to take this framework into account. The implementation of this methodology involves the choice of a minimum number of deprivations required in order for an individual to be identified as poor. This cutoff and the choice of a poverty measure to aggregate the data are two sources of arbitrariness in poverty comparisons. The aim of this chapter is twofold. We first explore properties that characterize an identification method which allows different weights for different dimensions. Then the chapter examines dominance conditions in order to guarantee unanimous poverty rankings in a counting framework. Design/methodology/approach – In the unidimensional poverty field, one branch of the literature is devoted to establishing dominance criteria that guarantee unanimous orderings at a variety of poverty thresholds and indices. This chapter takes this literature as a starting point, and investigates circumstances in which these ordering conditions may be applied in a weighted counting framework. Studies in Applied Welfare Analysis: Papers from the Third ECINEQ Meeting Research on Economic Inequality, Volume 18, 153–172 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2010)0000018010

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Findings – Necessary and sufficient conditions are obtained that guarantee that two vectors, which represent the weighted sum of the deprivations felt by each person, may be unanimously ranked regardless of the identification cutoff and of the poverty measure. Originality/value – Since most of the data available for measuring capabilities or dimensions of poverty is either ordinal or categorical, the counting approach provides an alternative framework that suits these types of variables. The implementation of the ordering conditions derived in this chapter is based on simple graphical devices that we call dimension deprivation curves. These curves become a useful way to check the robustness of poverty rankings to changes in the identification cutoff. They also provide a tool for determining nonambiguous poverty rankings in a wide set of multidimensional poverty indices that suit ordinal and categorical data.

1. INTRODUCTION In recent years there has been considerable agreement that poverty is a multidimensional phenomenon and great efforts have been made from both a theoretical and an empirical point of view, trying to assess multidimensional poverty.1 Following Sen (1976), poverty measurement should consist of a method to identify the poor and an aggregative measure. In a multidimensional framework, the identification of the poor usually incorporates two cutoffs. The first has to do with the traditional identification of the poor within each dimension using a dimension-specific poverty line. In the second step, a minimum number of deprived dimensions is required to be considered as a poor person. Thus a person is identified as poor if deprived in at least a given number of dimensions. The two extreme cases are referred to as the ‘‘union’’ and ‘‘intersection’’ approaches, respectively. Whereas the union procedure identifies someone who is deprived in at least one dimension as poor, the intersection definition requires a poor person to be deprived in all dimensions. Nevertheless, how to identify the multidimensional poor is still a debatable issue and it may be of interest to explore properties to be fulfilled in this step. Allowing different weights for different dimensions, Section 2 explores properties of the second stage of the identification procedure. Once the poor people are identified, the multidimensional headcount ratio, which is the percentage of poor people in the society, may be gauged.

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As regards the aggregation step, the majority of the indices introduced for the measurement of multidimensional poverty behave well only with cardinal variables, that is, with dimensions that are quantitative in nature. However, most of the data available to measure capabilities or dimensions of poverty are either ordinal or categorical. Consequently, only indices that deal well with qualitative variables should be used in empirical applications when this sort of data are involved. What Atkinson (2003) refers to as a counting approach focuses on the number of dimensions in which each person is deprived, and is an appropriate procedure that deals well with ordinal and categorical variables. Among others, Chakravarty and D’Ambrosio (2006), Bossert, D’Ambrosio, and Peragine (2007), Alkire and Foster (2007), and Bossert, Chakravarty, and D’Ambrosio (2009) propose indices based on a counting approach. Specifically, Chakravarty and D’Ambrosio (2006) introduce the possibility that the dimensions may be weighted differently and derive counting measures in the social exclusion field. We also assume a weighted counting framework. In turn, Alkire and Foster (2007) propose what they call the adjusted headcount ratio, defined as the average of the number of deprivations suffered by the poor. The multidimensional headcount ratio and the adjusted headcount ratio, appropriately modified to incorporate dimensional weights, will play an important role in this work. In general, the choice of either the identification cutoffs or the indices adds arbitrariness to poverty comparisons, and different selections can lead to contradictory results. For this reason it may be of interest to investigate conditions to guarantee that comparisons be unanimous to the different choices. There exists a branch of the literature devoted to establishing dominance criteria that provide unanimous orderings when comparisons are made at a variety of poverty thresholds and measures. Zheng (2000) provides a comprehensive survey of dominance conditions in the poverty unidimensional field. In this chapter, we take this literature as a starting point, and more specifically the papers by Shorrocks (1983), Foster (1985), and Foster and Shorrocks (1988a, 1988b). In particular, we investigate circumstances in which two vectors, which represent the number of weighted deprivations felt by each person, may be unanimously ranked regardless of the identification cutoff and of the poverty measure. In Section 3, we will show that if the ranking provided by the multidimensional headcount ratio is unambiguous over all admissible identification thresholds, then agreement is guaranteed over all counting poverty measures that satisfy the dimensional monotonicity property.2 A similar result is obtained with respect to the (weighted) adjusted headcount ratio: rankings provided by

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this latter index are equivalent to agreement over all counting measures that fulfill monotonicity and distribution sensitivity. These results are by no means surprising. Atkinson (1987) derives a similar conclusion with regard to the headcount ratio in the unidimensional poverty field. In turn, Foster and Shorrocks (1988a, 1988b) characterize the poverty orderings obtained from the Foster–Greer–Thobercke measures (Foster, Greer, & Thorbecke (1984)). The implementation of these conditions is based on two different types of curves that we call dimension deprivation curves, introduced in Section 3. The first one, which we call the FD curve, represents the multidimensional headcount ratio for all the admissible dimension cutoffs.3 The second type of curve, henceforth SD curves, represents in the same picture the headcount ratio, the adjusted headcount ratio, and the average deprivation share according to the proposal by Alkire and Foster (2007), appropriately modified to incorporate dimensional weights. Since the Lorenz curve was introduced in the literature, a number of cumulative curves have been widely used to check unanimous orderings in the inequality, poverty, and polarization fields.4 In this connection, we will show that the curves proposed in this chapter become a powerful tool for checking unanimous orderings according to a wide class of counting measures. They also avoid the choice of an arbitrary identification cutoff and offer a useful way to determine the bounds of the number of dimensions for which multidimensional comparisons are robust. As the multidimensional headcount ratio and the adjusted headcount ratio behave particularly well with ordinal and categorical data, the dimension deprivation curves play a key role in making poverty comparisons when data are ordinal. The chapter finishes with some concluding remarks.

2. A WEIGHTED COUNTING POVERTY APPROACH 2.1. Notation and Basic Definitions We consider a population of n  2 individuals endowed with a bundle of k  2 attributes considered as relevant to poverty measurement. The number of dimensions is given and fixed. Regarding the identification of the poor through the specification of a poverty line, let us consider zj 40 to be the minimum quantity of the jth attribute for a subsistence level. An individual i is deprived as regards attribute j if xij ozj .

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In a counting approach, poverty is measured by taking into consideration the number of dimensions in which people are deprived. Thus we assume that the dimensions are represented by binary variables and characteristics of individual i are identified by a deprivation vector xi 2 f0; 1gk , whose typical component j is defined by xij ¼ 1 when individual i is deprived in attribute j and xij ¼ 0 otherwise. Allowing one dimension to be more important than another, let w 2 Rkþþ be a vector of weights summing to k, whose jth component, wj is the weight assigned to attribute j. We assume that the vector of weights is given and fixed. sum of the dimensions in which person i Let us denote by di the weighted P is deprived, that is, d i ¼ 1jd wj xij , which represents the poverty score of individual i.5 Let D ½0; k be the set of all admissible scores. In general, D is a discrete set of [0, k] containing the value 0, which corresponds to a person nondeprived in any dimension, and the value k, when the person is deprived in all the dimensions. When all weights are equal to 1 (i.e., all dimensions are assumed to be equally important), D ¼ f0; 1; . . . ; kg. By contrast, if the weights are all different, then D is a discrete set with 2k elements. The vector d ¼ ðd 1 ; . . . ; d n Þ 2 Dn is referred to as the vector of weighted deprivation counts. This vector plays an important role in poverty measurement when ordinal data are involved. In fact this vector is invariant if the achievement levels and the poverty lines are transformed under the same monotonic transformations, and this is a crucial property when the achievements or capabilities are measured with ordinal variables. We will denote by d the permutation of d in which individuals’ poverty scores have been arranged in decreasing order, that is, di  diþ1 for i ¼ 1; . . . ; n. Hence people are ranked from the most deprived to the least. Let G ¼ [n1 Dn be the set of all admissible vectors of deprivation counts. We will say that the vector d u is obtained from the vector d by a 0  by a replication if d 0 ¼ ðd; d; . . . ; dÞ; by an increment permutation if d ¼ d; 0 (decrement) if d i 4d i (d 0i od i ) for some individual i, and d 0j ¼ d j for all jai. 2.2. The Identification of the Poor The first step in measuring poverty is to identify the poor people. Two main methods have been used in this stage in a multidimensional setting, referred to as the ‘‘union’’ and the ‘‘intersection’’ approaches, respectively.

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Whereas the union procedure identifies as poor someone who is deprived in at least one dimension, the intersection definition requires a poor person to be deprived in all dimensions. These methods present well-known drawbacks when the number of poverty dimensions is great. Whereas ‘‘almost nobody’’ is identified as poor with the intersection approach, ‘‘almost everybody’’ is poor with the union identification. There is an intermediate procedure that proposes establishing a cutoff in the number of dimensions. If a weighted sum of deprivations is assigned to each person, this score may be used in the identification step. Thus, a person is identified as poor if the number of weighted dimensions in which they are deprived is at least m, that is, d i  m, with 0om  k. Person i is nonpoor otherwise, that is, if d i om.6 For m ¼ min1jk wj , this method coincides with the union approach, whereas for m ¼ k, it is equivalent to the intersection approach. We will use rm to denote this procedure. In this framework the identification function is assumed to be the same for all individuals. The rm method is simple and intuitive, and it may be worth examining the conditions that lead to a rm identification method in a multidimensional setting. For doing so, first of all we assume that the function that identifies the poor satisfies a property of dichotomization. This is a strong property that makes sense in a counting approach since it assumes that identifying a person as poor depends only on each individual’s deprivation vector. This property is formalized as follows: Dichotomization: An identification procedure r is a dichotomized identification function if r : f0; 1gk ! f0; 1g links xi, the vector of deprivations of individual i, with an indicator variable such that rðxi Þ ¼ 1 if person i is identified as poor and rðxi Þ ¼ 0 if person i is not poor. In addition, we introduce a property for a dichotomized identification function. We think that a reasonable assumption is to require that if a person is considered as poor according to an identification method, then any other person deprived in equal or more weighted dimensions should also be considered as poor. We call this property Poverty Consistency and it is formulated as follows: Poverty Consistency. Let r be a dichotomized identification function. We say that r satisfies the poverty consistency property if given P a person i 0 Þ ¼ 1 for all person iu such that d i ¼ Þ ¼ 1 then rðx with rðx i i 1jd wj xij  P d i0 ¼ 1jd wj xi0 j .

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The following proposition characterizes the rm identification method. Proposition 1. A nontrivial dichotomized identification function r fulfils the poverty consistency property if and only if there exists some m 2 ð0; k P such that rðxi Þ ¼ 1 if d i ¼ 1jd wj xij  m and rðxi Þ ¼ 0 otherwise. Proof. Since r is a nontrivial identification method, there exists a person i such that rðxi Þ ¼ 1. Let m ¼ mini0 ¼1; ... ;n fd i0 =rðxi Þ ¼ 1g. For definition rðxi0 Þ ¼ 0 if d i0 om and for poverty consistency, rðxi0 Þ ¼ 1 if d i0  m. The sufficiency of the proof is clear. Q.E.D. As a consequence of this proposition, the only dichotomized identification method that is poverty consistent is rm for some m 2 ð0; k. Throughout this chapter the poor are supposed to be identified according to a rm procedure. Let us denote by Qm and qm, respectively, the set and number of poor identified using the dimension cutoff m. For each vector d of weighted deprivation counts, we define the censored vector of deprivation counts, denoted by d(m), as follows: d i ðmÞ ¼ d i if d i  m, and d i ðmÞ ¼ 0 if d i om.

2.3. Aggregating Deprivations with a Counting Measure The second step in poverty measurement is the aggregation of the poverty scores of the poor people. In what follows, a counting poverty measure P is a nonconstant function whose typical image, denoted by Pm ðdÞ, represents the level of poverty in a society with a vector of weighted deprivation counts d and where the poor are identified according to a rm procedure. The following four properties are the counterparts for a counting measure of the basic properties assumed in the poverty field. First of all, since poverty measurement is concerned with poor people, it is usually demanded that a poverty index should not change under the improvements of the nonpoor people. In a counting approach, improvements are reflected in a decrease in the number of the weighted deprivations. Then, the poverty focus property may be formulated as follows. Poverty Focus (PF): For any m 2 ð0; k, Pm remains unchanged if the poverty score of a nonpoor person decreases. It may be worth noting that PF ensures that Pm ðdÞ ¼ Pm ½dðmÞ. In order to narrow down the shape of the measure, it may be interesting to identify some types of transformations that seem to have an effect on the poverty level, and to require the index to be consistent with them.

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If one believes that all the individuals and all the dimensions are essential in measuring poverty, then it appears intuitive to demand that if the poverty score of any poor individual decreases, then the overall poverty should decrease. Note that the decrement in the poverty score of a poor person may lift them out of poverty.7 This property may be is formulated as follows: Dimensional Monotonicity (MON): For any m 2 ð0; k, Pm ðd 0 ÞoPm ðdÞ if d 0i od i for some individual i with d i  m, and d 0j ¼ d j for all jai. According to Sen (1976), a poverty measure should be sensitive to distribution among the poor and greater weight should be attached to the poorer person. Consequently, a decrease in poverty due to a decrease in the poverty score of a poor person should be greater the higher the score of the person is. Let us consider two poor individuals, i and j, such that the poverty score of individual i is higher than that of individual j, that is, d i 4d j . Let us assume that it is possible to decrease the two scores by the same amount. MON ensures that the poverty level decreases under the two transformations. Nevertheless, the next axiom goes beyond MON and demands that the decrease in poverty under the former decrement (that of the poverty score of the poorer), should be higher than that under the latter. As in MON, the two individuals involved in the transformation are allowed to lift out of poverty.8 This axiom may be formalized as follows. Distribution Sensitivity (DS): For any m 2 ð0; k and h40: Pm ðdÞ  Pm ðd 1 ; . . . ; d i  h; . . . ; d j ; . . . ; d n Þ4 Pm ðdÞ  Pm ðd 1 ; . . . ; d i ; . . . ; d j  h; . . . ; d n Þ if ðd 1 ; . . . ; d i  h; . . . ; d j ; . . . ; d n Þ; ðd 1 ; . . . ; d i ; . . . ; d j  h; . . . ; d n Þ 2 Dn d i 4d j  m.

and

The following property establishes that no other characteristic apart from the number of weighted dimensions in which a person is deprived matters in defining a counting poverty index. This principle is much stronger than its counterpart in the unidimensional field since it implicitly entails a trade-off between the dimensions. For instance, when all the dimensions are weighted equally this property implies that it does not matter in which particular dimensions people are deprived and, somehow, all of them become interchangeable. A similar conclusion may be obtained if the weights are different. 0  Symmetry (SYM): For all m 2 ð0; k, Pm ðdÞ ¼ Pm ðd 0 Þ if d ¼ d.

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Finally, the following condition allows the comparisons of populations of different sizes. Replication Invariance (RI): For all m 2 ð0; k, Pm ðdÞ ¼ Pm ðd 0 Þ if d 0 ¼ ðd; d; . . . ; dÞ. We define the following two inclusive classes of counting poverty measures: P1 ¼ fP counting poverty measure=P satisfies PF; MON; SYM and RIg P2 ¼ fP counting poverty measure=P satisfies PF; MON; DS; SYM and RIg Clearly P2 P1 , and as will be shown, the inclusion is strict. The first counting poverty measure introduced in the literature is the multidimensional headcount ratio, denoted by H, which is the percentage of poor people in the society. In other words, for each m 2 ð0; k identification cutoff, H m ¼ qm =n is the percentage of the population whose poverty scores are higher than or equal to m. This index, usually used to measure the incidence of poverty, is unable to capture the intensity, is not distribution sensitive and violates MON, that is, it does not change if a person already identified as poor becomes deprived in an additional dimension. The (weighted) adjusted headcount ratio, M, is defined as the ratio of the number of weighted deprivations suffered by the poor P to the total number of weighted deprivations, that is, M m ðdÞ ¼ ½ 1in d i ðmÞ=nk. In contrast to the headcount ratio this index satisfies MON, although it does not belong to class P2 since it violates DS. More information about poverty can be incorporated by using the weighted average deprivation share across the poor denoted by A, introduced by Alkire and Foster (2007) when all the dimensions are equally weighted. This is defined as the mean among the poor, of the P weighted sum of the deprivations suffered by the poor, that is, Am ðdÞ ¼ ½ 1in d i ðmÞ=qm k. This index captures the intensity of poverty. Furthermore, it holds that M m ðdÞ ¼ H m Am ðdÞ.

3. COUNTING POVERTY ORDERINGS This section is concerned about how to rank two vectors of weighted deprivation counts in order to evaluate whether poverty is higher in one society than in another. Poverty rankings may be reversed depending on the identification threshold, or on the measure selected. Thus, in order to avoid

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contradictory results, poverty orderings require unanimous rankings for a set of identification cutoffs, or a class of poverty measures. As it is impossible to check unanimity for infinite pairwise comparisons, ordering conditions are derived to characterize unanimous agreement. Following the literature, given a counting poverty measure, P, we define the partial ordering with respect to P, denoted by "P , in the set of vectors of deprivation counts, by the rule9 d 0 "P d if and only if Pm ðd 0 Þ  Pm ðdÞ for all m 2 ð0; k In this section, we will examine the partial poverty orderings with respect to the multidimensional headcount ratio, H, and the weighted adjusted headcount ratio, M. 3.1. Poverty Ordering with Respect to the Multidimensional Headcount Ratio, H, and the FD Curve As already mentioned, given a cutoff of the number of dimensions, m, the multidimensional headcount ratio, H m , gauges the percentage of poor people according to the identification procedure rm . For any vector of deprivation counts it is possible to consider the graph of H as a function of this dimension cutoff, ranked in decreasing order. We will refer to this curve as the FD curve associated with the vector d, and its ordinates are computed as follows: FDðd; pÞ ¼ H kp ; p 2 ½0; k The following example helps to clarify this. Let us consider a society of 10 individuals endowed with 4 attributes weighted equally. Let us assume that the vector of deprivation counts is d ¼ ð4; 3; 3; 2; 2; 1; 1; 1; 0; 0Þ. The FD curve for this vector is displayed in Fig. 1. Some interesting properties of this curve may be mentioned. The FD curve is an increasing step function that is right-continuous. The horizontal axis displays the identification cutoffs ranked in decreasing order, and in the vertical axis, by definition, the multidimensional headcount ratio, Hm, is recovered. Two limiting curves correspond with the extreme situations: if nobody is deprived, the curve coincides with the horizontal axis; whereas, if everybody is deprived in all dimensions, the curve becomes the parallel line to the horizontal axis through the point (0,1). It is clear from the graph that, for two vectors of deprivation counts with the same population size, d; d 0 2 Dn , if d 0 "H d, that is, if H m ðd 0 Þ  H m ðdÞ for all m 2 ð0; k, then the FD curve of d must be below or to the left of the

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H1

H2

H3

H4 p=0 m=4

Fig. 1.

p=1 m=3

p=2 m=2

p=3 m=1

p=4

Plotting the Identification Cutoffs and the Headcount Ratio: The FD Curve.

FD curve of d u. And the reverse is also true: if the FD curve of d is below or to the left of the FD curve of du then H m ðd 0 Þ  H m ðdÞ for all m 2 ð0; k, and d 0 "H d. We get the following proposition. Proposition 2. For any d; d 0 2 Dn vectors of deprivation counts, the following statements are equivalent: (i) (ii) (iii) (iv) (v)

FDðd; pÞ  FDðd 0 ; pÞ for all p 2 ½0; k; H m ðdÞ  H m ðd 0 Þ for all m 2 ð0; k; 0 di  di for all i ¼ 1; . . . ; n; 0  d may be obtained P P fromd0 by a finite sequence of increments; 0  1in jðd i Þ  1in jðd i Þ for all continuous, increasing functions j : ½0; k!R

Proof. In the appendix. This proposition shows that when the FD curve of a vector of deprivation counts lies above or to the right of the curve of other with the same population size, or equivalently, when these two vectors can be ordered with respect to H, then one may be obtained from the other by a finite sequence of permutations and/or increments. Consequently, any poverty measure belonging to class P1 will rank these two vectors exactly in the same way. Moreover, since both H and the measures belonging to class P1 are replication invariant, the result also holds for vectors with different population sizes.

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The reverse is also true. PIn fact, consider the following class of counting measures: Pðd; mÞ ¼ 1=nf 1in c½d i ðmÞg, with c : ½0; k ! R, a continuous strictly increasing convex function. It is quite simple to show that P function j : ½0; k ! R belongs to class P1. Given any continuous increasing P and 40, then the measures P ðd; mÞ ¼ 1=n½ 1in ðc þ jÞðd i ðmÞÞ also belong to class P1. Consequently, given two vectors d and du with P ðd; mÞ  P ðd 0 ; mÞ, when  ! 0 we get statement (v) in Proposition 2 and have the following result, that links the ordering with respect to H with first-degree stochastic dominance: Proposition 3. For any d; d 0 2 G vectors of weighted deprivation counts: FDðd 0 ; pÞ  FDðd; pÞ for all p 2 ½0; k if and only if Pm ðd 0 Þ  Pm ðdÞ for all P 2 P1 and for all identification cutoff m 2 ð0; k. This proposition reveals that, although H fails to satisfy MON, the ordering with respect to H is equivalent to agreement over all counting measures satisfying MON. Consequently, if the FD curves of two vectors of deprivation counts do not intersect, then all poverty counting measures satisfying MON will lead to the same verdict. By contrast, when the curves intersect, there are two possibilities in order to obtain unanimous ranking: either the set of measures is restricted, as shown in Section 2.2, or the admissible cutoffs are limited, as will be developed in Section 2.3.

3.2. Poverty Ordering with Respect to the Adjusted Headcount Ratio, M, and the SD Curve One interesting feature of the FD curve introduced in the previous section is that, given a vector d and a dimension threshold m, it is straightforward to prove that the area beneath the curve of the censored vector, FD½dðmÞ, is equal to d M m . Thus, even if a conclusive poverty verdict could not be reached with the H ordering, it would be possible to get unanimous rankings with respect to M. As usual, we propose constructing the SD curve, for any vector d, plotting the headcount ratio against the adjusted headcount ratio, that is, pairs of points ðH m ; M m Þ. We also plot two extreme points ð0; 0Þ as the start of the curve, and ð1; M 1 Þ, as the end of the curve. Then we join the dots. Fig. 2 shows the SD curve associated with the vector d in the previous example.

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A1 M1 M2 M3 M4 0 0

H4

H3

H2

H1

1

cumulative population share

Fig. 2.

Plotting the Headcount Ratio and the Adjusted Headcount Ratio: The SD Curve.

It may be worth noting that for any vector of deprivation counts d, ranked from the highest poverty score to the lowest, the SD curve can equivalently be defined in the following way: for each integer p ¼ 0; . . . ; n  1 the ordinate of the curve is computed as the cumulative of the sum of poverty scores of the first p people divided by the total number of deprivations that could possibly be experienced by all people. At intermediate points the curve is determined by linear interpolation. Thus, the ordinates of the SD curve are computed as follows: SDðd; 0Þ ¼ 0  p 1 X d i ; p ¼ 1; . . . ; n ¼ SD d; n nk 1ip   pþy 1 X ¼ SD d; ðd i þ yd pþ1 Þ; p ¼ 0; . . . ; n  1; y 2 ½0; 1 n nk 1ip The ordinates of this curve are replication invariant, and are also invariant to permutations of d. The graph, as displayed in Fig. 2, begins at the origin, and is a continuous nondecreasing concave function.

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There are two boundaries that correspond to the extreme situations of minimum and maximum deprivation. If nobody is deprived, the curve coincides with the horizontal axis. By contrast, if everybody is deprived in all dimensions, the curve becomes the diagonal line. By construction, on the vertical axis the curve shows the percentage of the weighted sum of deprivations experienced by the percentage of the most deprived population, which is displayed on the horizontal axis. Thus, the maximum value of the curve corresponds to the percentage of the deprivations experienced by all the people, that is, the adjusted headcount ratio according to the union procedure. The point at which the curve becomes horizontal yields the percentage of people deprived in at least one dimension: the headcount ratio according to the union procedure. Less obvious, perhaps, is the meaning of each point at which the slope of the curve changes. It should be noted that when people with the same poverty scores are accumulated, the slope does not change. By contrast, adding a person whose poverty score is less than the existing ones makes the slope decrease. Thus, each of the points at which the slope changes, yields the percentage of people deprived in at least, say, m weighted dimensions, or H m . The vertical axis on the other hand displays, by definition, the adjusted headcount ratio M m . For instance, the first point at which the slope changes shows the percentage of deprived dimensions suffered by the population deprived in all dimensions. In other words, according to the intersection procedure, the headcount ratio (on the horizontal axis) and the adjusted headcount ratio (on the vertical axis) are recovered at this point. The weighted average deprivation share across the poor, Am , is also represented in the graph by the slope of the ray from (0,0) to ½p; SDðpÞ. The following proposition is based on the results established by Marshall and Olkin (1979, Propositions 4.A.2 and A.B.2) for vectors with the same number of components: Proposition 4. For any d; d 0 2 Dn vectors of deprivation counts, the following statements are equivalent: SDðd 0 ; pÞ  SDðd; pÞ for all p 2 ½0; 1; 0 M m ðdÞ  MP m ðd Þ for all m 2 ð0; k; P  0 1ip d i  1ip d i for all p ¼ 1; . . . ; n; d u may be obtained from d by a finite sequence of permutations, increments and/or transformations of the form TðzÞ ¼ ðz1 ; . . . ; zi þ ; zn Þ with h40 and zi  zj ; h; P. . . ; zj h; . . .P 0 jð d Þ  (v) i 1in 1in jðd i Þ for all continuous, increasing and convex functions j : ½0; k ! R.

(i) (ii) (iii) (iv)

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Proof. In the appendix. An implication of this proposition is the result below. Proposition 5. For any d; d 0 2 Dn vectors of deprivation counts and for any measure P 2 P2 , if SDðd 0 ; pÞ  SDðd; pÞ for all p 2 ½0; 1 then Pm ðd 0 Þ  Pm ðdÞ for all m 2 ð0; k. Proof. In the appendix. Consequently, when the SD curve of a vector du lies above the curve of another, d, with the same population size, any poverty measure belonging to class P2 will rank these two vectors in exactly the same way. In addition, as both the deprivation curves and the measures P 2 P2 are invariant under replication, the result also holds for vectors with different population sizes. The reverse is also true and the proof is completely similar to the corresponding result in the previous section. So we get: Proposition 6. For any d; d 0 2 G vectors of deprivation counts: SDðd 0 ; pÞ  SDðd; pÞ for all p 2 ½0; 1 if and only if Pm ðd 0 Þ  Pm ðdÞ for all P 2 P2 and for all identification cutoff m 2 ð0; k. Then, this result reveals that although M, the dimension adjusted headcount ratio, violates DS, if two vectors of deprivation counts can be unanimously ranked by M m at all dimension cutoffs, then all poverty counting measures satisfying DS will rank societies in the same way.

3.3. Poverty Ordering when the Curves Intersect When the dimension deprivation curves introduced in the two previous sections intersect, it is still possible to establish dominance conditions by restricting the set of identification cutoffs. In fact, even if the curves of two vectors cross, there exists a threshold m 2 ð0; k that corresponds with the identification cutoff after which the intersection occurs. In other words, m ensures that the curves do not intersect for all m 2 ðm ; k, which becomes the relevant set for the cutoffs. A simple way to establish dominance conditions in these cases is to base comparisons on the censored vectors, and to modify the results derived in the previous sections accordingly. Taking into consideration the respective censored vectors, denoted by dðm Þ and d 0 ðm Þ, we get the following proposition.

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Proposition 7. For any d; d 0 2 G vectors of deprivation counts: (i) FDðd 0 ðm Þ; pÞ  FDðdðm Þ; pÞ for all p 2 ½0; k if and only if Pm ðd 0 Þ  Pm ðdÞ for all P 2 P1 and for all identification cutoff m 2 ðm ; k. (ii) SD½d 0 ðm Þ; p  SD½dðm Þ; p for all p 2 ½0; 1 if and only if Pm ðd 0 Þ  Pm ðdÞ for all P 2 P2 and for all identification cutoff m 2 ðm ; k. The implication of this proposition is that, even when the dimension deprivation curves intersect, they allow us to obtain robust conclusions in a wide set of counting measures restricting the set of identification cutoffs. Since not all the admissible cutoffs are equally meaningful in poverty measurement, this result may be quite useful in empirical applications: when two deprivation vectors cannot be unanimous ranked for all cutoffs, concentrating on the poorest people can lead to conclusive verdict.

4. CONCLUDING REMARKS A counting approach that concentrates on the number of dimensions in which each person is deprived is an appropriate procedure to measure multidimensional poverty with ordinal and categorical variables. The choice of a cutoff to identify the poor, and a poverty measure to aggregate the data are two sources of arbitrariness and different selections may lead to contradictory conclusions. In this chapter we have characterized the identification procedure and have derived dominance conditions in order to obtain unanimous rankings in a wide set of counting measures, and a set of identification cutoffs. The implementation of these conditions is based on two different types of dimension deprivation curves, which guarantee unanimous rankings of vectors of deprivation counts when they do not intersect. And, even if the curves cross, additional results are derived that lead to conclusive verdicts by restricting the admissible cutoffs in the identification of the poor. Thus, these curves become a useful way to determine the boundaries of the number of dimensions for which counting poverty comparisons are robust and have been shown to play a key role in making poverty comparisons when the data are ordinal. Policy makers should choose the dimension-specific poverty line and the weight attached to any dimension.

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NOTES 1. A comprehensive survey on multidimensional poverty can be found in Chakravarty (2009). 2. This property demands that the measure should decrease if the number of dimensions, in which a poor person is deprived, decreases. 3. This curve is quite similar to the deprivation distribution profile proposed by Subramanian (2009). However, two main differences can be pointed out. On the one hand, we propose to represent this cumulative curve as a step function that is rightcontinuous. On the other hand, to our knowledge, S. Subramanian does not derive dominance conditions in his paper. 4. Among them the TIP curves proposed by Jenkins and Lambert (1997), the poverty curves in Foster and Shorrocks (1988b), the polarization curve introduced by Foster and Wolfson (2010), and the proposal of Shorrocks (2009) to derive unemployment indices. 5. Bossert et al. (2009) characterize this first stage of aggregation of the characteristics of each individual. 6. Assuming that all the dimensions are equally weighted, this intermediate identification method is followed by Mack and Lansley (1985), Gordon, Nandy, Pantazis, Pemberton, and Townsend (2003), and Alkire and Foster (2007) among others. 7. This is the counterpart of what Donaldson and Weymark (1986) refers to as the strong monotonicity axiom in the unidimensional poverty field. Zheng (1997) discusses different types of monotonicity axioms and their relationships. 8. This principle has the same spirit as the Transfer Sensitivity Axiom introduced by Sen (1976) as long as progressive transfers among the poverty scores make sense. A discussion about the relationship between these axioms may be found in Zheng (1997). 9. We follow Atkinson (1987) and adopt the weak definition of a partial ordering. Although not all the results derived in this chapter hold for the other two levels (the semistrict and the strict ones), similar conditions could be also obtained in these two cases.

ACKNOWLEDGMENTS The author is very grateful to Professor Bhuong Zheng whose suggestions greatly contributed to improving this chapter. She also acknowledges the hospitality of the Oxford Poverty and Human Initiative Centre (OPHI) of the University of Oxford where she was visiting when the first version of this chapter was prepared. Preliminary versions were presented in the OPHI seminar and in the LGS6 meeting (Japan, 2009). The author wants to thank Sabina Alkire, Armando Barrientos, James Foster, Miguel Nin˜o Zarazu´a, Masood Awan, Maria Emma Santos, Xiaolin Wang, and the rest of the participants in these events for their comments, and her colleague

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Ana Urrutia for her help and support. This research has been partially supported by the Spanish Ministerio de Educacio´n y Ciencia under project SEJ2009-11213, cofunded by FEDER and by Basque Government under the project GIC07/146-IT-377-07.

REFERENCES Alkire, S., & Foster, J. (2007). Counting and multidimensional poverty measurement. OPHI Working Paper no. 2007-7. Available at http://www.ophi.org.uk/pubs/OPHI_WP7.pdf Atkinson, A. B. (1987). On the measurement of poverty. Econometrica, 55, 244–263. Atkinson, A. B. (2003). Multidimensional deprivation: Contrasting social welfare and counting approaches. Journal of Economic Inequality, 1, 51–65. Bossert, W., Chakravarty, S., & D’Ambrosio, C. (2009). Multidimensional poverty and material deprivation. ECINEQ Working Paper no. 2009-129. Available at www.ecineq.org/ milano/WP/ECINEQ2009-129.pdf Bossert, W., D’Ambrosio, C., & Peragine, V. (2007). Deprivation and social exclusion. Economica, 74, 777–803. Chakravarty, S. R. (2009). Inequality, polarization and poverty: Advances in distributional analysis. In: J. Silber (Ed.), Economic studies in inequality, social exclusion and well-being (Vol. 6). New York: Springer. Chakravarty, S. R., & D’Ambrosio, C. (2006). The measurement of social exclusion. Review of Income and Wealth, 523, 377–398. Donaldson, D., & Weymark, J. A. (1986). Properties of fixed-population poverty indices. International Economic Review, 27, 667–688. Foster, J. E. (1985). Inequality measurement. In: H. Peyton Young (Ed.), Fair Allocation. Providence, RI: American Mathematical Society. Foster, J. E., Greer, J., & Thorbecke, E. (1984). A class of decomposable poverty measures. Econometrica, 52, 761–766. Foster, J. E., & Shorrocks, A. (1988a). Poverty orderings. Econometrica, 56, 173–177. Foster, J. E., & Shorrocks, A. (1988b). Poverty orderings and welfare dominance. Social Choice and Welfare, 5, 179–198. Foster, J. E., Wolfson, M. C. (2010). Polarization and decline of the middle class: Canada and the U.S. Journal of Economic Inequality, 8, 241–245. Gordon, G., Nandy, S., Pantazis, C., Pemberton, S., & Townsend, P. (2003). Child poverty in the development world. Bristol, UK: The Policy Press. Jenkins, S., & Lambert, P. (1997). Three ‘I’s of poverty curves, with an analysis of UK poverty trends. Oxford Economic Papers, 49, 317–327. Mack, J., & Lansley, S. (1985). Poor Britain. London: George Allen and Unwin Ltd. Marshall, A. W., & Olkin, I. (1979). Inequalities: Theory of majorization and its applications. New York: Academic. Sen, A. K. (1976). Poverty: An ordinal approach to measurement. Econometrica, 44, 219–231. Shorrocks, A. (1983). Ranking income distributions. Economica, 50, 3–17. Shorrocks, A. (2009). On the measurement of unemployment. Journal of Economic Inequality, 7(3), 311–327.

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Subramanian, S. (2009). The deprivation distribution profile: A graphical device for comparing alternative regimes of multidimensional poverty. Discussion Paper, 12. Madras Institute of Development Studies. Available at www.mids.ac.in/mids_ds.htm Zheng, B. (1997). Aggregate poverty measures. Journal of Economic Surveys, 11(2), 123–162. Zheng, B. (2000). Poverty orderings. Journal of Economic Surveys, 14(4), 427–466.

APPENDIX Proof of Proposition 2. The equivalence between (i) and (ii) follows from the definition of FD curve. To prove that (ii) implies (iii), let us suppose 0 that there exists i 2 f1; . . . ; ng such that dj  dj for all j ¼ 1; . . . ; i  1 and 0 di 4di . Taking m ¼ d i , we find that H m ðdÞ4H m ðd 0 Þ. That (iii) implies 0 0 (ii) is clear. After (iii), writing di ¼ di þ ðdi  di Þ we have (iv). That (iv) implies (v) is straightforward. If (v) holds, since the function j1 ðzÞ ¼ 0 0 maxfz  d1 ; 0g is continuous and increasing we find that d1  d1 . Let 0 us suppose that there exists i 2 f1; . . . ; ng such that dj  dj for all j ¼ 0 1; . . . ; i  1 and di 4di . If we consider the continuous increasing function

j ðzÞ ¼

8 > > < > > :

1 ðz 

0 di Þ=ðdi

0



0 di Þ

if z  di if d0 i ozodi ; 0 if z  di

P P    0 we find that 1in j ðd i Þ  i4 1in j ðd i Þ ¼ i  1. Then (v) implies (iii) and the proof is complete. Q.E.D. To prove Proposition 4 we use the following result established by Marshall and Olkin (1979, propositions 4.A.2 and A.B.2): Lemma 1. For any d; d 0 2 Rnþ the following conditions are equivalent: P P  0 (i) 1ip d i  1ip d i for all p ¼ 1; . . . ; n; (ii) d may be obtained from du by successive applications of a finite number of T transforms of the form T 1 ðzÞ ¼ T 1 ½z1 ; . . . ; zi1 ; lzi þ ð1  lÞzj ; ziþ1 ; . . . ; zj1 ; lzj þ ð1  lÞzi ; zjþ1 ; . . . ; zn  where 0  l  1; and/or of the form T 2 ðzÞ ¼ T 2 ðz1 ; . . . ; zi1 ; azi ; ziþ1 ; . . . ; zn Þ, where 0 P P ao1;  0 (iii) 1in jðd i Þ  1in jðd i Þ for all continuous, increasing and convex functions j : ½0; k ! R.

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Proof of Proposition 4. From the definitions of SD curve and the weighted adjusted headcount ratio, it is clear that (i), (ii), and (iii) are equivalent. From Lemma 1, (iii) is also equivalent to (v). Moreover, according to (ii) in Lemma 1, under the same hypothesis, d may be obtained from du by d ¼ T 2 ðd 0 Þ, that is, by a decrement; and/or by d ¼ T 1 ðd 0 Þ. Note that for l ¼ 0, T1 reduces to a permutation. For the rest of values of l, as permutations are allowed, we may assume, without loss of generality that zi 4zj and l 2 ½1=2; 1. Defining h ¼ ð1  lÞðzi  zj Þ40, T1 may rewritten as T 1 ðzÞ ¼ T 1 ðz1 ; . . . ; zi  h; . . . ; zj þ h; . . . ; zn Þ, with zi  h  zj þ h, and T1 is the inverse of the T transformation of (iv). Q.E.D. Proof of Proposition 5. From Proposition 4 (iv) holds. Then it is enough to prove that if d 0 ¼ ðd 1 ; . . . ; d i þ h; . . . ; d j  h; . . . ; d n Þ with h40 and d i  d j then for any m 2 ð0; k, Pm ðd 0 Þ  Pm ðdÞ. If d j  m, Pm ðd 0 Þ4Pm ðdÞ since Pm satisfies DS. If d i þ h  m4d j , Pm ðd 0 Þ4Pm ðdÞ by MON. Otherwise, Pm ðd 0 Þ ¼ Pm ðdÞ by PF. Q.E.D.

CHAPTER 8 TESTING FOR MOBILITY DOMINANCE Ye´le´ Maweki Batana and Jean-Yves Duclos ABSTRACT This chapter proposes tests for stochastic dominance in mobility based on the empirical likelihood ratio. Two views of mobility are considered, either based on measures of absolute mobility or based on transition matrices. First-order and second-order dominance conditions in mobility are first derived, followed by the derivation of statistical inferences techniques to test a null hypothesis of nondominance against an alternative of mobility dominance. An empirical analysis, based on the US Panel Study of Income Dynamics (PSID), is performed by comparing four income mobility periods ranging from 1970 to 1990.

1. INTRODUCTION Two sources of income mobility are usually considered in the literature. The first arises from a temporal reallocation of incomes across individuals, and the second comes from changes in total income (Fields & Ok, 1996). Ever since Prais (1955), a variety of mobility measures have been proposed to capture the effect of these on the income distribution. These include relatively elementary measures based on statistics such as the coefficient of Studies in Applied Welfare Analysis: Papers from the Third ECINEQ Meeting Research on Economic Inequality, Volume 18, 173–195 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2010)0000018011

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correlation between individuals at different dates (see, for instance, Atkinson, Bourguignon, & Morrisson, 1992; Peters, 1992; Bjorklund & Jantti, 1997; Chadwick & Solon, 2002; or Corak, 2006) as well as more elaborate measures based on transition matrices and other measures of dynamic processes. Some of the studies have adopted a relative concept, treating mobility as a reranking in which individuals change position (Shorrocks, 1978a; Bartholomew, 1996). Others have treated mobility as an absolute concept in which any change in individuals’ incomes from the initial situation is equivalent to greater mobility (Fields & Ok, 1996; also see D’Agostino & Dardanoni, 2005 for a discussion). A further group of studies is based on specific interpretations of mobility, notably reflecting its implications for social welfare. Shorrocks (1978b) and Maasoumi and Zandvakili (1986), for instance, measure mobility as the ratio of income inequality averaged over several years relative to mean inequality in each year; see also Chakravarty, Dutta, and Weymark (1985), Atkinson (1983), Markandya (1984), Conlisk (1989), and Dardanoni (1993). Although a number of studies have addressed the measurement of mobility, relatively few have examined how and whether normatively robust comparisons of mobility can be made across countries or time periods – see Atkinson (1983), Conlisk (1989), Dardanoni (1993), Mitra and Ok (1998), Benabou and Ok (2001), and Formby, Smith, and Zheng (2003) for some of the more important exceptions. In addition, aside from Fields, Leary, and Ok (2002) who explicitly integrate the cumulative density function to derive conditions for stochastic dominance, and Formby, Smith, and Zheng (2004) who perform comparisons between two vectors of mobility measures (using composite hypotheses), most studies tend to focus on comparing estimates of mobility indices without implementing explicit statistical testing procedures. This chapter tries to fill some of this gap by demonstrating the use of empirical likelihood ratio procedures to test for the existence of stochastic dominance in mobility. Two types of mobility measures are retained: absolute measures and those based on transition matrices. These measures have the advantage of being relatively easy to reconcile with tools such as the cumulative density function, a feature which is convenient for the application of the empirical likelihood methods recently proposed by Davidson and Duclos (2006). The remainder of this chapter is organized as follows. Section 2 presents mobility measures and a theoretical framework for their partial ranking. Here we expand on several examples of the two types of measures retained

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as well as on conditions for stochastic dominance in mobility. Section 3 describes methods of statistical inference for ranking mobility based on empirical likelihood ratios and resampling procedures. Section 4 provides an illustration of the methods to robust comparisons of mobility across time in the United States. The final section concludes.

2. MOBILITY MEASURES AND STOCHASTIC DOMINANCE The concept of income mobility captures the extent to which the income distribution changes over time across a given set of individuals. Let an initial distribution of income be denoted by x ¼ ðx1 ; x2 ; . . . ; xi ; . . . ; xn Þ 2 Rnþ where xi is the income of the ith individual in a population of size n. Let yi be the income of individual i at a later date (or the income of a descendent of i if we were to think of intergenerational mobility), such that a new income distribution y ¼ ðy1 ; y2 ; . . . ; yi ; . . . ; yn Þ 2 Rnþ is generated by the transformation process (x-y). A mobility measure can be defined as a continuous function M : Rnþ  Rnþ ! R. If two other distributions xu and yu are characterized by the transformation process (xu-yu), we say that process (x-y) is more mobile than the process (xu-yu) if M (x, y)ZM(xu, yu). Several such measures of mobility exist. This chapter considers two classes of them. They reflect two sorts of mobility movements, to wit, structural (or growth) mobility and exchange (or lateral) mobility. Structural mobility increases with a general fall or rise in incomes, whereas exchange mobility captures changes in positions between individuals.1 Measures of absolute mobility fall into the first class, whereas transition matrices are part of the second class.

2.1. Measures of Absolute Mobility Fields and Ok (1996) propose measuring absolute income mobility using individual-specific temporal distances between incomes. A generic measure they propose is defined as: !1=a n X a jxi  yi j 8 x; y 2 Rnþ ; and a40 (1) M d ðx; yÞ ¼ i¼1

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More generally, let Di ¼ jxi  yi j, with i ¼ 1, y , n. Di helps assess the contribution of individual i to total mobility. Consider a class of functions Md ¼ Md (D) ¼ MD(D1, y , Di, y , Dn). We can draw on Willig (1981), Shorrocks (1983), and Foster and Shorrocks (1988) in the context of welfare measurement to establish conditions for dominance in such types of mobility measures. First, analogous to the property of monotonicity (let us refer to it as HA) of welfare functions (conferred by the Pareto principle), we can postulate that a rise in Di must, ceteris paribus, entail an increase in mobility. Second, we can postulate a symmetry assumption (HB) that imposes that Md (D) ¼ Md(CD), where C is a permutation matrix. This says that a reranking of the mobility contributions of individuals leaves total mobility unchanged. Let Mþ d represent the class of mobility functions that satisfy assumptions HA and HB. Let D and Du be two profiles of individual mobility contributions with cumulative distribution functions (cdf) given by F(z) and G(z), respectively. Let QF (p) and QG (p) be the corresponding quantile functions, given by the inverse of the cdfs. By analogy with the welfare dominance conditions shown inter alia in Duclos and Araar (2006), Theorem 1 shows conditions for first-order dominance in mobility. Theorem 1. (First-order mobility dominance) The following conditions are equivalent: (i) M d ðDÞ  M d ðD0 Þ 8 M d 2 Mþ d (ii) FðzÞ  GðzÞ 8 z 2 ½0; þ1½ (iii) QF ðpÞ  QG ðpÞ 8 p 2 ½0; 1. Second-order dominance conditions are obtained when an assumption then be the HC of Schur-concavity of the function Md is added. Let Mþþ d class of mobility functions that are consistent with assumptions HA, HB, and HC. Define F 2(z) and G2(z) as: Z z Z z F 2 ðzÞ ¼ FðtÞdt and G2 ðzÞ ¼ GðtÞdt (2) 0

0

Also define the generalized Lorenz curves GLF (p) and GLG (p) as: Z p Z p QF ðqÞdq and GLG ðpÞ ¼ QG ðqÞdq GLF ðpÞ ¼ 0

(3)

0

with pA[0,1]. Theorem 2 shows conditions for second-order dominance in mobility.

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Testing for Mobility Dominance

Theorem 2. The following conditions are equivalent: (i) M d ðDÞ  M d ðD0 Þ 8 M d 2 Mþþ d (ii) F 2 ðZÞ  G2 ðZÞ 8 Z 2 ½0; þ1½ (iii) GLF ðpÞ  GLG ðpÞ 8 p 2 ½0; 1. For the class of mobility functions proposed by Fields and Ok (1996), the Schur-concavity assumption is verifiable for ar1. Mitra and Ok (1998), for their part, focus on a class of Schur-convex functions by setting aZ1. They develop an approach to partial rankings in which (x-y) is more mobile than (xu-yu) if: n X

Dai ðx; yÞ 

n X

i¼1

Dai ðx0 ; y0 Þ 8 a  1

(4)

i¼1

Mitra and Ok (1998) then derive another, more easily testable, condition for a partial ranking. Arranging D and Du in descending order j, (x-y) is more mobile than (xu-yu) if and only if: s X j¼1

Dj ðx; yÞ 

s X

Dj ðx0 ; y0 Þ 8 s ¼ 1; . . . ; n

(5)

j¼1

This is an alternative approach to that of the generalized Lorenz dominance one in Theorem 2, where the Dis are arranged in increasing order. Here, we weigh the most mobile individuals more heavily, which is compatible with the Schur-convexity assumption implied by the choice of aZ1. If we let ar1, we have Schur-concavity, and an ‘‘egalitarian mobility transfer’’ is desirable. In this case, the usual generalized Lorenz approach is preferred.

2.2. Measures Based on Transition Matrices 2.2.1. The Transition Matrix For simplicity, let income dynamics be described by a discrete Markov process with Z states. Let P ¼ [Pkl] be a nonsingular (Z  Z) transition matrix (i.e., P d is strictly positive for sufficiently large values of the integer d ), such that there exists a vector of steady-state probabilities p that uniquely solves the equation pt ¼ ptP (t designates the transpose). The element Pkl represents the probability that an individual initially in state k will end up in state l.

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178

Let us return to the process (x-y), where x ¼ (x1,x2, y , xi, y, xn) and y ¼ (y1,y2, y , yi, y , yn). Assume that xi and yi are bounded above by  respectively. Let Ok be one of the Z the strictly positive values x and y,  representing a class of rank k, and Gl be partitions of the interval ½0; x,  representing a class of rank l. one of the Z partitions of the interval ½0; y, DefineP pi as the probability associated with each pair (xi, yi), such n that i¼1 pi ¼ 1. The expression for the conditional probability Pkl is given by: Pn Prð½xi 2 Ok  \ ½yi 2 Gl Þ p Iðx 2 Ok ; yi 2 Gl Þ Pni i ¼ i¼1 (6) Pkl ¼ Prðxi 2 Ok Þ i¼1 pi Iðxi 2 Ok Þ where Pr stands for probability and I(  ) is an indicator function taking the value one when the argument is true and zero otherwise. We also observe the following marginal probabilities: pk ¼

n X

pi Iðxi 2 Ok Þ and pl ¼

i¼1

n X

pi Iðyi 2 Gl Þ

(7)

i¼1

The vectors of these marginal probabilities are represented, respectively, by p(k) and p(l), with p(k) ¼ p(l) ¼ p at the steady state. 2.2.2. Mobility and Social Welfare The dominance conditions derived by Atkinson (1983) allow mobility to be ranked for a class of welfare functions satisfying a certain set of properties. Let us consider the case of two periods – specifically, a process (x-y). Let F(x, y) be the bivariate cdf and F(x) and F(y) be the marginal cdf. A social welfare function defined over the two time periods can be defined as: Z x Z y Uðx; yÞdFðx; yÞ dxdy (8) W¼ 0

0

Let G(x, y) represent the bivariate cdf of another process (xu-yu). G(x) and G(y) are the marginal cdf, and assume for now that F(x) ¼ G(x) and F(y) ¼ G(y). If a mobility process is described by the transition matrix P, we have pt ðlÞ ¼ pt ðkÞP or pl ¼

Z X k¼1

Pkl pk 8 l ¼ 1; . . . ; Z

(9)

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where p(k) and p(l) are two (Z  1) vectors. Let xk and yl be incomes of rank k and l, respectively. Expected welfare in Eq. (8) becomes: W¼

Z X Z X

Uðxk ; yl ÞPkl pk

(10)

k¼1 l¼1

A distribution F stochastically dominates a distribution G if the expected utility level F generates is at least as high as that for G for all utility functions in some class of U. A first-order class of U is made of the U functions for which UxW0, UyW0 and Uxyr0. Atkinson (1983) shows that, for identical marginal distributions [i.e., F(x) ¼ G(x) and F(y) ¼ G(y)], the process (x-y) first-order dominates the process (xu-yu) if and only if: Fðxk ; yl Þ  Gðxk ; yl Þ 8 k; l ¼ 1; . . . ; Z

(11)

It is also possible to derive a second-order dominance condition that is weaker than the above first-order condition. A distribution F second-order dominates a distribution G if the expected utility level it generates is at least as high for all utility functions U such that UxW0, UyW0, Uxyr0, Uxyo0, Uxxy, UxyyZ0, and Uxxyyr0. This is equivalent to F 2 ðxk ; yl Þ  G2 ðxk ; yl Þ 8 k; l ¼ 1; . . . ; Z

(12)

where 2

Z

x

Z

y 2

Z

x

Z

Fðs; tÞdtds and G ðx; yÞ ¼

F ðx; yÞ ¼ 0

0

y

Gðs; tÞdtds 0

(13)

0

A difference between p(k) and p(l) would allow incorporation of structural (or growth) mobility. Normalizing incomes such as to equalize the marginal distributions, i.e., p(k) ¼ p(l) ¼ p, we can isolate pure mobility. This is what is done in the illustration below – see also Dardanoni (1993) and Formby et al. (2003) for alternative approaches to this exercise.

3. METHODS OF STATISTICAL INFERENCE The tests for dominance in mobility derived in this section extend the use of the empirical likelihood ratio statistics developed in Davidson and Duclos (2006). They are based on an intersection–union approach that makes it possible to test directly for strict dominance of one distribution over another. For measures of absolute mobility, the dominance

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180

condition pertains to a univariate distribution, and the Davidson and Duclos method is thus directly applicable. A comparison of measures based on transition matrices requires an extension to the two-dimensional case; this has been suggested and partly investigated by Batana and Duclos (2008). 3.1. Inference with Measures of Absolute Mobility Let two distributions of individual mobility D and Du derive from the process (x-y) and (xu-yu). Also let the distributions of D and Du be denoted by F(Z) and G(Z), respectively, with sample analogues (with sample sizes equal to NF and NG, respectively) given by: NG NF 1 X 1 X ^ ^ FðZÞ ¼ IðDs  ZÞ and GðZÞ ¼ IðD0h  ZÞ N F s¼1 N G h¼1

(14)

The problem of maximizing the unconstrained empirical likelihood function (ELF) is as follows: max

NF X

pFs ;pG h s¼1

log pFs þ

NG X

log pG h subject to

h¼1

NF X

pFs ¼ 1;

s¼1

NG X

pG h ¼ 1

(15)

h¼1

0 where pFs and pG h represent, respectively, the probabilities of Ds and Dh F G occurring. The solution to this problem yields ps ¼ 1=N F and ph ¼ 1=N G , so that the maximized value of the unconstrained likelihood is:

d UC ¼ N F log N F  N G log N G ELF

(16)

Strict dominance in mobility of G by F, i.e., stochastic dominance in mobility of (xu-yu) by (x-y), implies that G(Z)WF(Z) for all Z. This condition is violated if there exists a point Z for which G(Z)rF(Z). To test the null hypothesis of nondominance of G by F, the natural null to be tested is therefore H 0 : GðZÞ4FðZÞ for atleast one Z

(17)

against the alternative hypothesis H 0 : GðZÞ4FðZÞ for all Z

(18)

The condition of dominance of Du by D in the samples is met when ^ ^ FðZÞ  GðZÞ8Z. When that condition holds, the ELF maximization problem (15) can then be recast to impose a nondominance constraint.

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Testing for Mobility Dominance

This constraint is given by: NG X

0 pG h IðDh  ZÞ 

NF X

pFs IðDs  ZÞ

(19)

s¼1

h¼1

for some Z. Maximization of Eq. (15) subject to Eq. (19) gives the d C ðZÞ, and leads to the following constrained empirical likelihood ELF expressions: pG h ¼

IðD0h  ZÞ ½1  IðD0h  ZÞ þ y f

(20)

pFs ¼

IðDs  ZÞ ½1  IðDs  ZÞ þ N y N f

(21)

where y¼

N  N G ðZÞ N  M G ðZÞ and f ¼ N G ðZÞ þ N F ðZÞ M G ðZÞ þ M F ðZÞ

N G ðZÞ ¼

NG X

IðD0h  ZÞ and N F ðZÞ ¼

NF X

IðDs  ZÞ

s¼1

h¼1

M G ðZÞ ¼ N G  N G ðZÞ; M F ðZÞ ¼ N F  N F ðZÞ; and N ¼ N G þ N F The statistic LR(Z) is given by the difference between the value of the d UC Þ and the value of the maximum unconstrained maximum likelihood ðELF d likelihood ½ELFC ðZÞ constrained at Z. For values of Z for which we have ^ ^ in the samples that GðZÞ  FðZÞ, the constraint is not binding and, ^ ^ consequently, LR(Z) is nil. When GðZÞ4 FðZÞ for some Z, the constraint d UC Þ, so that LR(Z) d becomes binding and ðELFC Þ is then less than ðELF assumes a strictly positive real value: ( LRðZÞ ¼

0

^ ^ if GðZÞ  FðZÞ

40

otherwise

) (22)

YE´LE´ MAWEKI BATANA AND JEAN-YVES DUCLOS

182

LR(Z) is then given by: 9 8 N log N  N G log N G  N F log N F > > > > > > > > > > þN ðZÞ log N ðZÞ þ N ðZÞ log N ðZÞ G G F F > > = < LRðZÞ ¼ 2 þM G ðZÞ log M G ðZÞ þ M F ðZÞ log M F ðZÞ > > > > > ½N G ðZÞ þ N F ðZÞ log½N G ðZÞ þ N F ðZÞ > > > > > > ; : ½M ðZÞ þ M ðZÞ log½M ðZÞ þ M ðZÞ > G

F

G

(23)

F

The final test statistic LR for testing nondominance is obtained by minimizing LR(Z) over all values of Z: LR ¼ min LRðZÞ Z

(24)

One could use the methods of Davidson and Duclos (2006) to show that LR follows asymptotically a w2 distribution under the null of nondominance. Finite sample refinements can, however, be profitably obtained by bootstrapping the value of LR. This is done by computing the statistics LRb(Z) and LRb for each of b ¼ 1, y , B (B is set to 399 in the illustration below) bootstrap samples of both distributions simultaneously. These bootstrap samples are generated with the probabilities given by Eq. (29), where Z ¼ arg min LRðZÞ Z

(25)

The p-value of the bootstrap test is then given by the proportion of the statistics LRb that is greater than LR. To test for second-order dominance, we consider condition (iii) from Theorem 2. This is dominance in terms of generalized Lorenz curves. Unlike in the case of first-order dominance, we here reject dominance of G by F if GLF (p)rGLG(p) for some p. The constraint (19) becomes: NF X s¼1

pFs Ds IðDs  Z^ F ðpÞÞ ¼

NG X h¼1

0 0 ^ pG h Dh IðDh  Z G ðpÞÞ

(26)

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Testing for Mobility Dominance

where Z^ F ðpÞ and Z^ G ðpÞ correspond to sample quantiles at percentile p. The Lagrangian, L, is given by: L¼

X

log

s

m

pFs

( X

þ

X

log

pG h

þ lF 1 

X

! pFs

þ lG 1 

s

h

pFs Ds I s ½Z^ F ðpÞ 

X

s

X

! pG h

h

)

0 ^ pG h Dh I h ½Z G ðpÞ

ð27Þ

h

where lF, lG, and m 2 R are the Lagrange multipliers. These equations cannot be solved analytically. However, starting from the first-order conditions, we can derive numerical procedures to find a solution. The solutions use lF þ lG ¼ N F þ N G ¼ N

pG h ¼

1 N  lF 

mD0h I h ½Z^ G ðpÞ

and pFs ¼

(28)

1 lF þ mDs I s ½Z^ F ðpÞ

(29)

^ and solve for l^ F and m: ^ ¼ arg min  ðl^ F ; mÞ lF ;m2R



X

X

logflF þ mDs I s ½Z^ F ðpÞg

s

logfN  lF  mD0h I h ½Z^ G ðpÞg

ð30Þ

h

Once estimated, l^ F and m^ allow calculation of the probabilities and the likelihood ratio for each pair ½Z^ F ðpÞ; Z^ G ðpÞ. However, Z^ F ðpÞ and Z^ G ðpÞ are endogenous since both depend respectively on pFs and pG h . Thus, from some initial pair ½Z^ F ðpÞ0 ; Z^ G ðpÞ0 , the estimated probabilities are used to compute a new pair ½Z^ F ðpÞ1 ; Z^ G ðpÞ1 . The solution to problem (30) is then iterated and new probabilities are reestimated by using the expressions in (29). The process is stopped at the ith iteration when the differences between ½Z^ F ðpÞi1 ; Z^ G ðpÞi1  and ½Z^ F ðpÞi ; Z^ G ðpÞi  become numerically insignificant. Both the generalized Lorenz and the Mitra and Ok (1998) approaches are considered for such a test in the illustration below.

YE´LE´ MAWEKI BATANA AND JEAN-YVES DUCLOS

184

3.2. Inference with Transition Matrices Consider PK PL a transition matrix P ¼ [Pkl]. Consider also the expression k¼1 l¼1 pk Pkl . If we replace Pkl and pk by their expressions from Eqs. (6) and (7), we obtain: K X L X

pk Pkl ¼

k¼1 l¼1

K X L X n X

pi Iðxi 2 Ok;yi 2 Gl Þ 8 K; L ¼ 1; 2; . . . ; Z

(31)

k¼1 l¼1 i¼1

Denote Ok ¼ [Xk1, Xk] and Gl ¼ [Yl1, Yl], with O1 ¼ ½0; X 1 ;  and G1 ¼ [0, Y1], and GZ ¼ ½Y Z1 ; y.  We set each Ok and OZ ¼ ½X Z21 ; x, Gl to contain the same sample proportions. Xk and Yl (k, l ¼ 1,y, Z) are then the Z quantiles retained for each dimension. Eq. (31) can be rewritten as: K X L X k¼1 l¼1

pk Pkl ¼

n X

pi Iðxi  X k;yi  Y l Þ 8 k; l ¼ 1; 2; . . . ; Z

(32)

i¼1

Let two cumulative bivariate densities, F and G, describe the two distributions of (x-y) and (xu-yu), respectively, with Ok ¼ [Xk1, Xk] and Gl ¼ [Yl1, Yl] being associated to F and O0k ¼ ½X 0k1 ; X 0k  and G0l ¼ ½Y 0l1 ; Y 0l  being associated to G. We distinguish between two cases, depending on how the transition matrix is calculated. The first case corresponds to the situation in which we are only concerned with pure, or exchange, mobility (see Atkinson, 1983 and Dardanoni, 1993). Here, we impose the steady-state and pure mobility ^ X^ 0 Þ ¼ Gð ^ Y^ 0 Þ ¼ pk and k ¼ 1,y, Z, ^ Y^ k Þ ¼ Gð ^ X^ k Þ ¼ Fð assumptions that Fð k k ^ and Ys ^ are the sample quantiles and the F^ and G^ are the where the Xs empirical (or sample) distributions of F and G. In this first case, those sample quantiles are determined from the joint distributions of x and y in ^ both F^ and G. The second case integrates growth, as suggested by Formby et al. (2003). Here, the growth that is considered is that observed in the dynamics of each population separately. The normalization imposed here is that the two distributions F and G have the same initial quantiles in the x dimension. The values of these quantiles are then fixed and used to assess the degree of mobility in the y dimension. Here therefore, the equality that is imposed ^ X^ 0 Þ, with X^ k ¼ Y^ k and X^ 0 ¼ Y^ 0 for k ¼ 1,y, Z. For ^ X^ k Þ ¼ Gð is that Fð k k k simplicity, we focus below on the first case; the extension to the second case will also be briefly mentioned. According to the Atkinson (1983) and Dardanoni (1993) mobility dominance criteria, the constraint that F does not dominate G in the

185

Testing for Mobility Dominance

samples implies, for at least one pair (k, l), that NG X

0

0

0 ^ 0 ^ pG j Iðxj  X k ; yj  Y l Þ 

j¼1

NF X

pFi Iðxi  X^ k yi  Y^ l Þ

(33)

i¼1

0 0 X^ k and Y^ l are the sample quantiles (generated by the empirical distribu^ that correspond to the same percentiles as the sample quantiles X^ k tion G) ^ ^ The testing strategy follows a similar and Y l for the empirical distribution F: procedure to that of the previous section. The maximization results yield:

pG j ðk; l Þ ¼

0 0 0 0 Iðx0j  X^ k ; y0i  Y^ l Þ ½1  Iðx0j  X^ k ; y0i  Y^ l Þ þ y f

(34)

pFi ðk; l Þ ¼

Iðxi  X^ k ; yi  Y^ l Þ ½1  Iðxi  X^ k ; yi  Y^ l Þ þ Ny N f

(35)

with y¼

N G ðk; l Þ ¼

N  N G ðk; l Þ N  M G ðk; l Þ ;f ¼ N G ðk; l Þ þ N F ðk; l Þ M G ðk; lÞ þ M F ðk; l Þ

NG X j¼1

0 0 Iðx0j  X^ k ; y0j  Y^ l Þ; N F ðk; l Þ ¼

NF X

Iðxi  X^ k ; yi  Y^ l Þ

i¼1

M G ðk; l Þ ¼ N G  N G ðk; l Þ; M F ðk; l Þ ¼ N F  N F ðk; l Þ and N ¼ N G þ N F The likelihood ratio is again first obtained by considering the difference between the unconstrained and the constrained ELF for some fixed (k, l ). This equals: 9 8 N log N  N G log N G  N F log N F > > > > > > > > > > þN ðk; l Þ log N ðk; l Þ þ N ðk; l Þ log N ðk; l Þ G G F F > > = < (36) LRðk; lÞ ¼ 2 þM G ðk; l Þ log M G ðk; l Þ þ M F ðk; l Þ log M F ðk; l Þ > > > > > > ½N G ðk; l Þ þ N F ðk; l Þ log½N G ðk; l Þ þ N F ðk; l Þ > > > > > ; : ½M ðk; l Þ þ M ðk; l Þ log½M ðk; l Þ þ M ðk; l Þ > G F G F 0 0 As above, the pairs ðX^ k ; Y^ L Þ and ðX^ k ; Y^ L Þ are endogenous, making it necessary to perform several iterations. For each iteration, the constraint ^ X^ 0 Þ ¼ Gð ^ Y^ 0 Þ ¼ pk for k ¼ 1; . . . ; Z is imposed. ^ Y^ k Þ ¼ Gð ^ X^ k Þ ¼ Fð that Fð k k

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186

The test statistic LR is given by minimizing LR(k, l ) over all possible values of (k, l ). The bootstrap p-value is obtained as above, by computing bootstrap LR statistics on samples generated under the probabilities given by Eqs. (34) and (35) at the value of (k, l ) that minimizes LR(k, l ) in the initial samples. For the case of growth mobility, the above expression remains the same except that the constraints imposed on the ^ X^ 0 Þ; X^ k ¼ Y^ k ; and X^ 0 ¼ Y^ 0 for ^ X^ k Þ ¼ Gð empirical quantiles is now that Fð k k k k ¼ 1; . . . ; Z. To test for second-order mobility dominance, we use the following useful expression (see Atkinson & Bourguignon, 1982) for F2 (x, y): F 2 ðx; yÞ ¼

Z

x

Z

y

ðx  sÞðy  tÞFðs; tÞdtds 0

(37)

0

The nondominance constraint (33) then becomes NG X

0 0 0 ^0 ^0 ^0 0 ^0 pG j ðX k  xj ÞðY l  yj ÞIðxj  X k; yj  Y l Þ

j¼1



NF X

pFi ðX^ k  xi ÞðY^ l  yi ÞIðxi  X^ k ; yi  Y^ l Þ

ð38Þ

i¼1

The rest of the procedure is analogous to the first-order procedure described above. The solutions to this problem are computed numerically also using a similar procedure to that for testing second-order absolute mobility dominance.

4. EMPIRICAL ILLUSTRATION Our data comes from the US Panel Study of Income Dynamics (PSID) surveys. The analysis is restricted to those individuals aged 24–49 that are not self-employed. Four periods are used to compare temporal mobility in the United States: 1970–1975, 1975–1980, 1980–1985, and 1985–1990. For instance, mobility in the 1970–1975 period is assessed by comparing the income status of our individuals in 1970 to their status in 1975. All incomes are evaluated at 1990 prices.

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Testing for Mobility Dominance

4.1. Absolute Mobility

(1975−80)−(1970−75) (1975−80)−(1980−85) (1985−90)−(1970−75) (1985−90)−(1980−85)

0.4 0.3 0.1

0.2

US dollars (1,000)

0.5

0.6

Tests for first-order mobility dominance show overall an absence of dominance in mobility across the four periods. For second-order dominance tests, both generalized Lorenz and Mitra and Ok (1998) approaches are considered. Fig. 1 shows four differences between the generalized Lorenz curves of distances [recall (3)]. These curves illustrate four possible dominance relationships, for dominance of the periods 1970–1975 and 1980–1985 by the periods 1975–1980 and 1985–1990. All of the differences are positive in the sample. The four curves in the figure intersect, a feature that thus excludes the presence of other dominance relations. As to the Mitra and Ok (1998) approach (Fig. 2), which cumulates distances from the largest to the lowest, only the distribution for 1975–1980 dominates in the sample all others – the differences between the 1975–1980 curve and the three other curves being positive. The three other curves intersect excluding as above the existence of additional dominance relations. Statistical inference confirms most of these relations. In all of the following tables, statistical inference can be made by comparing the reported p-values to conventional critical levels, which we implicitly set here

0.1

Fig. 1.

0.2

0.3

0.4

0.5 centiles

0.6

0.7

0.8

0.9

Differences in the Generalized Lorenz Curves of the Income Distances Across Time.

YE´LE´ MAWEKI BATANA AND JEAN-YVES DUCLOS

0.5

0.6

(1975−80)−(1970−75) (1975−80)−(1980−85) (1975−80)−(1985−90)

0.3

0.4

US dollars (1,000)

0.7

0.8

0.9

188

0.1

Fig. 2.

0.2

0.3

0.4

0.5 centiles

0.6

0.7

0.8

0.9

Differences Between the Curves Implied by the Mitra and Ok (1998) Approach.

Table 1. Results of the Statistical Tests on Differences in the Generalized Lorenz Curves of the Income Distances Across Time. G is not Dominated by F 1970–1975 1970–1975 1980–1985 1980–1985

vs. vs. vs. vs.

1975–1980 1985–1990 1975–1980 1985–1990

LR Ratio

p-Value

4.653 2.898 3.362 1.108

0.000 0.010 0.025 0.208

and denote that the statistics are significant at 5% and 1% levels, respectively.

to 5% level. The p-values that are shown are the probability that the null hypothesis of nondominance of G by F be wrongly rejected. The rejection of this null (when the p-value is low) implies that F statistically dominates G in mobility (F displays greater mobility). Tables 1 and 2 show indeed that five of the seven previously identified relations are significant at the conventional 5% level. The two remaining

189

Testing for Mobility Dominance

Table 2.

Results of the Statistical Tests on Differences in the Curves of the Mitra and Ok (1998) Approach.

G is not Dominated by F

LR Ratio

p-Value

1970–1975 vs. 1975–1980 1980–1985 vs. 1975–1980 1985–1990 vs. 1975–1980

1.089 2.859 3.018

0.163 0.008 0.000

denotes that the statistics are significant at 1% level.

dominance relations do not appear to be significant, as evidenced by their p-values that exceed 5%. 4.2. Transition Matrices The transition matrices were estimated by grouping the population into five income classes, as in Formby et al. (2004). These income classes are defined by the vector of quintiles x and y, for the first-period and the second-period income distributions, respectively. For the first case (exchange mobility), we normalize the distributions by fixing F(x) ¼ F(y) ¼ G(x) ¼ G(y)P¼ p P ¼ (0.2, 0.2, 0.2, 0.2, 0.2). The tests L1 are then performed on the matrix ½ K k¼1 l¼1 0:2ðQkl  Pkl Þ, excluding the element k ¼ K and l ¼ L which are equal to zero by definition. In this first case, it proves impossible to find dominance relations across periods. For the second case, we only set F(x) ¼ G(x) ¼ p ¼ (0.2, 0.2, 0.2, 0.2, 0.2). The quantiles computed for the initial x period are used to define income classes for the PL1y period. In this second case, the test is performed on P second the matrix K k¼1 l¼1 0:2ðQkl  Pkl Þ since only the elements l ¼ L are equal to zero by definition. The appendices provide further details for each of these two types of normalizations. The estimated transition matrices within each of the three periods 1970–1975, 1980–1985, and 1985–1990 are then estimated as: 2 3 0:480 0:297 0:118 0:075 0:032 6 0:190 0:369 0:254 0:125 0:061 7 6 7 6 7 ^P19701975 ¼ 6 0:079 0:190 0:308 0:301 0:118 7 6 7 6 7 4 0:021 0:057 0:186 0:401 0:423 5 0:014 0:021 0:075 0:165 0:598

YE´LE´ MAWEKI BATANA AND JEAN-YVES DUCLOS

190

2

^ 19801985 P

0:554 6 0:210 6 6 ¼6 6 0:145 6 4 0:049 0:049

2

^ 19851990 P

0:449 6 0:169 6 6 ¼6 6 0:081 6 4 0:041 0:034

0:271 0:415

0:100 0:248

0:061 0:092

0:216

0:314

0:238

0:090 0:061

0:183 0:067

0:316 0:189

0:278 0:363

0:138 0:322

0:078 0:138

0:120

0:265

0:321

0:062 0:029

0:148 0:054

0:389 0:163

3 0:014 0:043 7 7 7 0:120 7 7 7 0:330 5 0:619

3 0:057 0:077 7 7 7 0:140 7 7 7 0:418 5 0:663

Here two relations show dominance in the samples: both the periods 1975–1980 and 1985–1990 dominate in the samples the 1980–1985 period. However, the inference results (see Table 3) show that these dominance relations are not statistically significant. To test for second-order dominance, we maximize as above the empirical likelihood without and with constraint (38), and normalizations of the quantiles also being carried out in the same manner as above. The sampling estimates suggest six potential dominance relations to be formally tested, including two relations in the case of exchange mobility and four in the case of growth mobility. As we see in Table 4, none of these dominance relations proves to be statistically significant since all of the p-values are too high to reject the null of nondominance. Statistical significance is, however, obtained if the dominance relations are restricted by dropping the last columns of the transition matrices. Two dominance relations – one for each case – are then statistically significant at a 5% level (Table 5), with the 1985–1990 period dominating the period 1975–1980 in both cases. Table 3.

First-Order Growth Mobility Dominance.

G is not Dominated by F

LR Ratio

p-Value

1980–1985 vs. 1975–1980 1980–1985 vs. 1985–1990

0.409 0.205

0.376 0.459

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Testing for Mobility Dominance

Table 4. G is not Dominated by F

Second-Order Mobility Dominance. LR Ratio

p-Value

Exchange mobility 1975–1980 vs. 1970–1975 1975–1980 vs. 1985–1990

0.135 0.023

0.650 0.808

Growth mobility 1970–1975 vs. 1985–1990 1975–1980 vs. 1970–1975 1975–1980 vs. 1985–1990 1980–1985 vs. 1985–1990

0.499 0.241 1.364 0.174

0.501 0.376 0.330 0.702

Table 5.

Second-Order Dominance (Restricted to the First Four Quintiles of Income).

G is not Dominated by F

LR Ratio

p-Value

Exchange mobility 1970–1975 vs. 1985–1990 1975–1980 vs. 1970–1975 1975–1980 vs. 1980–1985 1975–1980 vs. 1985–1990

0.088 0.135 0.057 1.900

0.822 0.120 0.970 0.010

Growth mobility 1970–1975 vs. 1985–1990 1975–1980 vs. 1970–1975 1975–1980 vs. 1985–1990 1980–1985 vs. 1985–1990

0.744 0.769 2.080 0.591

0.506 0.313 0.036 0.719

denotes that the statistics are significant at 5% level.

5. CONCLUSION A number of mobility measures have been proposed in the literature. This chapter proposes procedures for testing for whether mobility is robustly greater in a distribution A than in a distribution B over different possible classes of mobility measures. For this, it draws on the frameworks for making partial orderings of mobility proposed by Atkinson (1983), Conlisk (1989), Dardanoni (1993), Mitra and Ok (1998), Formby et al. (2003), and others. Following in the footsteps of Maasoumi and Trede (2001) and Formby et al. (2004), this chapter proposes statistical tests for comparing mobility, with the difference being that the procedure relies here on nulls of nondominance. Two types of comparisons are made. The first is based on

192

YE´LE´ MAWEKI BATANA AND JEAN-YVES DUCLOS

absolute mobility measures. The second uses transition matrices. The tests are performed on the null hypothesis of nondominance versus the alternative hypothesis of dominance. The analysis covers both first- and second-order stochastic dominance. Illustrations are performed from the US PSID data by comparing four 5-year periods between 1970 and 1990. There are no first-order absolute mobility dominance relationships across these periods because the sample dominance curves intersect. Two approaches are taken to assess secondorder dominance: the classical generalized Lorenz approach and the Mitra and Ok (1998) approach. Our analysis identifies several dominance relations in the samples, most of which end up being statistically significant. For measures based on transition matrices, two first-order dominance relations (both in the case of growth mobility) are observed in the sample, none of which being statistically significant. Several second-order dominance relations exist in the samples, only two of which being statistically significant. These results suggest that it may sometimes be difficult to obtain rankings of distributions that are robust over wide classes of mobility indices, and that it can also be important to perform statistical tests of such rankings since sample rankings of mobility may not always be strong enough to infer population ones.

NOTE 1. See Fields and Ok (1999) for a more detailed discussion.

ACKNOWLEDGMENTS We are grateful to Canada’s SSHRC, to Que´bec’s FQRSC, and to the Programme canadien de bourses de la francophonie for financial support. This work was also carried out with support from the Poverty and Economic Policy (PEP) Research Network, which is financed by the Government of Canada through the International Development Research Centre (IDRC) and the Canadian International Development Agency (CIDA), and by the Australian Agency for International Development (AusAID). We are also grateful to Andrew Heisz for useful comments.

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193

REFERENCES Atkinson, A. B. (1983). The measurement of economic mobility. In: A. B. Atkinson (Ed.), Social justice and public policy. London: Harvester Wheat-sheaf. Atkinson, A. B., & Bourguignon, F. (1982). The comparison of multidimensional distributions of economic status. Review of Economic Studies, 49, 183–201. Atkinson, A. B., Bourguignon, F., & Morrisson, C. (1992). Empirical studies of earnings mobility. Switzerland: Harwood Academic Publishers. Bartholomew, D. J. (1996). The statistical approach to social measurement. San Diego: Academic Press. Batana, Y. M., & Duclos, J.-Y. (2008). Multidimensional poverty dominance: Statistical inference and an application to West Africa. CIRPE´E Working Paper 08-08, CIRPE´E, Universite´ Laval, Que´bec. Benabou, R., & Ok, E. A. (2001). Mobility as progressivity: Ranking income processes according to equality of opportunity. NBER Working Paper 8431, NBER, Cambridge, MA. Bjorklund, A., & Jantti, M. (1997). Intergenerational income mobility in Sweden compared to the United States. American Economic Review, 87, 1009–1018. Chadwick, L., & Solon, G. (2002). Intergenerational income mobility among daughters. American Economic Review, 92, 335–344. Chakravarty, S. R., Dutta, B., & Weymark, J. A. (1985). Ethical indices of income mobility. Social Choice Welfare, 2, 1–21. Conlisk, J. (1989). Ranking mobility matrices. Economics Letters, 29, 231–235. Corak, M. (2006). Generational income mobility. Review of Income and Wealth, 52, 477–486. D’Agostino, M., & Dardanoni, V. (2005). The measurement of mobility: A class of distance indices. Paper presented to the Society for the Study of Economic Inequality, July, Palma de Mallorca, Spain. Dardanoni, V. (1993). Measuring social mobility. Journal of Economic Theory, 61, 372–394. Davidson, R., & Duclos, J.-Y. (2006). Testing for restricted stochastic dominance. IZA Discussion Paper no. 2047, IZA, University of Bonn, Bonn. Duclos, J.-Y., & Araar, A. (2006). Poverty and equity: Measurement, policy and estimation with DAD. Berlin and Ottawa: Springer and IDRC. Fields, G. S., Leary, J. B., & Ok, E. A. (2002). Stochastic dominance in mobility analysis. Economics Letters, 75, 333–339. Fields, G. S., & Ok, E. A. (1996). The meaning and measurement of income mobility. Journal of Economic Theory, 71, 349–377. Fields, G. S., & Ok, E. A. (1999). The measurement of income mobility: An introduction to the literature. In: J. Silber (Ed.), Handbook of inequality measurement. Dordrecht: Kluwer Academic Publishers. Formby, J. P., Smith, W. J., & Zheng, B. (2003). Economic growth, welfare and the measurement of social mobility. In: Y. Amiel & J. A. Bishop (Eds), Research on economic inequality (Vol. 9, pp. 105–111). Bingley, UK: Emerald. Formby, J. P., Smith, W. J., & Zheng, B. (2004). Mobility measurement, transition matrices and statistical inference. Journal of Econometrics, 120, 181–205. Foster, J. E., & Shorrocks, A. F. (1988). Poverty orderings and welfare dominance. Social Choice Welfare, 5, 179–198.

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Maasoumi, E., & Trede, M. (2001). Comparing income mobility in Germany and the United States using generalized entropy mobility measures. Review of Economics and Statistics, 83, 551–559. Maasoumi, E., & Zandvakili, S. (1986). A class of generalized measures of mobility with applications. Economics Letters, 22, 97–102. Markandya, A. (1984). The welfare measurement of changes in economic mobility. Economica, 51, 457–471. Mitra, T., & Ok, E. A. (1998). The measurement of income mobility: A partial ordering approach. Economic Theory, 12, 77–102. Peters, H. E. (1992). Patterns of intergenerational mobility in income and earnings. Reviews of Economics and Statistics, 74, 456–466. Prais, S. J. (1955). Measuring social mobility. Journal of the Royal Statistical Society, 118, 56–66. Shorrocks, A. F. (1978a). The measurement of mobility. Econometrica, 46, 1013–1024. Shorrocks, A. F. (1978b). Income inequality and income mobility. Journal of Economic Theory, 19, 376–393. Shorrocks, A. F. (1983). Ranking income distributions. Economica, 50, 3–17. Willig, R. D. (1981). Social welfare dominance. American Economic Review, 71, 200–204.

APPENDICES Appendix A: Normalizing the Data to Consider Exchange Mobility Let XK and YK be the empirical quintiles from the distribution F, and X 0K , and Y 0K be the ones from the distribution G, with K, L ¼ 1, y , 5. We wish ~ yÞ ~ so that their respective to normalize the observations ðx0 ; y0 Þ of G into ðx; quintiles be given by XK and YK. To do this, we can apply the formula: x~ j ¼ X K1 þ ðx0j  X 0K1 Þ

X K  X K1 ; 8 j ¼ 1; . . . ; N G with X 0K1 ox0j  X 0K X 0K  X 0K1 (A.1)

We set X 0 ¼ X 00 ¼ 0. The same procedure can be applied for estimating y~j : y~j ¼ Y L1 þ ðy0j  Y 0L1 Þ

Y L  Y L1 ; 8 j ¼ 1; . . . ; N G with Y 0L1 oy0j  Y 0L Y 0L  Y 0L1 (A.2)

Again, we set Y 0 ¼ Y 00 ¼ 0: After these transformations, the new ~ yÞ ~ drawn from G have the same quintiles as those obtained observations ðx; for F.

Testing for Mobility Dominance

195

Appendix B: Normalizing the Data to Consider Growth Mobility Here, only the initial income quintiles are considered, namely XK for F and X 0K for G, with K ¼ 1, y , 5. These quintiles are used for determining the final income classes. Let the proportions of individuals in the final income classes be respectively given by PK and P0K . We need to transform ðx0 ; y0 Þ into ~ yÞ, ~ so that the initial income quintiles x~ be equal to XK. Moreover, this ðx; transformation should let the proportions P0K unchanged. We first follow the procedure described in Eq. (A.1) for computing x~ j . The y~j are then determined as follows: y~j ¼ X K1 þ ðy0j  X 0K1 Þ

X K  X K1 ; 8 j ¼ 1; . . . ; N G with X 0K1 oy0j  X 0K X 0K  X 0K1 (B.1)

CHAPTER 9 DISTRIBUTIONAL CHANGE, REFERENCE GROUPS, AND THE MEASUREMENT OF RELATIVE DEPRIVATION Jacques Silber and Paolo Verme ABSTRACT This chapter attempts to explicitly integrate the idea of reference group when measuring relative deprivation. It assumes that in assessing his situation in society an individual compares himself with individuals whose environment can be considered as being similar to his. By environment we mean the set of people with a similar set of observable characteristics such as human capital, household attributes, and location. We therefore propose to measure relative deprivation by comparing the actual income of an individual with the one he could have expected on the basis of the level of these characteristics. We then aggregate these individual comparisons by computing an index of ‘‘distributional change’’ that compares, on a non anonymous basis, the distributions of the actual and ‘‘expected’’ incomes. At the difference of other approaches to relative deprivation, our measure takes into account not only the difference between the actual and ‘‘expected’’ individual incomes but also that between the actual and ‘‘expected’’ individual ranks. We applied our approach to Moldova, the poorest country in Europe, using a survey that Studies in Applied Welfare Analysis: Papers from the Third ECINEQ Meeting Research on Economic Inequality, Volume 18, 197–217 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2010)0000018012

197

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JACQUES SILBER AND PAOLO VERME

covered a period of six years (from 2000 to 2005). We then observed that our measure of deprivation is well suited to study wage deprivation across genders and is able to proxy subjective deprivation in living standards reported by survey respondents better than conventional measures of relative deprivation.

I have become increasingly convinced that relative deprivation actually has little to do with envy. Rather, it is fundamentally about the link between context and evaluation. Robert H. Frank, From the Preface to Falling Behind. How Rising Inequality Harms the Middle Class (University of California Press, 2007).

1. INTRODUCTION In The Wealth of Nations, Adam Smith (1776) wrote that ‘‘By necessaries I understand not only the commodities which are indispensably necessary for the support of life, but whatever the custom of the country renders it indecent for creditable people, even of the lowest order, to be without.’’ More recently, Frank (2007) writes that ‘‘Evidence suggests that, relative to the mix of goods that would maximize our health and happiness, we spend too much on context-sensitive goods and too little on goods that are relatively insensitive to context.’’ In emphasizing context rather than envy, Frank stresses in fact the importance of the concept of ‘‘reference group.’’ He thus writes that ‘‘ y a house of a given size is more likely to be viewed as spacious the larger it is relative to other houses in the same local environment’’ (Frank, 2007). Marx (1847) himself wrote that ‘‘A house may be large or small; as long as the neighboring houses are likewise small, it satisfies all social requirement for a residence. But let there arise next to the little house a palace, and the little house shrinks to a hut. The little house now makes it clear that its inmate has no social position at all to maintain, or but a very insignificant one; and however high it may shoot up in the course of civilization, if the neighboring palace rises in equal of even in greater measure, the occupant of the relatively little house will always find himself more uncomfortable, more dissatisfied, more cramped within his four walls.’’ This point was also stressed by Runciman (1966) who structured in a theory of social justice an idea initially put forward by Stouffer, Suchman, De Vinney, Star, and Williams (1949). Thus, Runciman (1966) wrote that ‘‘The questions to ask

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are first, to what group is a comparison being made? Second, what is the allegedly less well-placed group to which the person feels that he belongs?.’’ In the latter quotation, Runciman clearly does not limit the concept of relative deprivation to that of ‘‘context’’ since he considers that an individual sees himself as belonging to a group but also as making a comparison with the situation of some other group(s). Economists (e.g., Yitzhaki, 1979; Hey & Lambert, 1980) seem, however, to have translated Runciman’s ideas in a rather narrow way that amounts more or less to identify relative deprivation with envy (with respect to individuals with a higher income), although sometimes (see, Berrebi & Silber, 1985) both the feeling of deprivation with respect to those with a higher income and that of satisfaction with respect to those with a lower income are taken into account. Also Bossert and D’Ambrosio (2006) noted that the reference group considered by Yitzhaki (1979) can be seen as a subset of a larger reference group that includes all individuals: ‘‘The reference group includes all agents the individual compares itself to in general (and, thus, not only when considering matters of deprivation), whereas the comparison group is the subset of this set containing those who are richer.’’ Whatever the specific way in which relative deprivation is measured (on this topic, see also Kakwani, 1984), economists clearly did not devote much attention to the concept of ‘‘reference group.’’ The purpose of this chapter is precisely to explicitly integrate the idea of reference group when measuring relative deprivation, following recent attempts to integrate in the notion of reference group other dimensions in addition to welfare (Clark & Oswald, 1996; Verme & Izem, 2008). We will assume that in assessing his situation in society, an individual compares himself with individuals whose environment can be considered as being similar to his. By environment we mean not only what Frank (2007) called ‘‘local environment’’ in one of the quotations given previously but also other aspects such as the ‘‘professional environment’’ of an individual or his ‘‘family environment’’ (background). As stressed by Schaefer (2008), ‘‘relative deprivation is the conscious experience of a negative discrepancy between legitimate expectations and present actualities.’’ We believe that a good proxy for these ‘‘legitimate expectations,’’ that is, for the reference group of an individual, is the set of people with similar observable characteristics such as human capital, household attributes, and location. We therefore propose to measure relative deprivation by comparing the actual income of an individual with the one he could have expected on the basis of the level of these characteristics. Ferrer-i-Carbonell (2005) took somehow a similar approach when she defined an individual’s reference

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group as all the individuals who belong to the same age group, have similar education, and live in the same region. We, however, aggregate these individual comparisons by computing an index of ‘‘distributional change’’ that compares, on a non anonymous basis, the distributions of the actual and ‘‘expected’’ incomes. At the difference of other approaches to relative deprivation, our measure takes into account not only the difference between the actual and ‘‘expected’’ individual incomes but also that between the actual and ‘‘expected’’ individual ranks. We applied our approach to Moldova, the poorest country in Europe, using a survey that covered a period of six years (from 2000 to 2005). We then observed that our measure of deprivation is well suited to study wage deprivation across genders and is able to proxy subjective deprivation in living standards reported by survey respondents better than conventional measures of relative deprivation. It should be interesting to note that if relative deprivation is indeed a function of the gap between actual and ‘‘expected’’ individual incomes, the latter being somehow formed ‘‘in relation to standards for allocating rewards’’ (Shepelak & Alwin, 1986), we may be led to accept Berger, Zelditch, Anderson, and Cohen (1972) statement according to which ‘‘as a consequence of beliefs about what is typically the case, expectations y come to be formed about what one can legitimately claim ought to be the case.’’ A similar idea was indeed formulated earlier by Heider (1958) who argued that ‘‘tradition represents the existing reality made solid by a long history in which it becomes identified with the just, the ethical the ‘should be’ y and the ‘is’ takes on the character of the ‘ought’.’’ Such a view certainly goes in the direction of our findings that stress that it is not the existing income inequality that matters for relative deprivation feelings but the comparison of actual with ‘‘expected’’ incomes. This chapter is organized as follows. Section 2 defines our new measure of relative deprivation. Section 3 gives an empirical illustration based on data for Moldova, whereas Section 4 offers concluding comments.

2. A NEW APPROACH TO MEASURING RELATIVE DEPRIVATION Assume yi is the income of individual i, and Xi a vector of his personal characteristics. We may then write that yi ¼ a þ bX i þ i

(1)

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where ei includes the effect of unobserved factors on the income of individual i as well as the impact of measurement errors. Let us now define the ‘‘predicted’’ or ‘‘expected’’ income yPi of individual i as ^ i yPi ¼ a^ þ bX

(2)

where a^ and b^ are estimates of a and b. Call now si and vi the shares of individual i in the total actual and expected income of the society. However, given that E(ei) ¼ 0, the average incomes yi and yPi are identical and hence are the total values of the actual and expected incomes, respectively. Using an algorithm originally proposed by Silber (1989), we may compute the Gini indices I G;fyi g and I G;fyPi g of the sets of incomes {yi} and {yPi}, respectively, as    0 1 . . . G½. . . ðsi Þ . . . I G;fyi g ¼ . . . (3) n 

I G;fyPi g

  0 1 . . . G½. . . ðvi Þ . . . ¼ ... n

(4)

where ½. . . ð1=nÞ . . . 0 is a row vector of ‘‘population shares’’ whose elements are all equal to (1/n), the share of each individual in the whole population, and ½. . . ðsi Þ . . . and ½. . . ðvi Þ . . . are, respectively, column vectors whose typical elements are, respectively, the shares of individual i in the total amount of actual and expected incomes. Finally, the term G in Eqs. (3) and (4), called G-matrix by Silber (1989), is a n  n square matrix whose typical element ghk is equal to 0 if h ¼ k, to 1 if h  k, and to þ1 if h k. Note that in Eqs. (3) and (4), the shares si and vi have to be ranked by decreasing values (decreasing incomes). Call now wi the share in the total amount of expected incomes that individual i would have obtained, had his rank in the distribution of the ‘‘expected’’ incomes ({yPi}) been the same as his rank in the distribution of the actual incomes ({yi}). Following earlier work by Cowell (1980) on the concept of distributional change, Silber (1995) suggested then to measure the degree of ‘‘distributional change’’1 between the distributions {yi} and {yPi} as

    0      0  1 1 . . . G½. . . ðsi Þ . . .  . . . G½. . . ðwi Þ . . . ... ... (5) J GP ¼ n n

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The subindices G and P in JGP indicate that this index first is derived from the Gini index, and second, that it is ‘‘population-weighted’’ since each individual receives the same weight (1/n). Given the linearity of the G-matrix operator, Silber (1995) then showed that Eq. (5) could be also expressed as



  0     0  1 1 . . . G½. . . ðsi Þ . . .  . . . . . . G½. . . ðvi Þ . . . ... J GP ¼ n n

    0      0  1 1 ð6Þ . . . G½. . . ðvi Þ . . .  . . . . . . G½. . . ðwi Þ . . . þ ... n n Expression (6) indicates clearly that the index JGP is an index measuring somehow the degree of income mobility of the individuals between their actual situation in the distribution {yi} and their hypothetical situation in the distribution {yPi}. More precisely, JGP includes two components. The first one is called ‘‘structural mobility’’ (the first expression on the RHS of (6)) and measures the difference between the inequality (Gini index) of the distribution of the actual incomes ({yi}) and that of the ‘‘predicted’’ incomes ({yPi}).2 The second component called ‘‘exchange mobility’’ (the second term on the RHS of (6)) measures the amount of ‘‘reranking’’ that takes place when one compares the position of the individuals in the distribution of the actual and predicted incomes (for more details on these two concepts of income mobility, see Fields & Ok, 1999).3 Total deprivation in the population is thus a function of two elements. The first one measures the gap between the inequality of actual incomes (measured by the Gini index) and that of the expected incomes. One may recall here that the traditional approach to total (relative) deprivation identifies it with the Gini index of actual incomes (see Yitzhaki, 1979; Kakwani, 1984) and thus implicitly assumes that a comparison is made with a hypothetical perfectly equal distribution, whereas we take as reference the distribution of the expected incomes. The second component of total deprivation is a positive function of the amount of reranking that takes place when comparing actual and expected incomes. It is hence assumed that the greater the amount of such reranking, the higher the total deprivation. Since any amount of such reranking can be shown to be the sum of a certain number of income swaps, what is implicitly assumed here is that in any income swap, the individual whose rank decreases is in absolute value more (negatively) affected than the individual with whom he swapped incomes and whose rank improved, a change that should positively affect the latter individual.4

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Silber (1995) defined also what he called an ‘‘income-weighted’’ measure of distributional change expressed as: J GI ¼ ½. . . ðsi Þ . . . 0 G½. . . ðwi Þ . . .

(7)

the shares si and wi in Eq. (7) being ranked this time by decreasing values of the ratios (wi/si). As indicated by Silber (1995), a graphical interpretation of the index JGI may be obtained by plotting the cumulative values of the shares si and wi, respectively, on the horizontal and vertical axes. Here in plotting these cumulative values one has to rank the individuals by increasing, rather than decreasing, values of the ratios (wi/si). The graph obtained is in fact what Kakwani (1980) has called a relative concentration curve, whose slope, like that of a Lorenz curve, is nondecreasing. Note that it is easy to prove that the index JGI is in fact equal to twice the area lying between this relative concentration curve and the diagonal. The JGP index can be interpreted as a measure of the distributional change observed when comparing the actual income of the individuals with that predicted on the basis of their personal characteristics. As mentioned previously, this distributional change is a function, first, of the difference between the inequality based on the actual incomes and that computed on the basis of the predicted incomes, and, second, of the difference between the ranking of the individuals according to their actual income and that derived from their predicted income. However, as the explanatory power of the regression for the predicted values increases, two effects are at work. On the one hand, the Gini index of the incomes yi will get closer to that of the predicted incomes yPi and this will reduce JGP. On the other hand, the correlation between the incomes yi and yPi will also increase and with it the correlation between the rank of yi and that of yPi, thus reducing the reranking effect, and hence the second component of JGP on the RHS of Eq. (6). It can be shown (see Silber, 1995) that the distributional change index JGP will be greater, the greater the number of income swaps (leading to reranking) between individuals, and the impact of an income swap on JGP will be greater, the greater the difference between the swapped incomes as well as between the ranks of the individuals who swap their income (this is the exchange mobility component of JGP). Similarly, the index JGP increases with the number of transfers having taken place between a richer and a poorer individual (assuming no reranking) and the impact of such a transfer will be greater, the greater the amount transferred (this is the structural mobility component of JGP).

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3. AN EMPIRICAL ILLUSTRATION 3.1. Data Sources To illustrate the indexes proposed, we use the Moldova Household Budget Survey (MHBS). Moldova stands out in Europe as the country that experienced the deepest recession of the postwar period with a combined loss in output of over 60% between 1990 and 1996. The recession has also been accompanied by a rapid growth in poverty and inequality. The World Bank (2004) estimated that in 1999 about 71% of the population was below the poverty line whereas the Gini coefficient rose from an estimated figure of 0.24 in 1988 to 0.37 in 1997 (World Bank, 1999). Not surprisingly, if one takes the World Values Surveys database that covers the period 1981–20045 and estimates average life satisfaction by country and year, one will find that the lowest life satisfaction scores ever recorded by the surveys worldwide were those of Moldova in 1996. Since 2000, Moldova has reversed its fortunes and enjoyed sustained output growth estimated at around 7% per year on average. This has contributed to reduce poverty to a headcount ratio below 30% of the population by 2004, whereas inequality continued to remain relatively high with a Gini coefficient pivoting around a value of 0.35. Such epochal swings in output, poverty, and inequality are expected to be reflected in significant changes in the subjective evaluation of living conditions and make of Moldova a unique case for the study of relative deprivation. The MHBS is administered by the National Bureau of Statistics (NBS) of the Republic of Moldova. The survey, initiated in 1997, is the product of a joint effort of the Moldovan NBS and the World Bank, it has been revised and improved on several occasions, and today is considered as one of the most comprehensive and reliable surveys available for transitional economies. Of a population of 3.6 million people, the MHBS covers 6,000 households every year interviewed in monthly blocks of 500 households each. The sample is a rotating one and includes a panel component of about 25% of households with tenure per household of four years. The questionnaire is very rich and comparable to the World Bank Living Standards Measurement Surveys. It also includes questions on subjective estimations of living conditions that can be used to assess the performance of the relative deprivation indexes proposed.

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3.2. A Simple Example As a first illustration of the indexes proposed, we restrict the sample to male heads of households, aged 25–55 in 2006 (2,248 observations) and focus on one indicator of welfare, household per capita consumption ( yi). Predicted values ( yPi) are estimated with an OLS regression based on a set of regressors that we thought define well the reference group.6 These are age (years), education level (dummies for each level),7 marital status (dummies for each category),8 social group (dummies for each category),9 district (dummies for each district), and urban and rural areas (dummy for urban areas). By selecting these variables, we are implicitly assuming that individuals select the reference group based on the characteristics described by the listed variables and they are able to observe all and only these characteristics. This is evidently a normative choice made by the researcher and based on the knowledge of the local population.10 Table 1 reports the estimations of the two indexes together with their components, including structural and exchange mobility. Bootstrap standard errors and confidence intervals are also reported.11 Relative deprivation for male heads of household in 2006 is estimated at around 23%. The greatest part is explained by structural mobility contributing for about 59% of the index, whereas reranking (exchange mobility) contributes for the remaining part. Both components are evidently important in determining relative deprivation. The Silber (1995) income-weighted measure of distributional change (JGI) provides a higher estimate of relative deprivation for the same group of people. The Gini of the incomes yi – which is the equivalent of the Yitzhaki (1979) measure of relative deprivation (divided by the mean) – provides the highest estimate of relative deprivation. As already discussed, this is due to the construction of the other two indexes. This is illustrated in Table 2 where we test (by removing one regressor at the time) how the JGP and JGI indexes behave as the explanatory power of the regression for the estimation of the predicted values decreases. As anticipated, both indexes converge toward the Gini of the incomes yi.

3.3. Relative Deprivation by Population Subgroup In this second example, we restrict the sample to men and women of working age (1,737 men and 1,548 women in age 25–55) and we consider as a measure of welfare individual wages. The purpose is to show the

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Table 1.

GY GYP GW FGP PERM JGP JGI

Relative Deprivation Indexes and their Components.

Gini y Gini yp (predicted y) Gini w (Gini yp with yp sorted by y) Structural mobility (GY-GYP) Exchange mobility (GYP-GW) Distributional Change (FGPþPERM) Distributional Change Income weighted

Index

Bootstrap Standard Error

z

Normal Based [95% Confidence Interval]

0.3537 0.2183 0.1240

0.0064 0.0035 0.0049

54.9 62.4 25.4

0.3410 0.2114 0.1145

0.3663 0.2252 0.1336

0.1354 0.0943 0.2296

0.0066 0.0032 0.0069

20.5 29.1 33.2

0.1224 0.0879 0.2161

0.1483 0.1006 0.2432

0.2971

0.0058

50.9

0.2857

0.3086

Source: MHBS 2006. Sample: Male heads of household in age 25–55. Welfare measure: Household consumption per capita per month.

Table 2.

reg1: reg2: reg3: reg4: reg5: reg6:

Relative Deprivation Indexes with Reduced Equations for the Estimation of the Predicted Values.

‘‘y ¼ ageþeducatþmaritalþsoc_groupþterritþurb_rur’’ ‘‘y ¼ ageþeducatþmaritalþsoc_groupþterrit’’ ‘‘y ¼ ageþeducatþmaritalþsoc_group’’ ‘‘y ¼ ageþeducatþmarital’’ ‘‘y ¼ ageþeducat’’ ‘‘y ¼ age’’

R2

JGP

JGI

0.2536 0.2534 0.0981 0.0749 0.0686 0.0000

0.2296 0.2298 0.2980 0.3112 0.3136 0.3538

0.2971 0.2972 0.3284 0.3343 0.3352 0.3537

Source: MHBS 2006. Sample: Male heads of household in age 25–55. Welfare measure (y): Household consumption per capita per month. Age: given in years. Educat: see Footnote 6. Marital: see Footnote 7. Soc_group: see Footnote 8. Territ: Dummies for administrative districts. Urb_rur: Dummies for urban areas.

application of the indexes to the study of gender bias in terms of wage deprivation. The introduction of the notion of reference group through the estimation of the predicted values allows the researcher to model empirically alternative assumptions about the identification of the reference group. For example, we could estimate relative deprivation based on the assumption that both men and women consider as a reference group both genders (joint predictions) and estimate the predicted values with one equation for both genders. Alternatively, we could assume that individuals compare themselves only with their own gender and estimate the predicted values

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Table 3.

JGP Males – Joint predictions Females – Joint predictions Males – Separate predictions Females – Separate predictions JGI Males – Joint predictions Females – Joint predictions Males – Separate predictions Females – Separate predictions

Relative Deprivation by Gender.

Index

Bootstrap Standard Error

z

Normal Based [95% Confidence Interval]

0.253

0.006

40.86

0.241

0.265

0.223

0.009

25.2

0.206

0.241

0.229

0.007

32.28

0.215

0.243

0.222

0.007

31.24

0.208

0.236

0.325

0.007

48.24

0.311

0.338

0.308

0.006

48.75

0.295

0.320

0.316

0.006

50.6

0.304

0.329

0.296

0.007

39.52

0.282

0.311

Deprivation Gap (((IndexM/IndexF)  100)100)

13.2

3.4

5.6

6.7

Source: MHBS 2006. Sample: Men and women in age 25–55 with salary W0. Welfare measure ¼ monthly salary. No. of observations: 1,548 males and 1,737 females.

separately for men and women (separate predictions). In Table 3, we report the estimations of the JGP and JGI indexes with the respective standard errors, z-scores, and confidence interval under the two assumptions described.12 According to the population-weighted index (JGP), males are more deprived than females and this is true whether we consider the joint or separate predictions. However, the gender gap (estimated as a ratio between the male and female indexes) is much higher if predictions are made jointly (13.2%) than separately (3.4%). In fact, if estimations are made jointly, the lower and upper bounds of the estimates for men and women are nonoverlapping providing a rather strong indication that the gender difference is very significant. Instead, the relative deprivation indices of the two genders are not significantly different when separate predictions are made, since the actual value of the index for one of the genders falls within the confidence interval of the index for the other gender.

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According to the income-weighted index (JGI), men are also more deprived than women but the difference this time is significant in both cases, that of joint and separate predictions, since the actual value of the index for one of the genders falls always outside the bounds of the confidence interval of the index for the other gender. What this exercise shows is that making different assumptions about gender selection of the reference group can lead to quite different estimates of relative deprivation. And making difference assumptions about the selection of the reference group is economically justified by the nature of the society under study. For example, it could be more appropriate to assume that in very conservative societies with low levels of female education and labor market participation, each gender derives its proper sense of deprivation from the comparison with members of the same gender. On the contrary, in modern societies with equal labor force participation across genders, it could be more appropriate to assume that men and women compare themselves with both genders. Ignoring considerations about the self-selection mechanism of the reference group could lead to very biased estimates of relative deprivation.

3.4. Relative Deprivation Over Time In this section, we look closer at the developments in welfare and inequality in Moldova between 2000 and 2005 and then check how the indexes proposed behave in describing changes in relative deprivation induced by changes in welfare and inequality. As already described, between 2000 and 2005, Moldova experienced rapid output growth estimated at around 7% per year on average. This growth was clearly translated in improved household living conditions and a sharp reduction in poverty between 2000 and 2003. However, starting from 2003, poverty reduction has stalled and household mean income declined rather than increased. This is explained by a combination of pro-rich growth patterns combined with a lack of growth in labor-intensive sectors and by a decrease in public and private transfers as compared to the period 2001–2003.13 The growth incidence curves (Ravallion & Chen, 2003)14 depicted in Fig. 1 show well how growth in household consumption turned from positive values across the distribution during the period 2000–2003 to negative and pro-rich values during the period 2003–2005. Between 2000 and 2001, all households have enjoyed strong growth in consumption and

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30

2001-2002

20

2000-2001

10

growth

2002-2003

0

2003-2004

-10

2004-2005

0

20

40

60

80

100

percentiles

Fig. 1.

Growth Incidence Curve – Household Expenditure per Capita (2000–2005). Source: Constructed from MHBS (2000–2005).

this growth has been rather evenly distributed. In this case, we should not expect major changes in inequality, although relative positions in welfare and rank within the reference group may have changed leading to a change in relative deprivation. Between 2002 and 2003, the growth incidence curve turned propoor and inequality declined, whereas relative deprivation may have followed a different path depending on how relative rank and consumption changed within the reference groups. Moreover, the selfdefinition of reference group may also be mobile over time causing a further effect on relative deprivation. From 2003 onward, the growth incidence curve becomes pro-rich and growth rates are for the quasi-totality of the distribution negative. During this period, inequality increases and average household consumption decreases. Consider now Table 4 where we report mean values (m) of our consumption measure (y), the Gini coefficient (Gyi), the Yitzhaki measure of relative deprivation (mGy), and our two alternative measures of relative deprivation (JGP and JGI).15 All measures are calculated using the same population group and welfare measure of the first example (male heads of household in age 25–55 and household consumption per capita).

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Table 4. Relative Deprivation over Time.

2000 2001 2002 2003 2004 2005

Mean Consumption

Gini

Yitzhaki RD

RD Population Weight

m

Gyi

mGyi

JGP

JGI

0.392 0.387 0.369 0.355 0.364 0.382

64.7 76.2 91.5 101.5 102.6 101.6

0.241 0.237 0.249 0.242 0.228 0.223

0.335 0.322 0.311 0.308 0.302 0.307

165 197 248 286 282 266

Base year: 2000 (¼100) 2000 100.0 2001 119.4 2002 150.3 2003 173.3 2004 170.9 2005 161.2

100.0 98.7 94.1 90.6 92.9 97.4

100.0 117.9 141.5 157.0 158.7 157.1

100.0 98.3 103.3 100.4 94.6 92.5

RD Income Subjective Weight Deprivation

100.0 96.1 92.8 91.9 90.1 91.6

DEPR 2.443 2.367 2.292 2.201 2.181 2.145 100.0 96.9 93.8 90.1 89.3 87.8

Source: MHBS 2000–2005. Sample: Males head of household in age 25–55. Welfare measure: Household consumption per capita per month.

In Table 4, we also report the average response to a question contained in the MHBS on living conditions asking respondents: ‘‘How do you assess your household living conditions?’’ Replies to this question included a one to five scale where one corresponded to ‘‘very good’’ and five to ‘‘very bad.’’ Due to the small number of observations for the answer ‘‘very good,’’ we grouped replies into three categories: (1) good or very good, (2) satisfactory, and (3) bad or very bad. We then calculated the annual average of this measure and reported it in Table 4 under the heading ‘‘DEPR.’’ This average can be regarded as a measure of average actual deprivation with increasing values depicting increasing deprivation. We can use this measure as a benchmark to compare the performance of our indexes vis-a`-vis the more traditional Yitzhaki measure of relative deprivation. In line with Fig. 1, real household consumption per capita increases between 2000 and 2003 very significantly and declines from 2003 onward showing a hump-shaped development. On the contrary, the Gini (Gy) declines between 2000 and 2003 and increases between 2003 and 2005 following a U-shaped trend. We observe therefore an inverse relation between growth and inequality. The actual measure of subjective deprivation (DEPR) follows a very different path from either growth or inequality. This measure declines

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continuously throughout the period, between 2000 and 2003 and also between 2003 and 2005. We cannot argue therefore that either growth or inequality alone can explain changes in subjective deprivation. We should expect instead that changes in subjective deprivation are the outcome of more complex mechanisms. Combining information on mean income and inequality as suggested by Yitzhaki in his measure of relative deprivation (mGy) bring us even further from the path followed by actual deprivation. The Yitzhaki relative deprivation measure is constantly increasing throughout the period with the exception of 2005 as opposed to actual deprivation (DEPR) that is constantly decreasing. This is explained by the fact that the Yitzhaki measure is equivalent to the Gini multiplied by mean income (Yitzhaki, 1979) and that these two components moved in opposite directions and at different speed between 2000 and 2005. Combining mean and inequality estimates by multiplying these two measures as suggested by Yitzhaki does not seem to improve the approximation of real subjective deprivation for a society. Instead, the JGP and JGI indexes that we proposed seem to follow actual deprivation much more closely than the Yitzhaki index. Both measures are continuously declining throughout the period with one exception for the JGP measure in 2001 and one exception for the JGI measure in 2005. The two measures are also more closely associated with real subjective deprivation than with the mean or the Gini taken separately indicating that they are not proxies of either of these measures. The time series used here is too short to provide conclusive remarks on the indexes proposed and we do not pretend to have found measures that can closely approximate real feelings of deprivation. However, we have some preliminary evidence that including a mechanism for the selection of the reference group into a measure of relative deprivation can lead to rather different conclusions and improved estimates on relative deprivation as compared to the more traditional Yitzhaki measure. This is an important insight for future research on the measurement of relative deprivation.

4. CONCLUDING REMARKS This chapter proposed a new approach to the measurement of relative deprivation. It suggested linking the extent of individual relative deprivation to the gap existing between individual actual and ‘‘expected’’ incomes, the latter being defined on the basis of basic individual characteristics such as

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age, education, marital status, the region where one lives, and other factors. These gaps between actual and ‘‘expected’’ incomes were then aggregated via a measure of distributional change that takes into account not only differences between actual and expected individual incomes but also differences between actual and expected individual ranks. When we applied this approach to the study of wage discrimination across genders, we found it useful to better understand how men and women may attain different degrees of deprivation depending on the reference group they consider. When we applied the proposed indexes to the study of deprivation over time, we found that these indexes are better suited than the Yitzhaki index to capture real deprivation and can provide rather different information on the evolution of deprivation over time. Such findings may thus vindicate Runciman (1966) intuition that ‘‘The questions to ask are first, to what group is a comparison being made? Second, what is the allegedly less well-placed group to which the person feels that he belongs?’’

NOTES 1. When the two distributions to be compared are income distributions at two different time periods, ‘‘distributional change’’ is in fact another name for ‘‘income mobility.’’ 2. Note that the structural mobility component is equivalent to the index proposed by Verme and Izem (2008) divided by mean income. 3. We could have also used an alternative breakdown where the first term would have been the reranking component comparing the actual ranking in the distribution {yi} with that the individual would have had in the distribution {yi}, had he kept his ranking in the distribution {yPi}. A Shapley-type decomposition procedure (see Shorrocks, 1999 and Sastre & Trannoy, 2002, for more details on this procedure) would take into account the two possible breakdowns. 4. In their analysis of mobility, Silber and Weber (2005) assume that exchange mobility may be considered as having a neutral, positive, or negative impact on individual welfare. We have in fact assumed here that as a whole exchange mobility has a negative impact on welfare and, as a consequence, it is positively correlated with deprivation. 5. Wired at http://www.jdsurvey.net 6. The results of the regression on the basis of which the indices presented in Table 1 were computed are given in Annex A1. 7. Categories include: (1) higher education, (2) technical colleges, (3) completed secondary, (4) incomplete secondary, (5) primary, (6) no primary, and (7) illiterate. The classification changed slightly in 2005 and 2006. The classification above was reconstructed using homogeneous categories.

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8. Categories include: (1) never married, (2) married, (3) widow, and (4) divorced or separated. The classification changed in 2004 and the classification above takes changes into account. 9. Categories include: (1) farmers, (2) hired workers in agriculture, (3) hired workers in nonagriculture, (4) self-employed, (5) pensioners, and (6) others. 10. One referee has argued that ‘‘comparing the real income with the expected one is perhaps not a good measure of the individual’s feelings of deprivation. For instance, let us assume that I am a black person, and according to my race my real income coincides with my expected income. Let us assume that the white people’s incomes are much higher. Nevertheless, according to the authors, I do not feel relative deprivation, because I receive what corresponds to my social group.’’ The question is therefore to correctly define the reference group. There is certainly no clear-cut solution and this is why in our empirical investigation we examined two possibilities, one where the reference group included (in addition to other characteristics) only individuals of the same sex and one where we assumed that individuals compared themselves with both males and females. 11. The question of estimation of the standard error for the Gini index has received considerable attention in recent years and several methods have been proposed. One possibility is to use the ‘‘Delta’’ method based on the central limit theorem. This is used for example by the statistical package DAD (see Duclos & Araar, 2006 for a description of possible applications to distributional indexes) and could potentially be extended to the first of our two indexes, but it is unclear how it could be used for the second index. A second method is the one proposed by Giles (2004) who shows that the standard deviation of the Gini can be obtained by simply estimating the P weighted least squares regression of i ¼ yþni where i ¼ rank and P y ¼ ni¼1 iyi = ni¼1 yi (the Gini index stripped of its constants). This method cannot be applied to our indexes because the weighted least square regression implies taking the square root of the unit values that in our case can be negative. A third possibility is to use bootstrap or jackknife estimations. These are simple to estimate and most statistical packages use readymade routines but they are computationally heavy. Very recently, Davidson (2008) reviewed the various methodologies and proposed an alternative method. This last paper also finds the bootstrap method to be a rather efficient estimator as compared with other methods. Based on the findings of this recent literature and on a small test, we opted to use a bootstrap method. Using our sample, we tested bootstrapping on the Gini index comparing the outcome of this method with the one of the Delta method in-built in the Stata DASP package prepared by Duclos and Araar (2006). We found bootstrapping to reach a very close approximation of the standard deviation derived from the delta method after only 50 replications and we finally decided to settle for this method. Naturally, this result applies to our sample, which is quite large. The estimation of the standard error of the indexes proposed for small samples should be reconsidered in the light of the discussion offered by Davidson (2008). 12. The results of the regression on the basis of which the indices presented in Table 3 were computed are also given in Annex A1. 13. For a discussion of welfare trends in Moldova during the period considered, see Verme (2008).

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14. The growth incidence curve plots the growth rate in household consumption by quantile with consumption sorted in ascending order. 15. Standard errors and confidence intervals for all measures are reported in Annex A2.

REFERENCES Berger, J., Zelditch, M., Anderson, B., & Cohen, B. (1972). Structural aspects of distributive justice: A status value formulation. In: J. Berger, M. Zelditch & B. Anderson (Eds), Sociological theories in progress (2, pp. 119–146). Boston: Houghton Mifflin. Berrebi, Z. M., & Silber, J. (1985). Income inequality indices and deprivation: A generalization. Quarterly Journal of Economics, C, 807–810. Bossert, W., & D’Ambrosio, C. (2006). Reference groups and individual deprivation. Economics Letters, 90, 421–426. Clark, A. E., & Oswald, A. J. (1996). Satisfaction and comparison income. Journal of Public Economics, 61, 359–381. Cowell, F. A. (1980). Generalized entropy and the measurement of distributional change. European Economic Review, 13, 147–159. Davidson, R. (2008). Reliable inference for the Gini index. Mimeo. Available at http://russelldavidson.arts.mcgill.ca/articles/ Duclos, J., & Araar, A. (2006). Poverty and equity. New York: Springer. Ferrer-i-Carbonell, A. (2005). Income and well-being: An empirical analysis of the comparison income effect. Journal of Public Economics, 89, 997–1019. Fields, G. S., & Ok, E. A. (1999). The measurement of income mobility: An introduction to the literature. In: J. Silber (Ed.), Handbook on income inequality measurement. Boston: Kluwer Academic Publishers. Frank, R. H. (2007). Falling behind. How rising inequality harms the middle class. Berkeley, CA: University of California Press. Giles, D. E. A. (2004). Calculating a standard error for the Gini coefficient: Some further results. Oxford Bulletin of Economics and Statistics, 66(3), 425–433. Heider, F. (1958). The psychology of interpersonal relations. New York: Wiley. Hey, J. D., & Lambert, P. J. (1980). Relative deprivation and the Gini coefficient: Comment. Quarterly Journal of Economics, XCV, 567–573. Kakwani, N. C. (1980). Inequality and poverty, methods of estimation and policy applications. New York: Oxford University Press. Kakwani, N. C. (1984). The relative deprivation curve and its applications. Journal of Business and Economic Statistics, 2, 384–394. Marx, K. (1847). Wage labour and capital (translated in 1891 by F. Engels, chapter on ‘‘Relation of wage-labour to capital’’. Available at http://www.marxists.org/archive/marx/works/ 1847/wage-labour/ch06.htm Ravallion, M., & Chen, S. (2003). Measuring pro-poor growth. Economics Letters, 78(1), 93–99. Runciman, W. G. (1966). Relative deprivation and social justice. London: Routledge and Kegan Paul. Sastre, M., & Trannoy, A. (2002). Shapley inequality decomposition by factor components: Some methodological issues. Journal of Economics (Suppl. 9), 51–89. Schaefer, R. T. (2008). Racial and ethnic groups (11th ed., p. 69). Harlow, UK: Pearson Education.

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Shepelak, N. J., & Alwin, D. F. (1986). Beliefs about inequality and perceptions of distributive justice. American Sociological Review, 51(1), 30–46. Shorrocks, A. F. (1999). Decomposition procedures for distributional analysis: A unified framework based on the shapley value. Mimeo, University of Essex. Silber, J. (1989). Factor components, population subgroups and the computation of the Gini index of inequality. Review of Economics and Statistics, LXXI, 107–115. Silber, J. (1995). Horizontal inequity, the Gini index and the measurement of distributional change. Research on Economic Inequality, VI, 379–392. Silber, J., & Weber, M. (2005). Gini’s mean difference and the measurement of absolute mobility. Metron – International Journal of Statistics, LXIII(3), 471–492. Smith, A. (1776). An Inquiry into the Nature and Causes of the Wealth of Nations. Reprinted as The Wealth of Nations. New York: Penguin Classics, 1986. Stouffer, S. A., Suchman, E. A., DeVinney, L. C., Star, S. A., & Williams, R. A. (1949). The American soldier (1). Adjustment during army life. Princeton, US: Princeton University Press. Verme, P. (2008). Social assistance and poverty reduction in Moldova 2001–2004. An impact evaluation. World Bank Policy Research Working Paper No. 4658. Washington, DC. Verme, P., & Izem, R. (2008). Relative deprivation with imperfect information. Economics Bulletin, 4(7), 1–9. World Bank. (1999). Moldova: Poverty assessment. World Bank Country Study No. 19926. World Bank. (2004). Recession, recovery and poverty in Moldova. The World Bank, Report No. 28024. Yitzhaki, S. (1979). Relative deprivation and the Gini coefficient. Quarterly Journal of Economics, XCIII, 321–324.

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ANNEXES Annex A1.

OLS Estimations for Predicting Income (Tables 1 and 3). Table 1

Age Education – primary Education – incomplete secondary Education – secondary Education – secondary professional Education – special vocational Education – higher

Marital status – married Marital status – Married unofficial Marital status – widow Marital status – divorced Labor status – employees in agriculture Labor status – employees in nonagriculture Labor status – self-employed Labor status – pensioners Labor status – others Rural areas Household size Territorial dummies Constant Observations R2

2.868 2.092 316.3 402.3 640.9 317.9 812.1 316.9 853.7 316.4 991.6 317.5 1197 318.8 317.9 113.3 382.0 179.4 52.78 166.6 132.8 142.7 45.99 56.92 122.7 50.07 504.9 163.9 25.97 83.36 252.2 58.79 436 509.6 No Yes 954.4 786.2 2248 0.254

Table 3

Table 3

Table 3

Gender – joint

Male

Female

5.129

8.432

1.939 339.7 679.2 263.4 622.2 172.7 622.1 16.11 621.8 81.37 622 429.2 622.4

3.206 600.9 1062 655.1 981.9 823.7 982 906.2 980.1 1045 981.3 1472 982.5

221.4 68.38 138.8 145.6 100 100.5 31.41 85.76 24.38 68.04 184.0 60.47 9.232 220.4 33.97 73.21 5.898 97.13 174.2 366.2 10.05 13.3 Yes 805.7 817 3285 0.278

288.2 105.7 124.9 212 190.8 231.1 150.8 173.6 1.628 135.6 237.1 128 232 591.1 45.76 146.9 15.8 226 2.386 677.9 29.97 22.42 Yes 276.2 1390 1548 0.350

0.0755 2.287 1235 854.8 1134 769.4 1063 770.4 962.8 770.6 816.3 770 503.6 770.5 165.3 88.53 20.55 199.9 154.5 112 134.7 99.51 82.87 76.38 41.69 64.49 30.34 208.1 86.8 77.43 21.37 95.56 278.3 382.7 9.144 15.66 Yes 1430 949 1737 0.292

Standard errors below coefficients. po0.01, po0.05, po0.1. Reference categories: education: preprimary; marital status: single; labor status: farmers.

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Annex A2.

Relative Deprivation Indexes (Table 4).

Coefficient

Bootstrap Standard Error

z

FGP 2000 2001 2002 2003 2004 2005

0.124 0.134 0.153 0.142 0.135 0.136

0.011 0.007 0.007 0.006 0.007 0.008

11.08 18.26 22.94 22.41 19.04 16.59

0.102 0.119 0.140 0.129 0.121 0.120

0.146 0.148 0.166 0.154 0.149 0.152

PERM 2000 2001 2002 2003 2004 2005

0.116 0.104 0.096 0.100 0.093 0.087

0.004 0.003 0.003 0.004 0.003 0.003

28.29 36.83 29.23 24.21 30.33 29.14

0.108 0.098 0.089 0.092 0.087 0.081

0.124 0.109 0.102 0.108 0.099 0.093

JGP 2000 2001 2002 2003 2004 2005

0.241 0.237 0.249 0.242 0.228 0.223

0.009 0.007 0.007 0.007 0.010 0.009

26.34 32.94 37.03 33.31 23.82 25.05

0.223 0.223 0.235 0.228 0.209 0.206

0.259 0.251 0.262 0.256 0.246 0.241

JGI 2000 2001 2002 2003 2004 2005

0.335 0.322 0.311 0.308 0.302 0.307

0.010 0.008 0.007 0.006 0.007 0.009

33.33 42.18 44.46 47.75 42.6 34.3

0.315 0.307 0.297 0.296 0.288 0.289

0.354 0.337 0.325 0.321 0.315 0.324

Gyi 2000 2001 2002 2003 2004 2005

0.392 0.387 0.369 0.355 0.364 0.382

0.010 0.007 0.008 0.008 0.008 0.008

38.76 52.46 48.69 45.16 46.73 46.11

0.372 0.373 0.354 0.339 0.349 0.366

0.411 0.402 0.384 0.370 0.380 0.398

Normal-Based [95% Confidence Interval]

CHAPTER 10 ECONOMETRIC IDENTIFICATION OF THE COST OF MAINTAINING A CHILD Martina Menon and Federico Perali ABSTRACT The chapter estimates the cost of maintaining a child, at different ages, the cost of being single, and the cost of additional adults present in a family, with the aim of making comparable the income levels of different households. The study investigates the issue of econometric identification of equivalence scales within a demand system modified to include demographic characteristics consistently with economic theory. It shows that a robust estimation of equivalence scales must take into formal consideration the problem of econometric identification. The estimate also puts forward all-encompassing demographic specifications to identify costs due to differences in needs, household lifestyles, and economies of scale.

1. INTRODUCTION Equivalence scales answer the question ‘‘what is the level of additional income needed by a family comprising two adults and a child compared to a family without children in order to enjoy the same level of economic Studies in Applied Welfare Analysis: Papers from the Third ECINEQ Meeting Research on Economic Inequality, Volume 18, 219–255 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2010)0000018013

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well-being?’’ If the demographic profile varies only in relation to the number of children, the equivalence scale corresponds to the cost of the characteristic ‘‘child’’ (Lewbel, 1989, 1991, 1993) associated with the presence of a child in the family; if the profile varies in relation to the number of elderly persons present in the family, then the equivalence scale represents the cost of an elderly person. The equivalence scale is an index that converts households of different composition into identical individuals thus making interhousehold comparisons possible. Welfare comparisons are implicitly made any time that we intend to establish, for example, whether a family is poorer than another or the most preferable policy scenario on the basis of the impact on a household’s well-being. The cost of household characteristics is therefore fundamental for the correct measurement of poverty and inequality and for the construction of indicators of the economic situation based on equivalent incomes, that is, incomes corrected for differences in household composition dividing by the equivalence scales. Further, the cost of living indices associated with household characteristics are an appropriate tool to account for household differences in designing tax schemes and to implement means tests incorporating fair criteria for accessing welfare programs. The estimation of equivalence scales has a special relevance in societies adopting a fiscal system based on household rather than individual incomes through the calculation of a quotient. In household-based fiscal systems, the tax brackets are calculated on equivalent incomes. This method incorporates the principle of horizontal equity recognizing that, at a given income, the larger household is relatively poorer and corrects the distortion implicit in fiscal systems based on separate taxation that penalizes the taxpayers, supporting a relatively large number of family members and families with a single wage earner. When the interest is to compare the costs associated with an individual-based fiscal system recognizing family allowances to account for differences in family composition and a fiscal system based on a quotient, we are in fact comparing a system adopting an equivalence scale expressed in absolute monetary terms and a system adopting an equivalence scale in relative terms. If the scale is measured correctly, the costs of the two systems should be similar. The validity of the comparison critically depends on the quality of the estimation of the scale. Another relevant measurement issue is estimation using stated or objective data. Recent studies carried out by Koulavatianos, Schro¨der, and Schmidt (2005, 2009) estimate the cost of a child by asking direct questions in an interview setting. Considering that both estimation strategies intend to

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estimate equivalence scales, it is crucial that the estimation techniques are as strong and accurate as possible if estimates are to be compared. These measurement concerns are the core of the motivation of the present study where the issue of the econometric identification of equivalence scales is addressed, not to be confused with the fundamental problem of identifying the cost of a child. The econometric identification issue addressed in this study concerns identifying the parameters of demographic-modifying functions used to estimate equivalence scales (Singh & Nagar, 1973; Muellbauer, 1977; Perali, 2003, p. 119). Interestingly, this problem is akin to the identification problem of the sharing rule within collective household models (Chiappori, Fortin, & Lacroix, 2002). The cost of a child should be thought of as the cost of maintaining a child (Ebert, 1997; Ebert & Moyes, 2003; Ebert & Moyes, 2009) as deduced from the expenditure for child necessities such as food, clothing, and housing. In line with Browning’s (1992) fundamental clarification of needs and expenditure related to the presence of children in a household, the cost of maintaining a child should be clearly distinguished from the cost of raising a child. The latter also includes other costs associated with nonnecessary expenditures for children, the value of time devoted by parents to children, and the value of other investments in child quality. For this reason, it is natural to think that the cost of raising or ‘‘producing’’ a child varies significantly with income, which is not necessarily the case for the cost of maintaining a child. While the costs of maintaining a child are useful to operate interhousehold comparisons, the estimates of the cost of raising children are appropriate to explain fertility choices and should not be used to operate interhousehold comparisons and to correct estimates of poverty and inequality. Estimating the cost of maintaining a child (cost of a child, here on) means comparing different households in different situations. Suppose, for example, that we are interested in comparing the cost of a child living in a poor household with the cost of a child living in a rich household. Certainly, a toy basket for a child in a poor household is smaller than the one for a child in a rich household. Furthermore, the content of the two baskets is very different. The basket of the less affluent family does not include expensive electronic toys, for example. The child living in a poor family does not take piano lessons. Also the clothing basket is likely to differ both in terms of size and quality. These considerations can be extended to other necessities such as food quality and the characteristics of the house in which they live. Rich parents, and their children, have more leisure. It follows that it is possible to compare children or people in rich and poor

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families, but it is crucial to confine the attention to ‘‘baskets’’ of similar dimensions containing necessary goods. In other words, it is fundamental to base comparisons only on expenditure for necessities comprising the cost for maintaining a child as we do in the present study by adopting a needs-based data selection rule. The objective of estimating equivalence scales strictly on a needs basis also involves dealing with two aspects. As shown by Blackorby and Donaldson (1991), the first concerns the fact that different households contain different numbers and types of people (adults, children, disabled people, and so on), and therefore have different preferences and needs. The second concerns the fact that there are economies of scale in household consumption due to public and semipublic consumption within the household. The study uses an extended concept of equivalence scales that models household heterogeneity controlling for differences in needs and differences in scale economies due to differing lifestyles and related household technologies or the sharing of household public goods. We also show that a modified cost function can host a collective specification (Chiappori, 1988, 1992; Chiappori et al., 2002). The proposed model encompasses demographic translating and scaling and Ray’s generalized scaling (Ray, 1996) where differences in needs are captured by a generalized scaling term and differences in economies of scale by demographic translating and scaling (Lewbel, 1985; Browning, Chiappori, & Lewbel, 2008; Lewbel & Pendakur, 2008). Therefore, the present study addresses the issue of econometric identification of the parameters of a demographically modified demand system used to estimate equivalence scales on a needs basis while separating differences in size from differences in economies of scale. The chapter is organized as follows: Section 2 introduces a general theoretical background at the basis of equivalence scales accounting for both differences in needs and household technologies. Section 3 presents the econometric specification of the encompassing demand system, focussing special attention on the demographic transformations capable of separating composition from household lifestyles and related technologies. Section 4 shows how the model is econometrically identified. Section 5 describes data and aggregation choices. Section 6 describes the estimation method, the results, and the appropriateness of the estimated cost of the characteristics associated with the presence of children of different ages, of additional adults, and of being single. The final section ‘‘Conclusions’’ discusses important practical aspects related to the econometric identification of a modified demand system and of the associated equivalence scales.

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2. AN EXTENDED THEORY OF EQUIVALENCE SCALES The equivalence scale is an index number that converts families of different composition into identical individuals accounting for the associated differences in needs. The scale depends on the quantity of ‘‘public goods’’ consumed by the family, which directly influences economies of scale, and on the distribution rule of both monetary and time resources within the family. Traditionally, it is assumed that resources are distributed equally across household members (Ebert & Moyes, 2003). It follows that comparing the cost of living of a given household, indexed with superscript 1, with the cost of a reference household, indexed with superscript 0, it is important to relate the estimates of the cost function to differences in lifestyle, scale economies deriving from the sharing of household public goods such as housing, and the rule governing the allocation of resources within the household. For example, a childless couple may belong to a young or old cohort. Likewise, the choice of a single or a couple as a reference household has relevant consequences in terms of lifestyles and associated household technologies, economies of scale, and sharing behavior. The cost associated with the characteristic d is therefore given by the ratio between the two cost functions keeping the level of utility, of prices, and of lifestyle constant: ESR ¼

Cðu; p; d 1 ; Z; s; fÞ Cðu; p; d 0 ; sÞ

(1)

where Z is the degree of public sharing of household goods, s the lifestyle, and f the intrahousehold rule governing the distribution of resources between adults and children. We assume that the reference household does not have economies of scale associated with the public sharing of the household goods and the number of members who enjoy the goods. The sharing rule between adults and children f for a reference household is not defined because there are no children. In comparing the reference and the given household, the lifestyle s is kept constant to ensure, for example, that the childless reference couple is of child-bearing age and does not have a lifestyle using household technologies, which is typical of elderly couples. Household economies of scale Z are produced by the public dimension of living together and increase proportionally to the size of the household (Lewbel & Pendakur, 2008). Some goods are fully public, as in the case of

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housing and heating; others are only partly public, such as listening to music either alone or in company, using the car to go to work or to go on vacation with the family, or the use of the telephone. Recently, Browning et al. (2008) have suggested estimating these economies of scale using scaling demographic functions a` la Barten (1964), which consider each good as potentially either private or public, to differing degrees. Cloth used only for one child may be considered private good. If they are handed down to a second child, to a certain extent, the cloth becomes public. It is as if the family had bought the clothes half price. In a sense, the family is getting more out of the same quantity of good. Similarly, eating with other members of the family rather than alone generates higher utility as if the food were of better quality and with a lower shadow price. The incidence of household economies of scale is also affected by the rule f, which governs the distribution of resources within the household (Perali, 2003; Arias, Atella, Castagnini, & Perali, 2003; Lise & Seitz, 2007). Further, it is also important to control the measurement of equivalence scales for differences in lifestyles s characterized by specific household technologies related, for example, to single- or double-income households, head of families employed in the public sector, and many other situations (De Santis & Maltagliati, 2003).

3. ECONOMETRIC SPECIFICATION OF THE DEMAND MODEL: THE ALMOST IDEAL QUADRATIC DEMAND SYSTEM MODIFIED A` LA LEWBEL–BARTEN–GORMAN This section describes the specification of a complete demand system allowing the researcher to identify equivalence scales under an econometric point of view. The model assumes that consumer preferences are PIGLOG (Gorman, 1976; Muellbauer, 1974; Deaton & Muellbauer, 1980) at the basis of the almost ideal demand system. The base model, which is linear in the logarithm of income, can be extended to a quadratic specification in the logarithm of income (Banks, Blundell, & Lewbel, 1997) if the model is applied to data that are sufficiently nonlinear. The base specification is expressed in terms of prices and income and must be extended to host other exogenous factors affecting demand such as demographic characteristics (Lewbel, 1985).

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3.1. The Demographic Transformation Describing Equivalence Scales and Household Technologies In general, demographic characteristics in modified cost functions interact multiplicatively both with prices and income while maintaining the theoretical plausibility of the model (Lewbel, 1985). The interaction with prices captures the Barten substitution effects (Barten, 1964).1 The interactions with income can involve only demographic characteristics or can involve a function of both prices and demographics. In this case, the function describes fixed costs (Gorman, 1976), which represent the sum of the values of the quantities committed to guaranteeing the household survival in cases of a full loss of earnings, and generate at the demand level a demographically varying translating term. Using the notation introduced by Lewbel (1985) and dividing the set of demographic characteristics into two subsets D ¼ (r, d), the following demographically modified cost function can be defined: y ¼ Cðu; p; r; dÞ ¼ f ½C n ðu; pn Þ; p; r; d and pni ¼ pi mi ðr; dÞ, where y ¼ C(u,p;r,d) is the income corrected by the equivalence scale, y ¼ C(u,p) is the observed income, and mi(r,d) is the Barten price scaling function for each good i. The functions miZ0 for all i and strictly positive for at least one i, and f are continuous and at least twice differentiable. The function f describes the interactions between both demographic variables and total expenditure, while all the f and mi functions allow interactions of demographic variables with prices. As shown by Lewbel (1985, Theorem 8), a plausible form for the modifying function f is: y ¼ yn ½BðrÞPT ðpn ; dÞ ¼ yn P T ðpn ; r; dÞ with y P T ðpn ; r; dÞ

The term P T ðpn ; r; dÞ ¼ BðrÞPT ðpn ; dÞ represents a subcost function comprising Lewbel’s income scale component B(r) and Gorman’s fixed cost component PT ðpn ; dÞ, which also scales income at the cost function level. At the level of related demand, the fixed cost term generates the translating demographic function that has only demographic characteristics as arguments. Gorman’s (1976) translating component PT ðpn ; dÞ depends on yn ¼

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prices and the demographic characteristics d of the household. According to this transformation, the Gorman effect PT(p,d) represents a price index that interacts with the income scaling term B(r) and controls for regional differences or for other household characteristics d not related to household composition r. Function B(r) is independent of prices. It includes variables related to household composition r from which the cost of household characteristic is derived. Note that rAB and dAPT, otherwise it is not possible to econometrically identify equivalence scales as shown in Section 4. Remarkably, the income scaling function B(r) is analogous to the sharing rule of collective household models (Chiappori et al., 2002). The question addressed in this study is the following: given the choice of the demographic transformation f, can we identify the parameters associated with the function B(r) separating differences in size from economies of scale Z captured by Barten prices pn and lifestyle effects s as described by the translating term PT ðpn ; dÞ? If the modified cost function y ¼ C(u,p;r,d) is known because all the demographic parameters are identified, it is then possible to derive the equivalence scale associated with the cost of living index related to demographic characteristic r (Pollak, 1989; Lewbel, 1991; Lewbel, 1997): ESR ¼

y1 Cðu; p; r1 ; d 0 ; Z; s; fÞ Cðu; pn ; d 0 ÞP T ðpn ; r1 ; d 0 Þ ¼ ¼ y0 Cðu; p; r0 ; d 0 ; sÞ Cðu; pn ; d 0 ÞP T ðpn ; r0 ; d 0 Þ

Equivalence scales suffer from a fundamental identification problem. Different equivalence scales can be consistent with the same preferences described by observing consumer behavior (Pollak & Wales, 1979; Pollak, 1991; Perali, 2007). This indeterminacy implies that comparisons can change arbitrarily and the observation of consumer behavior is not sufficient to learn anything about interhousehold comparisons. The same conditional demands q(p,y|r,d) can be derived from the class of cost functions Cðu; p; r; dÞ ¼ Gðu; p; dÞm0 ðp; r; dÞ, where Gðu; p; dÞ ¼ minfp0 qjI½U 0 ðqjr; dÞ; r; d  ug and Iðu; r; dÞ is any function monotone in u such that Uðq; r; dÞ ¼ I½U 0 ðqjr; dÞ; r; d. It follows that different equivalence scales are consistent with the same preferences, and equivalence scales are not identified. In the case of nonconditional preferences, demographic attributes affect the utility function directly through the term Iðu; r; dÞ and indirectly affect consumption choices and the level of direct utility U 0 ðqjr; dÞ (Perali, 2003). The presence of a child in the household induces a reallocation of

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expenditures if the level of income does not change, but the presence of a child per se affects the level of utility of the household, positively when the child smiles and negatively when the child cries. From consumption data, it is possible to identify only conditional preferences. Nonetheless, the fundamental identification problem does not imply that equivalence scales cannot be estimated uniquely. A household equivalence scale, or cost of characteristics index, is independent of the choice of the income or utility level on which interpersonal comparisons are based, namely base-independent IB (Lewbel, 1989, 1991) or exact (ESE) (Blackorby & Donaldson, 1991), if it depends on prices and demographic characteristics, but not on the income level chosen to make interhousehold comparisons. If two adjacent Engel curves referring to household typologies differing by a single characteristic are shape invariant (Pendakur, 1999; Perali, 2003), then the two Engel curves are also parallel. Equivalence scales are therefore exact in the sense that they are independent of the income level chosen for comparisons. It is important to stress that the IB/ESE property can be rejected, but the analysis of the observed demands is not sufficient to confirm the IB/ESE hypothesis because it is not possible to test if monotonic transformations are independent of household characteristics. As a consequence of the IB/ESE property, it is possible to separate the cost function in a subcost function G(u,p), equal for all households facing same prices, and in a function m0 ðp; r; dÞ grouping all demographicmodifying functions such as the Barten price scaling function m(p, d), Gorman’s translating function PT ðpn ; dÞ, and Lewbel’s income scaling function B(r) (Ferreira & Perali, 1992): y ¼ Cðu; p; r; dÞ ¼ Gðu; pÞ mðp; dÞ P T ðpn ; r; dÞ ¼ Gðu; pÞ m0 ðp; r; dÞ From this expression, if we deflate household income y by the (equivalence) scale factor m0 ðp; r; dÞ summarizing the needs specific to each household, we obtain a cardinal money measure of welfare u that is fully cardinally comparable (Perali, 1999, 2003): y ¼ Gðu; pÞ m0 ðp; r; dÞ corresponding to the definition of equivalent income. Being cardinally comparable, this measure is appropriate to implement interhousehold comparisons that are commonly made when identifying the beneficiaries of welfare policies or when measuring poverty and inequality.

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Thanks to the possibility to separate demographic information and the interaction terms between prices and demographic variables from the subcost function G(u,p), an IB/ESE equivalence scale can be written independently of the level of utility u chosen as reference: ESR ¼

y1 Cðu; p; r1 ; d 0 ; Z; s; fÞ ¼ y0 Cðu; p; r0 ; d 0 ; sÞ

¼

Gðu; pÞmðp; d 1 ÞP T ðpn ; r1 ; d 0 Þ Gðu; pÞmðp; d 0 ÞP T ðpn ; r0 ; d 0 Þ

¼

mðp; d 1 ÞP T ðpn ; r1 ; d 0 Þ mðp; d 0 ÞP T ðpn ; r0 ; d 0 Þ

The IB/ESE property allows equivalence scales to be recovered uniquely, but it does not contribute to solving the fundamental (economic) identification problem of equivalence scales because it does not add information related to nonconditional preferences. It is worth remarking that the above equivalence scale is independent of the choice of the reference household (Ebert & Moyes, 2003) because the ranking order of the distribution of welfare, expressed in terms of equivalent incomes, does not change as the reference household type changes. By separating the source of heterogeneity related to household size and composition r, appearing only in the term P T ðpn ; r; dÞ and not in mðp; dÞ, from other household characteristics d, we can control comparisons between households not only on the same price basis but also on the basis of similar demographic characteristics. These properties are incorporated in the specification of the demand system presented in the following section.

3.2. Specification of the Demand System Assume that the indirect utility function of the household is of the extended PIGLOG form (Banks et al., 1997): " #1  ln yn  ln aðp; dÞ 1 þ lðp; dÞ (2) lnVðy; p; r; dÞ ¼ bðp; dÞ where a(p,d) and b(p,d) are price aggregator functions and the logarithm of total expenditure is specified a` la Lewbel–Barten–Gorman as discussed before: ln yn ¼ ln y  ln P T ðpn ; r; dÞ

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and P T ðp; r; dÞ ¼ BðrÞPT ðpn ; dÞ The term l(p,d) is a differentiable function homogeneous of degree zero in prices p. When this function is independent of both prices and demographic characteristic d, the AIDS model is obtained, linear in income. Prices are scaled using Barten’s (1964) technique to obtain the shadow prices: ln pnj ¼ ln pj þ ln mj ðdÞ The vector of demographic characteristics d is an argument of the scaling function mj(d) describing household technology a` la Barten. Shadow prices correspond, in the dual space, with shadow quantities qnj ¼ qj =mj ðdÞ. The value of the scaling function mj ðdÞ ¼ qj =qnj reveals the individual differences in transforming the consumption of a certain good in utility units. The transformation technology differs both between households and between individuals in the same family. The income committed to survival, when, for example, a household head loses his/her job, is a fixed cost that translates the income comprising the sum of basic expenses for the single goods tj(d): ln P T ðp; r; dÞ ¼ ln BðrÞ þ ln PT ðpn ; dÞ ¼

R X

rk rk þ

k¼1

with BðrÞ ¼ exp

R X k¼1

! r k rk

and PT ðpn ; dÞ ¼ exp

N X

tj ðdÞ ln pnj

j¼1

N X

!2 tj ðdÞ ln pnj

j¼1

It is relevant to note that changes in lifestyles s and economies of scale Z are captured by the presence of demographic control variables transforming prices and incomes by means of household technologies a` la Barten and Gorman (Bollino, Perali, & Rossi, 2000; Perali, 2003). Further, note that this encompassing specification of the equivalence scale unifies the approach by Blacklow and Ray (1998), Lancaster, Ray, and Rebecca (1999), and Ray (1983) who use only the function B(r) and by Blundell and Lewbel (1991), Lyssiotou (1997, 2003), Pashardes (1995), and Phipps (1998) who estimate the scale by specifying only the translating term PT(p,d). The term related to fixed costs ln PT(p,d) is homogeneous of degree zero in p. Analogously to the Slutsky decomposition into the substitution and income effect, the household technology a` la Barten–Gorman, modifying the effective prices through the scaling substitution effect, rotates the budget

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constraint and translates the expenditure through the translating fixed cost effect. The equivalence scale function, B(r), scales both fixed costs and total expenditure at the same time. The cost function associated with the indirect utility function (2) is:   vðuÞbðp; dÞ (3) þ ln P T ðp; dÞ ln Cðu; p; r; dÞ ¼ ln aðp; dÞ þ 1  vðuÞlðp; dÞ In the tradition of the literature on demographic modifications of demand systems (Lewbel, 1985), prices are scaled, whereas income is translated. In the demographic transformation adopted in the present study, income is both scaled by the term B(r) to estimate the equivalence scale and translated by the term PT(p,d). The translating method used to introduce demographic information in the demand system is IB by construction (Perali, 2003). Considering that the main objective of the study is the estimation of the equivalence scale and not the estimation of household technologies and economies of scale associated with the different degree of sharing the public goods described by the interactions of demographic effects with prices, the Barten transformation has not been adopted here. In line with our objectives, it is important to concentrate on the issue of identifying the parameters of the B(r) income scaling function separately from the issue of identifying the price scaling function m(p,d). The latter issue has been already considered in the literature (Muellbauer, 1977; Ferreira & Perali, 1992; Perali, 2003). Hence, p ¼ p. The price aggregator is specified as a Translog function: ln aðpÞ ¼ ai þ

N X

gj ln pj þ

j¼1

N X N X

gij ln pi ln pj

i¼1 j¼1

while the price function is Cobb–Douglas: bðpÞ ¼

N Y

b

pi i

i¼1

The term l(p,d) is also independent of demographic characteristics because prices are not modified by demographic variables: lðpÞ ¼

N Y

pli i

i¼1

On the other hand, the translating demographic transformation is maintained.

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The application of Roy’s identity gives the system of demand equations expressed in shares: wi ¼ ai þ ti ðdÞ þ

N X

gi ln pi þ bi ½ln yn  ln aðpÞ þ

i¼1

li ½ln yn  ln aðpÞ2 bðpÞ (4)

Relationship (4) describes the specification of the estimated demand system. When the IB/ESE property is imposed, the equivalence scale for the QAIDS demand system demographically modified using Gorman translating described in Eq. (4) is the same regardless of the linear or quadratic in income specification: ESIB ðu; p; r; dÞ ¼ ¼

Cðu; p; r1 ; d 0 Þ Gðu; pÞ m0 ðp; r1 ; d 0 Þ ¼ Cðu; p; r0 ; d 0 Þ Gðu; pÞ m0 ðp; r0 ; d 0 Þ P T ðp; r1 ; d 0 Þ Bðr1 ÞPT ðp; d 0 Þ Bðr1 Þ ¼ ¼ P T ðp; r0 ; d 0 Þ Bðr0 ÞPT ðp; d 0 Þ Bðr0 Þ

ð5Þ

It should be remembered that where Barten substitution effects are absent, as in our case, the equivalence scales derived uniquely from the translated demographic effects and from the income scaling function is IB by construction. Given the specification of the equivalence scale adopted in the estimation, the cost of the characteristic index is obtained as follows: Cðu; p; r1 ; d 0 Þ Bðr1 Þ ¼ expðrk r1k Þ ¼ Cðu; p; r0 ; d 0 Þ Bðr0 Þ

(6)

where Bðr0 Þ ¼ 1 because for the reference family, being a childless couple, r0 ¼ 0 to ensure that the equivalence scale is independent of the choice of the reference household. Now the parameters of the generalized income scale term B(r) can be identified.

4. ECONOMETRIC IDENTIFICATION OF THE EQUIVALENCE SCALES Dealing with equivalence scales, it is important to distinguish between the issue of econometric identification (Lewbel & Pendakur, 2008) and the fundamental identification problem raised by Pollak and Wales (1979), due to the fact that two families with similar characteristics and

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conditional preferences in relation to characteristics for the consumption of goods may have different unconditional preferences and equivalence of scales. This section is concerned with the source of the econometric identification problem associated with the estimation of a demographically modified demand system a` la Lewbel–Gorman [Eq. (4)]. The system of Eq. (4) is reproduced here in the simplified linear version of an AIDS demand equation: wi ¼ ai þ tðdÞ þ bi ½ln x  BðrÞ  PT ðp; dÞ ¼ ai þ di d þ bi ½ln x  rr  di ðd ln pi Þ where tðdÞ ¼ di d; BðrÞ ¼ rr; PT ð p; dÞ ¼ di ðd ln pi Þ and wi denotes the budget share of good i ¼ 1, 2, p is the vector of associated market prices, x is total expenditure, d denotes a household characteristic such as the age of the household head, and r denotes a characteristic such as family size. The econometric identification of equivalence scales can be extended to the case of a demand system quadratic in total expenditure following the same line of proof described here for the linear case that we chose for the sake of clarity. We are not proposing the proof for the quadratic system, because it does not add useful information to the comprehension of the estimation problem. The number of goods in the basket has been limited to two for the sake of simplicity. The objective is to verify the identifying conditions for the parameter r argument of the scaling function B(r). Interestingly, this line of proof is similar to the one used by Chiappori et al. (2002) to prove the identification of the parameters of the sharing rule, because the sharing rule is a function that scales income as the function B(r) does and in Perali (2003, p. 119) for the identification of the demographic parameters of a Barten–Gorman model. Proposition 1. Given a structural functional form and the corresponding reduced form, both continuously differentiable, if there exists a one-toone correspondence between the elements of the Jacobian matrices, or the Hessian matrices, of the structural and the reduced form, then the demand equation wi is the solution of the utility maximization program and all parameters of the demand equation are identifiable.

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Proof. Consider the following functional structural specification and the associated reduced form Structural form w1 ¼ a1 þ d1 d þb1 ½ln x  rr  d1 ðd ln p1 Þ w2 ¼ a2 þ d2 d þb2 ½ln x  rr  d2 ðd ln p2 Þ

Reduced form w1 ¼ a0 þ a1 d þ a2 r þ a3 ln p1 þa4 d ln p1 þ a5 ln x w2 ¼ b0 þ b1 d þ b2 r þ b3 ln p2 þb4 d ln p2 þ b5 ln x

The one-to-one correspondence between the coefficients of the structural and reduced forms is found by differentiating the elements of the Jacobian matrix of the unrestricted reduced form and the elements of the Jacobian matrix of the structural form that describes the theoretical restrictions that link the reduced to the structural form. We begin by differentiating the first demand equation w1 as follows: Structural form @w1 =@d ¼ d1  b1 d1 ln p1 @w1 =@r ¼ b1 r @w1 =@ ln x ¼ b1 @w1 =@ ln p1 ¼ b1 ðd1 dÞ

Reduced form @w1 =@d ¼ a1 þ a4 ln p1 @w1 =@r ¼ a2 @w1 =@ ln x ¼ a5 @w1 =@ ln p1 ¼ a3 þ a4 d

Because the first and second elements of the Jacobian matrix are not linear in the parameters, we proceed with the second derivatives. The two nonzero elements of the Hessian matrix are equal to: Structural form @w1 @w1 ¼ ¼ b1 d1 ð@d@ ln p1 Þ ð@ ln p1 @dÞ

Reduced form @w1 @w1 ¼ ¼ a4 ð@d@ ln p1 Þ ð@ ln p1 @dÞ

Equating the corresponding elements of the Jacobian and Hessian matrices and solving for the structural parameters, we have b1 ¼ a 5 ;

r¼

a2 ; a5

d1 ¼ 

a4 a5

where the parameters of the equivalence scales in the structural form are a function of the parameters identified in the reduced form, and hence the former are identifiable as well.

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Differentiating twice the second demand equation w2, we obtain the following relationships: Structural form @w2 @w2 ¼ ¼ b2 d2 ð@d@ ln p2 Þ ð@ ln p2 @dÞ

Reduced form @w2 @w2 ¼ ¼ b4 ð@d@ ln p2 Þ ð@ ln p2 @dÞ

from which we derive the following identifying conditions: b2 ¼ b5 ;

r¼

b2 ; b5

d2 ¼ 

b4 b5

The overidentifying condition of the parameter associated with the equivalence scale is r¼

a2 b2 ¼ a5 b5

Inspection of the above conditions suggests the following remarks. Remark 1. Note that r ¼ a2 =a5 ¼ a2 =b1 ¼ b2 =b2 . It follows that the estimation of the parameter associated with the equivalence scale is inversely proportional to the size of the income parameter. It is therefore important, during econometric execution, to verify the effects related to the choice of the price level and of the deflating term ln a(p) on the dimension of the parameter associated with income, and, as a consequence, on equivalence scales. This effect is similar to one reported by Pashardes (1993) in relation to the distortion generated on the parameters associated with the use of the Stone index in substitution of the term ln a(p) that introduces nonlinearities in the estimation parameters. The problem is exacerbated in the quadratic specification because the deflating effect of the term ln a(p) can exert a strong scaling effect on the level of the income parameter. Remark 2. The identification proof shows that the demand system is estimable both in the structural and in the reduced form by estimating the reduced form in the first stage and obtaining in the second stage the parameters of the structure by applying the derived restrictions. This estimation technique is known as minimum distance estimation (MDE).3 Remark 3. Note that if the translating term describing Gorman fixed costs PT(p,d) includes the same variables related to household composition r present also in the term B(r), that is, if it is specified as PT(p,d,r),

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then the parameter associated with the variable r is not identifiable. Proof of this assertion follows the demonstration line provided in Proposition 1 step by step and therefore is not repeated here. As emphasized in Remark 1, the demonstration shows the importance of the dimension of the income parameter in determining the size of the equivalence scale. It is therefore critical to verify that the estimation of the parameter associated with income is stable and not biased, for example, by endogeneity problems of the income variable or specification problems of the price aggregator term ln a(p) when it acts as income deflator.

5. DATA DESCRIPTION The estimation of the complete demand system uses the household budgets collected by the Italian National Statistical Institute (ISTAT) in 2002. After excluding observations with household heads older than 65, the sample, corresponding to 30.75% of the total, comprises 19,045 observations. The households with heads of over 65 years of age have been excluded because they have significantly different expenditure behavior compared with younger households. The composition of expenditures includes expenditure flows and excludes expenditures on durable goods and expenditures on home rent. The purchase of durable goods is not frequent, though durables are used everyday like other goods. There is often a large degree of error in measuring the quantity and value of the daily use of durable goods. The inclusion of durables therefore may introduce a significant distortion in estimated parameters. Housing rent was excluded because the inclusion of rental values for homes in ownership could introduce significant distortions in the composition of expenditure for housing especially if one considers that 72.4% of the sample lives in their own home. As a consequence of this choice, the estimated demand system is conditional on the decisions made about the consumption of durable goods and housing.4 The conditioning has been modeled including dichotomic variables for the presence of home ownership. The equivalence scale describes the differences in the cost of living associated with the different socioeconomic characteristics of the households and with the related different levels of necessities. Hence, in line with the definition of the cost of maintaining a child, the aggregation of expenditure items in groups includes only the necessity components (Phipps, 1998) with the exception of the category ‘‘other goods.’’ For example, the

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group of food items does not include the expenditure for food away from home, and clothing does not include expenditures for furs and other luxury goods. The unnecessary components have been included in the category ‘‘other goods.’’ This aggregation allows for the computing of absolute equivalence scales corresponding to the difference between the cost of living of a given household and a reference household expressed in terms of the expenses for all necessary goods. Expenditures for necessities do not vary significantly as the level of income increases. The following table shows the components for each expenditure group. Interestingly, a similar reclassification of the basket of goods based on a basic needs approach has been adopted by ISTAT (2009) for the measurement of absolute poverty. Good categories included in the analysis

Goods excluded because not necessities and included in the category ‘‘other goods’’

Food Basic housing expenditures

Food away from home Gardeners, butlers

Basic clothing expenditures Basic transport and communications expenditures Basic education, recreation, and health (dentistry and medicines) expenditures Other goods and luxury goods

Furs Air tickets, taxi, and house removals Traveling abroad

Goods excluded because durables

Repairs, refurbishing, expenditures for second houses, and purchase of technologies Purchases of cars, cycles, motorcycles, and telephones Purchases of boats, cars, eyeglasses, and prothesis Silverware, radio, computers, and cameras

A limitation of ISTAT household budgets is the lack of information about quantities consumed by each household. If quantities and expenditures were known, then it would be possible to derive the associated household specific prices as unit values. This information is fundamental for demand studies aiming to estimate the welfare and utility levels necessary to derive cost functions for the estimation of equivalence scales. It is therefore necessary to estimate the unit values with an alternative procedure described in Atella, Menon, and Perali (2003) and Hoderlein and Mihaleva (2008).

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This technique attributes to ISTAT monthly price indices, published at the provincial level, the variability of unit values that incorporates the spatial differences in prices, and the objective and subjective differences in the quality of goods as they can be deduced from the socioeconomic characteristics of the household. The set of demographic characteristics D is divided into two subsets D ¼ {r, d}. The subset r includes the household characteristics for the estimation of equivalence scales: ri ¼ {r1 (number of children less than 5 years old), r2 (number of children of age between 6 and 13), r3 (number of adolescents aged between 14 and 18), r4 (number of adults beyond the couple), and r5 (single)}, with associated parameter vector qi ¼ (r1, y , r5). The complement subset d includes the demographic variables, ds ¼ {d1 ( ¼ 1 if resident in the north), d2 ( ¼ 1 if resident in the south or islands), d3 (age of the household), d4 ( ¼ 1 if the household head is an employee), d5 (level of education of the household head classified as low, average, or high), d6 ( ¼ 1 if the household lives in rural areas), d7 ( ¼ 1 if the wife is employed), and d8 ( ¼ 1 if the house is owned)} with associated vector of demographic parameters di ¼ (d1, y , d8). The reference household is a household living in the Center of Italy, which is the region excluded from the estimation, for which all variables of the r and d subset take the value of 0. In this construction and the chosen functional form, the value of the cost subfunction m(p,r0,d0) in Eq. (5) is equal to 1 and the independence of the equivalence scale from the chosen reference household is maintained. The budget shares for male, female, and children clothing, for education, and health are censored in a nonnegligible size. The proportion of zero outcomes is the following: male clothing 56%, female clothing 53%, children clothing 43%, nonassignable clothing 48%, education and recreation 16%, and health 49%. The realization of zero expenditures is in part explained by the short duration of the recall period of the survey design and in part by budget constraints (Pudney, 1990). The choice of the recall period for the expenditure of semidurable goods is one of the most important problems encountered by researchers designing an expenditure survey (Grosh & Glewwe, 2000). An excessively long recall period can lead to underestimation of the real expenditure. Given that the zero generating process is of a different nature for each expenditure category, the specification of bivariate models has been studied ad hoc for each censored expenditure category (Blundell & Meghir, 1987). Total expenditure computed from the attributed expenditures of the censored goods corrects in part also the measurement error often responsible for the endogeneity of total expenditure. Despite the correction for those measurement errors

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stemming from the lack of continuity in purchases, the endogeneity problem of total expenditure persists. According to the results of the Hausman-Wu test, total expenditure is endogenous with respect to all budget shares but for housing. Total expenditure has been corrected using the technique shown in Mroz (1987). Table A1 shows the definition of the variables used in the econometric analysis along with their mean, standard deviation, and minimum and maximum values. The subsequent descriptive tables illustrate the consumption habits of Italian households, expressed in terms of aggregate goods, since household typology, income, family size, and macroregion vary. Table A2 shows the different consumption habits of Italian household types. As can be reasonably expected, the comparison between consumption patterns reveals very different lifestyles. For example, it is interesting to note the differences between young and old couples without children. While young couples without children have a lower food share than couples with children, elderly couples spend more money on food than couples with children. This lack of monotonicity in the increase of the food share as family size gets larger is an apparent contradiction to the second Engel law demonstrating that, at the same level of household income, a large household has a higher food budget share (Perali, 2008). Furthermore, elderly childless couples have the lowest transportation and communications share of all household types. Because of these differences in consumption patterns, it is very important to control for ‘‘lifestyle’’ effects both in the specification of the econometric model and in the derivation of the household equivalence scale that should be independent of the choices both of the reference household and its lifestyle. If food share is considered a reliable indicator of welfare, plausibly monoparental households are relatively poorer. On the other hand, those who live alone and couples without young children are the household types with the highest budget shares on luxury goods. Table A3 shows the variation of the consumption shares by quintile of total expenditure. The first row related to the food share is in line with the first Engel law. Household expenditure, on the other hand, counts for relatively more in the budget of less affluent families. Expenditures for transportation and communications, clothing, and education, recreation, and health do not vary significantly as the level of income changes. This evidence is not surprising if we consider that the chosen expenditures are for necessary goods with the exception of the ‘‘Other Good’’ category, which includes luxury goods, which increases considerably with income.

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Table A4 shows the variation in consumption shares associated with household size and the macroregion of residence. The level of the food share for the North, Center, and South of Italy is in line with the second Engel law with the exception of the transition from the childless couple to the couple with one child because of the effect related to the different lifestyle of the young and elder childless couplet previously shown. Looking at the table, moving from the North to the Center and the South of Italy, we see that the level of the shares increases in line with the first Engel law, because the income levels decrease with latitude. In the North, housing expenditures are relatively higher because of heating. Similar considerations can be extended for the other goods. The presence of one or more children significantly changes both the household organization and the consumption pattern. The horizontal difference across levels of shares varies with the number of children and expresses a rough measure of household scale economies. These are present, as it is reasonable to expect, especially for housing and clothing expenditures. However, the effect is minimal. This is not surprising because only 3.7% of the sampled households have more than two children. Tables A2–A4 show the importance of conducting an estimation conditioning both for differences in lifestyles s and for the presence of scale economies Z. The possibility to control the estimates of equivalence scales for the rule governing the intrahousehold allocation of resources f as well requires a dedicated study, but is in principle estimable.

6. ESTIMATION METHOD AND RESULTS The adopted estimation technique is maximum likelihood. The share omitted to avoid singularity of the variance–covariance matrix is the other goods share. The demand system has been estimated with the restrictions of homogeneity and symmetry as maintained hypotheses and the conditions guaranteeing the econometric identification of equivalence scales as shown in Section 4. The joint estimation of the complete demand system comprising six goods ‘‘food, housing, clothing, transportation and communications, education, recreation and health, and other goods,’’ which exhaust total expenditure, was carried out using the quadratic specification of income transformed with the translating demographic modification corresponding to relationship (4). Table 1 shows estimated parameters. They are in general significantly different from zero. The statistical significance of the demographic parameters in most equations is evidence in favor of the importance of

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Table 1.

Estimated Parameters of the QAIDS Model with P T ¼ BðrÞPT ðp; dÞ. Food

Intercept Lnp(food) Lnp(Housing)

Housing

Transport and Communications

Clothing

Health, Education and Recreation

2.6494 0.0287 0.0495 0.0442 1.0396 0.1451 0.0408 0.0206 0.0398 0.0062

1.1740 0.0583 0.6543 0.0287 0.0481 0.0125 0.5554 0.0337

0.1165 0.0266 0.0062 0.0118 0.0374 0.0019 0.0079 0.0078 0.0601 0.0013

0.4915 0.0764 0.0127 0.0118 0.0230 0.0081 0.0008 0.0008

0.3291 0.0155 0.0193 0.0010

0.0143 0.0072 0.0022 0.0005

0.0629 0.0222 0.0114 0.0098 0.0209 0.0015 0.0060 0.0065 0.0054 0.0012 0.0197 0.0015 0.0152 0.0059 0.0013 0.0004

Lnp(transp&commun.) Lnp(clothing) Lnp(edurecr) Lnx (Lnx)2

R1 R3 Rural Age_cl Edu_cl Ts L_indj Ownership

Commodity-specific demographic function 0.0210 0.0130 0.0158 0.0020 0.0018 0.0011 0.0016 0.0006 0.0155 0.0152 0.0019 0.0117 0.0019 0.0011 0.0017 0.0007 0.0126 0.0062 0.0061 0.0013 0.0017 0.0011 0.0016 0.0006 0.0237 0.0100 0.0016 0.0094 0.0010 0.0006 0.0009 0.0003 0.0128 0.0038 0.0026 0.0036 0.0010 0.0007 0.0010 0.0004 0.0056 0.0031 0.0013 0.0047 0.0014 0.0009 0.0013 0.0005 0.0097 0.0022 0.0010 0.0009 0.0017 0.0011 0.0016 0.0006 0.0021 0.0033 0.0014 0.0021 0.0015 0.0009 0.0014 0.0005 Nch05 Nch613 0.1770 0.1512 0.0087 0.0063

Mean of log-likelihood

Nch1418 0.1647 0.0078

Adults_ag 0.0642 0.0049

0.0062 0.0006 0.0112 0.0006 0.0005 0.0006 0.0017 0.0003 0.0039 0.0004 0.0013 0.0005 0.0002 0.0006 0.0045 0.0005 Single 0.2263 0.0119

3.3245

Note: Standard errors are in italics. Nch05, no. of children 0–5; Nch613, no. of children 6–13; Nch1418, no. of children 14–18; Adults_ag, no. of additional adults.

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controlling for differing lifestyles and economies of scale when measuring equivalence scales. The parameters associated with both the linear and quadratic income terms are stable. Considering the attention paid to guaranteeing the exogeneity of incomes and to the specification of the ln a(p) price aggregator, the income parameters are not expected to be biased. As stated in Remark 1, these estimation features are crucial for a robust estimate of the equivalence scale. The fact that the parameters associated with the quadratic term are all statistically significantly different from zero suggests that the Engel space underlying the demand system has rank three. The parameters associated with the equivalence scale, presented in the last row of Table 1, are also significantly different from zero. As described in Eq. (6), equivalence scales correspond to the exponents of the parameters. Tables 2 and 3 describe the matrix of compensated price elasticities and expenditure elasticities and the matrix of the marginal impacts of demographic variables computed at the respective data means. The owncompensated price elasticities along the diagonal have the expected sign. Therefore, the demand system satisfies the regularity conditions at the data means. The Slutsky matrix is negative semidefinite. It is then possible to integrate the demand system and recover the cost function uniquely. The estimates can therefore be properly used to operate interhousehold comparisons because they comply with the requirements of welfare theory.

Table 2.

Elasticities of the QAIDS Model with P T ¼ BðrÞPT ðp; dÞ. Compensated Price Elasticities Food Housing Transport and communica tions

Food 1.223 0.263 Housing 0.656 0.998 Transport and 0.785 0.263 communications Clothing 0.106 0.194 Health, education, 0.192 0.043 and recreation Other goods 0.625 0.074 Estimated Observed

0.307 0.304

0.123 0.123

Clothing

Income Elasticity

Health Other education and goods recreation

0.514 0.429 1.748

0.034 0.155 0.122

0.063 0.035 0.149

0.349 0.103 0.430

0.488 0.662 1.243

0.248 0.300

0.273 0.054

0.054 0.695

0.060 0.192

0.813 0.965

0.505

0.034

0.112

1.351

2.004

0.100 0.100

0.171 0.173

Budget share means 0.201 0.099 0.202 0.099

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Table 3. Shares

Marginal Impacts of Demographic Variables.

North South Rural Age Class Education Wife Head Self- House of Level Working Employed Owned Household Condition Head

Food 0.071 0.052 0.043 Housing 0.104 0.122 0.049 Transport and 0.078 0.009 0.030 communications Clothing 0.021 0.119 0.013 Health, education, 0.062 0.112 0.004 and recreation Other goods 0.119 0.018 0.066

0.080 0.083 0.010

0.043 0.030 0.013

0.018 0.026 0.007

0.033 0.019 0.004

0.007 0.026 0.007

0.094 0.017

0.037 0.039

0.048 0.013

0.009 0.002

0.022 0.045

0.129

0.026

0.013

0.072

0.001

Note that income elasticities are less than one for all goods, in line with their nature as necessary goods, with the exception of ‘‘transportation and communications’’ and the ‘‘other goods’’ category that have an expenditure elasticity larger than one. This effect is not surprising especially for the category ‘‘other goods’’ because this expenditure aggregate comprises less necessary goods. The relative equivalence scales are presented in Table 4. The presence of a child less than 6 years old causes a maintenance cost increase of 19.4% in relation to the cost of living of a childless couple.5 With reference to an adult equivalent, the equally weighted member of a couple, a child under six costs 38.7%. The cost of maintaining a child of age between 6 and 13 increases the costs of a childless couple by 16.3% that corresponds to 32.6% of the cost of an adult equivalent. An adolescent costs 35.8% of an adult equivalent, while an extra adult, who can also be a child more than 18 years old living in the household of origin, costs 13.3% of the cost of an adult equivalent. For example, in relation to a six-member household comprising a married couple, three children distributed uniformly across the three age classes and an extra adult, it would give a household equivalence scale of 3.2 adult equivalents. The effective household size is almost halved. The difference between the real and effective family size measures the economies of scale that the comparison household obtains compared with a household comprising six adult equivalents. A household comprising a single person has a cost of almost 80% of the cost of a childless couple due mainly to the impossibility of sharing the fixed costs associated with the expenditure for the house including rent and other

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Econometric Identification of the Cost of Maintaining a Child

Table 4. Relative Equivalence Scales – Base ¼ Childless Couple. Base Childless couple ¼ 1 Adult equivalent ¼ 0.5

Childless couple ¼ 2 Adult equivalent ¼ 1

1.194 0.010 18.646

2.387

1.163 0.007 22.274

2.326

1.179 0.009 19.468

2.358

1.066 0.005 12.690

2.133

0.797 0.009 21.340

1.595

Child 0–5 SE t-stata Child 6–13 SE t-stata Child 14–18 SE t-stata Additional adult SE t-stata Single SE t-stata

Note: Standard errors and t-stat are in italics. a Tests the hypothesis that the scale is statistically significantly different from 1.

household public goods. With respect to an equivalent adult, the person living alone has a cost of living of about 60% more.6 The cost of a child can be expressed in monetary terms using the concept of absolute equivalence scale corresponding to the difference between the cost of living of the given household, for example, a couple with a child, and the cost of living of a childless reference couple. If we consider the average monthly expenditure for necessary goods of a childless couple in the sample including house rent of about 1,300 h,7 the cost of maintaining a child in absolute terms for the age classes defined in the study is {0–5,6–14,15–18} ¼ {252 h, 212 h, 233 h}. Table 5 shows an international comparison of equivalence scales. The problem found when comparing households also translates the comparison between societies or in the same society over time. However, if equivalence scales are effectively measuring differences in needs, it is plausible that differences in the estimates of the relative cost of maintaining a child in

1st Child

0.24

0.33 0.17 0.20 0.5 0.39

0.43 0.34 0.39 0.25 0.22 0.5 0.45

Merz, Faik – GER (1983)

Menon, Perali – IT. (This study) Subjective scales Van Praag et al. – NE (1982) Koulavatianos et al. – GER (1999) Expert scales OCSE Scale of international expertsc 0.5 0.34

0.15 0.20

0.36

0.13

0.03

0.21

0.42 0.24

0.44 0.65 0.13

3rd Child

0.5 0.39

0.19 0.21

0.36

0.23

0.23

0.26

0.42 0.23

0.40 0.37 0.19

Cost of a Childb

Extended linear Exp. system Extended linear Exp. system QAIDS

Translog

Quasi-utility AIDS Endogenous children – AIDS Extended AIDS Extended AIDS

Comments

b

The time lag between one child and the next corresponds to a hypothetic age profile: (o5, 5–10, W10). The cost of maintaining an under 18-year-old child, marked with an asterisk, is the mean of the cost of the characteristic ‘‘first, second, and third’’ child. c Merz, Garner, Smeeding, Faik, and Johnson (1994).

a

0.25

0.31

0.21

0.42 0.23

0.42 0.23

Ray – UK (1968–1979) Ray e Lancaster – AUS (1984–1988/ 1989) Phipps – CAN (1978, 1982, 1986, & 1992) Merz et al. – US (1986)

0.42 0.29 0.17

2nd Child

Household Typesa

International Comparison of the Cost of Maintaining a Child.

Consumption scales derived from complete demand systems McClements – UK (1972) 0.34 Blundell, Lewbel – UK (1970–1984) 0.18 Ferreira et al. – US (1987) 0.26

Author (Survey Year)

Table 5.

244 MARTINA MENON AND FEDERICO PERALI

Econometric Identification of the Cost of Maintaining a Child

245

different societies vary within a relatively small range irrespective of the method used and the peculiarities of the survey data used for the estimation.8 In other words, it is unlikely that in any society a child will be shown to cost as much as an adult. In general, the comparison of equivalence scales is complicated by the fact that the equivalence scales are not reported in terms of the same reference household. In some cases, the household chosen as the basis for comparison is a childless couple, in other cases, a single person. For comparison purposes, a different basis is required from the conventional bases used for cost of living indices (Atella, Caiumi, & Perali, 2001). According to the classification put forward by Buhman, Rainwater, Schmaus, and Smeeding (1988), household equivalence scales can be divided into scales based on the empirical data of household expenditure surveys and scales based on expert opinions about specific physiological needs or sociocultural necessities. The household scales derived from microdata can be further divided into subjective scales, based on individual perceptions about the minimum income necessary to enjoy the same level of utility as a reference family (Kapteyn & van Praag, 1976; Van Praag, 1991; Van Praag & Warnaar, 1997; Koulavatianos et al., 2005), and into objective scales based on consumption data. In line with the classification of Banks and Johnson (1993), the scales based on demand analysis can be further distinguished as (a) scales based on basic necessities, (b) scales that approximate the exact measure of welfare, such as the Rothbarth and Engel scales, and (c) scales based on the estimation of a complete demand system which are shown in Table 5 because they are comparable to the indices estimated in the present study. The comparison presented in Table 5 refers only to the cost of maintaining a child because the studies also estimating the cost of the characteristics ‘‘living alone,’’ or ‘‘being an elderly,’’ or other household characteristics are not as frequent. In constructing the table, we space three hypothetical children as follows o5, 5–10, and W10. When the cost of the child is not directly estimated in the studies consulted, the cost of the child is computed as the average of the cost of the first, second, and third child. The list of estimated equivalence scales, which is not complete, shows that the interval of variation for the cost of a child is [0.19, 0.69] compared with the cost of an adult equivalent corresponding to the member of a childless couple. Table 5 also shows that the equivalence scales estimated for the Italian case are comparable both between countries and in different periods. The Italian scale is slightly higher than the average of the set of estimates based on a complete demand system. Therefore, the estimates of the Italian case are coherent with other international estimates related to the cost of maintaining children.

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7. CONCLUSIONS This research estimates the cost of maintaining a child for different age classes, the cost of the characteristic ‘‘being single’’ or ‘‘being an adult member’’ of a household in order to make the income levels of households of different composition comparable. The estimated scales are derived using a method coherent with economic theory, analogous to the real cost of living index, based on a complete quadratic demand system plausibly modified to include demographic characteristics and in line with an extended theory of household equivalence scales. This chapter contributes to the existing literature on equivalence scales from two points of view: a) It clarifies important issues relating to the econometric identification of equivalence scales and the estimation procedure to guarantee robust estimates. b) It separates differences in needs, described by a generalized income scale, and differences in lifestyles/household technologies, as captured by demographic translating (fixed costs). The cost of maintaining a child below 6 years increases the cost of a couple without children by 19.4% and corresponds to 38.7% of the cost of an adult equivalent. The cost of maintaining a child of 6–13 and an adolescent corresponds to 32.6% and 35.8%, respectively, of the cost of an adult equivalent. An extra adult costs 13.3% compared with an adult equivalent, whereas the cost of a single person is 60% more than for a childless couple. It is important to recognize that estimated equivalence scales are a household concept referring to the welfare of the family and not to the individual welfare of household members. Equivalence scales assume that household resources are shared equally among all household members. But situations where resources are not equally distributed are in fact frequent. We can think of cases where one or both parents are selfish, or of extreme cases where one or both parents are alcohol or drug addicts. To overcome the often unrealistic assumption, many economists are shifting their attention to the estimation of individual demand systems derived from member-specific welfare functions within a collective framework (Arias et al., 2003; Borelli & Perali, 2003; Browning et al., 2008; Chiappori et al., 2002; Menon & Perali, 2008; Menon, Piccoli, & Perali, 2008). This extension may represent important progress because it could enable equivalence scales to be estimated accounting for intrahousehold allocation rules, making not only interhousehold but also interpersonal comparisons possible (Lewbel, 2003).

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As Sen (1983) has remarked, moving from interhousehold to interpersonal comparisons involves abandoning the assumption implicit in traditional equivalence scales of a ‘‘glued together’’ or ‘‘despotic’’ family where the parents’ indifference maps are considered as representative of each member’s preferences or a family where all members enjoy the same level of welfare. To accomplish this task, a more articulate welfare function would be necessary in order to describe the collection of unequal levels of welfare for each household member generated by a ‘‘mini’’ social choice problem. The knowledge of the rule governing the allocation of resources between adults and children (Bourguignon, 1999; Arias et al., 2003) answers Browning’s (1992) expenditure question and estimates the (full) cost of raising a child, given by the value of the amount of material and time resources invested on children, which is not directly observable. This income-dependent information is fundamental for explaining fertility choices (Lazear & Michael, 1988; Menon & Perali, 2006) and should not be confused with the cost of maintaining a child that is more plausibly independent of income. The specification of the demand system used in this study has the potential to include aspects related both to the publicness of household goods, with a Barten-type transformation of prices into individual-specific shadow prices, and the sharing rule for a collective specification of the demand system. This is next in our research agenda.

NOTES 1. The demographic transformations of a demand system can be grouped into two types: (a) modification without structure that consists in transforming the parameters associated with prices and incomes into linear functions of sociodemographic variables; this transformation is the same as adding to the demand system interaction variables obtained by multiplying either demographics and prices or demographics and income (Blundell, Pashardes, & Weber, 1993; Donaldson & Pendakur, 2004), (b) modification with structure consisting in defining modifying functions per se with arguments prices and demographic characteristics interacting with prices and incomes. This is the approach introduced by Barten (1964), Gorman (1976), Pollak and Wales (1981), and Lewbel (1985) and is the modifying technique followed in this work because it is deemed more interesting under a behavioral point of view. The two approaches can coexist within an encompassing model. The testing of the best functional specification under a statistical point of view will be considered in a future stage of the research program. 2. Note that the specification of the income scale function B(r) can also be written P as BðrÞn ¼ 1 þ R k¼1 rk rk considering that for sufficiently small parameters, ln BðrÞ  ln BðrÞn . This expression is generally adopted by Ray (1983), Lancaster et al. (1999), Blacklow and Ray (1998), and Perali (1999).

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3. For an application, see Blundell et al. (1993), Chiappori et al. (2002), Perali (2003), and Menon and Perali (2008). 4. An alternative option to the one undertaken in the paper is to estimate the service flows from durables converting one-time large expenses into small ones over time (Jorgenson & Slesnick, 1987; Slesnick, 1993). 5. It is worth remarking that equivalence scales can be translated in terms of equivalent adults evaluating each single component of the couple as equal to 1, not 0.5. This is equivalent to multiplying the scales by 2. The normalization in equivalent adults makes the scales comparable to the household dimension and per capita and equivalent incomes can therefore be put side by side. For example, suppose we are interested in comparing two households with the same income of 60 units and same household size of 6. If we do not have further information about the composition of the two households, then both households enjoy the same welfare level of 60 units. Suppose now that we know that family A comprises a couple with four children with a household equivalence scale of 4 and household B comprises a couple, an extra adult, and three children with an equivalence scale of 5. The more the number of equivalent adults is less than 6, the greater the economies of scale. Therefore, each member belonging to household A enjoys the same welfare level as a reference household comprising a single adult of 15 units, whereas household B enjoys a level of welfare per equivalent adult of 12 units. Note that the per capita income of the two households would be 10 units. In utility terms, the welfare of family A corresponds to 90 units for the six members, whereas the welfare level of household B is 72 units. Alternatively, to reach the same welfare level of household A, household B should enjoy 75 units of welfare. 6. In separate calculations, available on request from the authors, we also estimate Engel equivalence scales. The equivalence scales presented here are less than Engel scales as dictated by theory. Such coherence, which predicts that the theoretical scale be less than the upper limit represented by the Engel scale, is maintained for the different age classes. This degree of conformity with theory can be considered acceptable because the test is empirical and is basically intended to verify that the estimates are economically meaningful. 7. Considering that the expenditure for necessary goods in the sample used in this study does not vary as significantly as income, the macroregion, and different life cycles, the choice of a single level of expenditure on which to base the derivation of absolute equivalence scales is justified and is in line with the concept of independence of the base income chosen to implement interhousehold comparisons. The data shows that a household with two earners has expenditures for necessary goods of about 15% more. This difference can reasonably be attributed to differences in the quality of necessity goods. This may partly explain why the cost of maintaining a child can grow as income increases (Donaldson & Pendakur, 2004). 8. Equivalence scales based on a complete preference structure are in general base independent by construction. This property, which keeps the cost of a child constant along the income distribution of a society, also reduces the variability of the scales between countries and over time because it is less sensitive to changes in income distribution.

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ACKNOWLEDGMENTS The authors wish to thank the participants of III ECINEQ, Buenos Aires, July 21–24, 2009 and Andrea Brandolini, Gianfranco Cerea, Eugenio Peluso, Veronica Polin, Ranjan Ray, and an anonymous referee for their useful suggestions in the various phases of the development of the research, as well as Nicola Tommasi for his help in preparing the data. Any errors and omissions are the sole responsibility of the authors.

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APPENDIX. DATA Table A1. Variable

Descriptive Statistics of the Italian Sample, ISTAT 2002 (no. of Observations 19,045). Definition

Demographic characteristics Sex_cf ¼ 1, if head is male Age_cf Head age Fsize Family size Nch018 No. of children 0–18 Nch05 No. of children 0–5 Nch613 No. of children 6–13 Nch1418 No. of children 14–18 Adults_ag No. of additional adultsa Single ¼ 1, if single Tj ¼ 1, if head employed Ts ¼ 1, if wife employed L_dips ¼ 1, if wife is an employee L_indj ¼ 1, if head is self-employed Ownership ¼ 1, if house is owned Edu_cl Head educationb Age_cl Head age classesc Rural ¼ 1, if living in rural areas R1 ¼ 1, if living in the north R2 ¼ 1, if living in the center R3 ¼ 1, if living in the south

Mean

0.816 47.671 3.032 0.687 0.176 0.302 0.210 0.574 0.146 0.658 0.416 0.358 0.177 0.724 0.456 1.424 0.164 0.446 0.187 0.366

Budget shares, prices, and total expenditure Food Food 0.304 House Housing 0.123 Transp&commun. Transport and communications 0.202 Wcloth Clothing 0.099 Edurecr. Health, education and recreation 0.100 Other Other goods 0.173 Lnp(food) Food price, in log. 6.292 Lnp(housing) Housing price, in log. 5.208 Lnp(transp&commun.) Transportation and 5.505 communication price, in log. Lnp(clothing) Clothing price, in log. 5.000 Lnp(edurecr) Health, education, and recreation 5.246 price, in log. Lnp(other) Price of other goods, in log. 5.295 x Total expenditure in euro 1706.026 Lnx Log of total expenditure 7.308 a

Standard Deviation

10.5498 1.2688 0.9042 0.4464 0.5957 0.4734 0.8450

0.6579 0.7340

Minimum Maximum

0 19 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 65 10 7 3 4 4 7 1 1 1 1 1 1 2 2 1 1 1 1

0.1159 0.0612 0.0918 0.0367 0.0309 0.1297 0.3458 0.4175 0.5870

0.001 0.005 0.002 0.004 0.000 0.001 4.700 3.708 3.555

0.745 0.588 0.759 0.535 0.407 0.855 7.192 6.079 6.594

0.9477 0.7669

2.223 2.547

6.378 6.555

0.8633 933.3131 0.5243

3.073 151.205 5.019

6.850 10771.550 9.285

The additional adult is a dependent person over 18 years. The second member of a couple has not been considered as an additional adult. b ¼ 0 if head has a primary school diploma; ¼ 1 if secondary school; and ¼ 2 if high school diploma or college degree. c ¼ 0 if age head r35 years; ¼ 1 if between 36 and 45 years; ¼ 2 if W 46 years old.

0.248 0.116 0.212

0.104 0.100

0.220 1,021

0.29 0.143 0.200

0.069 0.096

0.202 2,653

0.172 1,503

0.089 0.095

0.323 0.133 0.187

0.179 1,496

0.129 0.101

0.283 0.117 0.190

0.164 3,229

0.114 0.108

0.303 0.113 0.197

Single o65 Childless Childless Couple Couple with Couple with Years Couple o45 W45 and o65 Young Older Children Children

0.165 1,861

0.094 0.097

0.314 0.130 0.200

Single Parental

Consumption Share per Household Type.

0.161 7,014

0.100 0.099

0.312 0.117 0.211

Multinuclear

0.173 19,045

0.099 0.100

0.304 0.123 0.202

Total

Note: The single parent household is the household without partner and/or with other adults and/or children; the multinuclear household comprises a couple with other adults and/or children of which at least one is over 18 years.

Food Housing Transport and communications Clothing Health, education, and recreation Other goods Frequency

Expenditure Categories

Table A2.

254 MARTINA MENON AND FEDERICO PERALI

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Econometric Identification of the Cost of Maintaining a Child

Table A3.

Consumption Share per Expenditure Quintiles.

Expenditure Categories

Food Clothing Transport and communications Clothing Health, education, and recreation Other goods

Table A4.

Expenditure Quintiles I

II

III

IV

V

0.353 0.159 0.202 0.109 0.084 0.093

0.339 0.138 0.202 0.112 0.098 0.110

0.323 0.124 0.199 0.112 0.106 0.135

0.300 0.114 0.204 0.106 0.106 0.170

0.254 0.100 0.199 0.091 0.097 0.259

Consumption Shares by Number of Children and Macro Region.

Macro Regions

Number of Children 0

1

2

3

Total

North

Food Housing Transport and communications Clothing Health, education, and recreation Other goods Total expenditure

0.268 0.135 0.191 0.089 0.105 0.211 1,460

0.268 0.13 0.198 0.101 0.106 0.196 2,167

0.271 0.122 0.199 0.101 0.111 0.195 1,523

0.288 0.118 0.187 0.098 0.116 0.193 247

0.27 0.129 0.196 0.098 0.108 0.2 5,397

Center

Food Housing Transport and communications Clothing Health, education, and recreation Other goods Total expenditure

0.299 0.121 0.2 0.09 0.097 0.193 466

0.286 0.115 0.21 0.105 0.103 0.18 862

0.298 0.111 0.216 0.102 0.105 0.168 726

0.306 0.112 0.207 0.102 0.111 0.162 88

0.293 0.115 0.21 0.101 0.103 0.178 2,142

South and Islands

Food Housing Transport and communications Clothing Health, education, and recreation Other goods Total expenditure

0.349 0.116 0.203 0.107 0.081 0.145 744

0.34 0.111 0.209 0.119 0.087 0.134 1,369

0.338 0.106 0.201 0.117 0.097 0.14 1,971

0.362 0.107 0.197 0.108 0.102 0.122 528

0.343 0.109 0.203 0.115 0.092 0.137 4,612

Total

Food Housing Transport and communications Clothing Health, education, and recreation Other goods Total expenditure

0.296 0.127 0.196 0.094 0.097 0.19 2,670

0.294 0.121 0.204 0.107 0.1 0.174 4,398

0.307 0.113 0.203 0.109 0.104 0.165 4,220

0.335 0.111 0.195 0.105 0.107 0.147 863

0.302 0.119 0.201 0.105 0.101 0.172 12,151

CHAPTER 11 RISING INCOMES AND NUTRITIONAL INEQUALITY IN CHINA John A. Bishop, Haiyong Liu and Buhong Zheng ABSTRACT Rising incomes in China have not led to a smaller degree of undernutrition as measured by percentage of population below calorie and protein recommended daily allowances. The weak relationship between income and nutrition is further demonstrated by our income elasticity estimates for calories and protein, which are generally zero. We do find that the percentage of fat in the calorie source is a normal good.

INTRODUCTION China has experienced a dramatic income growth over the past two decades. However, several economic factors confounded with the overall economic growth and consequently the low-income groups could have failed to improve their health and nutrition status. These factors include widening income inequality, rising food prices, and income uncertainty. In this chapter, we examine the combined effect on undernutrition of a secular

Studies in Applied Welfare Analysis: Papers from the Third ECINEQ Meeting Research on Economic Inequality, Volume 18, 257–266 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2010)0000018014

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increase in incomes with a one-time increase in food prices that occurred in the mid-1990s. Malnutrition is one of the most important measures of poverty and, therefore, it is crucial to examine nutrition intake in China during a period of rapid change.1 Clearly, eliminating malnutrition has its own intrinsic value; however, as pointed out by Alderman, Behrman, and Hoddinott (2005), among others, malnutrition can also be linked to productivity losses. These links include greater cognitive ability, physical stature and strength, greater school attendance, as well as the saving of resources used to combat the effect of past undernutrition. Direct measures of nutrition adequacy are difficult (see Osmani, 1992) to obtain; two types of proxies are used in practice, anthropometric measures relative to some reference standard and nutrient intake relative to some standard allowances. We use data on nutrient intakes to assess malnutrition. To examine the relationship between nutrition and income, we first present summary statistics including nutrient means, nutrient inequality (Gini coefficients), and proportions of the population below various fractions of the nutrient recommended daily allowances (RDAs). After presenting these summary measures, we use Kakwani’s (1977) method to estimate nutrient elasticities. In addition to income elasticities, we estimate elasticities with respect to food prices, women’s earnings, and women’s schooling.

ESTIMATING ELASTICITY WITH CONCENTRATION CURVES Kakwani (1977) provides a method to estimate nutrient-income elasticities at various percentiles of the income distribution. Let 0  F 1 ðpÞ  1 be the inverse c.d.f. of x, and without loss of generality, let t ¼ F 1 ðpÞ. Following Bishop, Chow, and Formby (1994), the Lorenz ordinates of x and the concentration ordinates of y can be written as follows: Z t Z 1 E½xI xt  1 1 xf ðxÞdx ¼ mx xI xt dFðxÞ ¼ (1) Lðt; xÞ ¼ mx E½x 0 0 where mx is the mean of x, I xt ¼ 1 if x  t, and I xt ¼ 0 otherwise, Z tZ 1 Z 1Z 1 E½yI xt  1 Cðt; yÞ ¼ m1 yf ðx; yÞdydx ¼ m yI xt f ðx; yÞdydx ¼ y y E½y 0 0 0 0 (2)

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Lðt; xÞ represents the proportion of income of x received by individuals with incomes x less than or equal to t. Cðt; yÞ indicates the proportion of calories (y) received by individuals with incomes x less than or equal to t. Comparing the concentration curve to the 451 line allows us to evaluate the goods’ income elasticity. If the concentration curve lies on the 451 line at any points along the curve, then the income elasticity equals zero at those points; if the concentration curve lies below the 451 line then the good is normal, and if, the concentration curve lies above the 451 line, then the good is inferior. We can also define the Gini and concentrations indices: given a continuous distribution F(x), the covariance definition of the Gini index is   Z 2 1 2 covfx; FðxÞg (3) xFðxÞdFðxÞ  1 ¼ Gx ¼ mx 0 mx and the associated concentration index for y ¼ g(x) is ! Z 2 1 2 Cy ¼ gðxÞFðxÞdFðxÞ  1 ¼ covfgðxÞ; FðxÞg my 0 my

(4)

CHINESE HEALTH AND NUTRITION SURVEY DATA The China Health and Nutrition Surveys (CHNS) were conducted by the Carolina Population Center at the University of North Carolina in 1989 (excludes children), 1991, 1993, 1997, 2000, and 2004. The data were collected on about 4,400 households (16,000 individuals) in nine provinces in China. Within each province, four counties were selected using a multistage, random cluster process. The provincial capital and a lower income city were also selected when feasible. Villages and townships within the counties as well as urban and suburban neighborhoods in the cities were randomly selected. We examine 3 years of CHNS, 1991, 1997, and 2004. We select 1991 and 2004 as the beginning and ending points of the complete data, respectively; 1997 is chosen because it follows the food price increases of the mid-1990s. The CHNS data provide detailed information on many variables of interest and are ideal for researchers who study income and nutrition inequality. In particular, the data have detailed demographic, economic, and nutritional information. Household income variables are constructed to include both earned (wages, farming and gardening income, and business

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income) and unearned (income derived from assets), as well as income from subsidies and bonuses (welfare subsidies and ration coupons). In this chapter, all income and price variables were deflated by the consumer price index to Chinese currency yuan measured in 1988. More importantly, all household members in 1991 and subsequent surveys provided individual data on dietary intake. The interviewers collected detailed household food consumption data during three consecutive days with a starting date randomly selected from Monday to Sunday. On each interview day, individuals were asked to report at-home individual consumption as well as all food consumed away from home for each day on a 24-h recall basis. The 1991 Food Composition Table for China was then utilized to convert food consumption to nutrient values for the dietary data. The RDAs are based on the principle that most, if not all, individuals of a specific population group should obtain an adequate nutrient intake to satisfy their requirements. In this chapter, we use a set of age and gender-specific RDAs sanctioned by the Chinese Nutrition Society (2000). For each specific age and gender group, recommended energy allowances, i.e., calorie intake, represent the average needs of individuals. In contrast, recommended protein allowances are high enough to meet an upper level of requirement variability among individuals within the groups.

CHANGES IN NUTRITIONAL STATUS IN RURAL AND URBAN CHINA We raise three questions in this chapter to evaluate changes in nutrition status during a period of rapid economic growth in China. First, have average nutrition intakes risen with rising incomes? Second, how has the inequality in nutrient intakes changed? Finally, how have the least wellnourished fared during this time period of rapid growth? Table 1 addresses the first question by presenting the changes in calorie and protein intakes, adjusted by the RDAs for both rural and urban China, 1991, 1997, 2004. In addition, the table reports the percent fat in diet, household income, and household size. A dramatic increase in household income along with shrinking household size implies each household member enjoys greater per capita income. This higher income growth is reflected in the growing percentage of calories that are consumed in the form of fat. While rising incomes allow both rural and urban individuals to consume a higher percentage of fat in their diets,2 equivalent calories and protein

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Table 1.

Changes in Nutritional Intakes Rural and Urban China: 1991, 1997, and 2004. Rural

Calorie/RDA Protein/RDA Percent fat HH income HH size

Urban

1991

1997

2004

1991

1997

2004

1.08 0.97 0.20 4,450 4.7

0.93 0.83 0.24 5,306 4.6

0.95 0.87 0.27 8,816 4.3

1.00 0.94 0.26 6,400 4.3

0.95 0.91 0.32 7,585 4.2

0.95 0.95 0.32 10,664 3.8

Note: Standard errors: calories B15; protein B0.5; %fat B0.005. The average daily allowances are 2,480 calories and 75 g of protein.

consumption have not grown with higher incomes. In 1991, average calorie consumption in both rural and urban areas was equal to or above the RDA. By 2004, average calorie consumption had fallen to 95 percent of the RDA. For protein, we find that the average rural individual consumed 97 percent of the RDA in 1991; however, this figure falls to 87 percent in 2004. Urban consumers fared somewhat better, maintaining 95 percent of the RDA between 1991 and 2004. Meng, Gong, and Wang (2004) also show declines in calorie and protein consumption during the 1990s. Meng et al. note that between 1993 and 1996 ‘‘food prices increased significantly and then stabilized (p. 3).’’3 This pattern of food prices is consistent with our finding that calorie and protein consumptions dropped between 1991 and 1997 and stabilized thereafter. While it appears that the levels on nutrient consumption have not grown along with income, a related question is how the inequality in nutrient intake has changed over time. Table 2 presents the familiar Gini coefficient of inequality for each of the three nutrient sources as well as for income. Our household income Gini coefficients verify the well-documented fact that income inequality in China has risen dramatically along with the economic reforms of the 1990s. The distribution of calories and protein is clearly more equal than that of income (smaller Gini’s), but in each case, inequality is increasing over time. Together the findings of Tables 1 and 2 imply that nutrient intakes are both declining and are more concentrated among the few. Table 3 investigates the status of the least well-off nutritionally by examining nutrient intakes of those individuals at or below the RDA. It is important to note that many researchers believe that the calorie cutoffs are set too high

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Table 2.

JOHN A. BISHOP ET AL.

Nutritional Intake Inequality Rural and Urban China: 1991, 1997, and 2004. Rural

Calorie Gini Protein Gini Fat Gini Income Gini

Urban

1991

1997

2004

1991

1997

2004

0.102 0.128 0.201 0.269

0.107 0.130 0.192 0.322

0.118 0.145 0.153 0.393

0.088 0.105 0.158 0.240

0.105 0.126 0.156 0.256

0.130 0.154 0.158 0.365

Note: Standard errors: 0.002–0.005.

Table 3. Fraction Below RDA: Rural and Urban China 1991, 1997, and 2004. Rural

Urban

1991

1997

2004

1991

1997

2004

Calories

1.0 RDA 3/4 RDA 2/3 RDA

0.437 0.119 0.060

0.629 0.242 0.157

0.622 0.266 0.169

0.534 0.164 0.079

0.628 0.238 0.145

0.638 0.301 0.186

Protein

1.0 RDA 3/4 RDA 2/3 RDA

0.609 0.267 0.162

0.748 0.410 0.286

0.719 0.401 0.286

0.631 0.260 0.153

0.689 0.335 0.216

0.631 0.318 0.224

Note: Standard errors: 0.005–0.007.

(see Osmani, 1992, p. 5; Scholl, Hediger, Schall, Khoo, & Fischer, 1994), and we report the undernutrition ratio at two-thirds and three-quarters of the established daily allowances, as well as at the RDA itself.4 Table 3 supports the notion that the RDA may well be ‘‘too high.’’ For example, three-quarters of rural residents (74.8 percent) are deemed to be below the RDA for protein in 1997. However, regardless of the cutoff chosen, a clear pattern emerges – the fraction of the population below the RDA grew dramatically between 1991 and 1997. Using the lowest cutoff, two-thirds of the RDA, we find that the fraction of both rural and urban residents below the cutoff approximately doubled between 1991 and 1997. By 2004, the fraction of rural residents below the calorie RDA increased from 6.0 to 16.9 percent and for urban residents from 7.9 to 18.6 percent. The share of both populations below two-thirds of the protein RDA, while

Rising Incomes and Nutritional Inequality in China

263

higher than those below the calorie RDA, did not increase as fast as for calories. We conclude that examining the average change in nutrient intakes over time (as in Table 1) underestimates the impact of rising food prices among the most nutritionally disadvantaged persons.

Nutrient Elasticities In the previous section, we saw that rising income over time appears not to improve nutritional outcomes. In this section, we ask a somewhat different question: Do lower income households spend a higher fraction of their income on nutrients? In other words, are equivalent calories and protein normal or inferior goods? To address this question, we present the Lorenzconcentration and Gini-concentration Index elasticities. As noted above, a comparison of the nutrient concentration curve (ordered by income) relative to the 451 line allows us to identify whether the nutrient is considered a normal or inferior good at various percentiles of the income distribution. The concentration index, as a summary measure, provides an overall indicator of the relationship between the nutrient and the income. Table 3 provides two illustrations of our approach. The table gives the calorie and percent fat concentration curve ordinates at deciles, ordered by household income for 2004. In the first example, calories, we observe that each concentration ordinate is very close its decile value – i.e., the concentration curves lies nearly on top the 451 line in the Lorenz curve unit square. Furthermore, each of the nine calorie test statistics is less than the 5 percent critical value of 1.96, or the 10 percent critical value of 1.67. The small test statistic on the concentration index, Cy, reinforces this finding, and we conclude that we cannot reject the null hypothesis of zero income elasticity. In the second example, we consider the income elasticity of calories consumed through fat (i.e., percent fat). In this case, seven of the nine test statistics are greater than 1.96, indicating that fat is a normal good. However, the concentration index is still quite small, suggesting that while percent fat is normal, the relationship with income is still quite weak. Table 4 provides a summary index of the elasticities as we vary the income concept. The first entry in any cell is for 1991, the second entry is for 1997, and the third entry is for 2004. A ‘‘1’’ implies that the nutrient is normal and a ‘‘0’’ denotes zero elasticity. The first row of Table 5 summarizes the results using household income as in examples of Table 3. The additional rows in Table 4 consider nutrient elasticities with respect to income adjusted by household size,5 food prices, and the socioeconomic

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Table 4.

Lorenz-Concentration Curve Income Elasticities Rural China, 2004.

Decile

Calories

Percent Fat

Concentration ordinate

z-statistics

Concentration ordinate

z-statistic

0.097 0.193 0.291 0.392 0.492 0.595 0.699 0.800 0.902 0.007 7,704

0.50 0.72 0.81 0.53 0.50 0.32 0.04 0.00 0.20 0.21 7,704

0.091 0.184 0.274 0.368 0.465 0.568 0.673 0.778 0.886 0.04

1.60 2.05 2.85 3.24 3.55 3.34 3.14 2.85 1.47 3.58

1 2 3 4 5 6 7 8 9 Cy N

Note: Protein consumption ordered by household income.

Table 5.

Elasticity Summary Rural and Urban China: 1991, 1997, and 2004. Rural

Household income Food price adjusted HH size adjusted Women’s schooling Women’s wages

Urban

Calories

Protein

Fat

Calories

Protein

Fat

0,0,0 0,0,0 0,0,0 0,0,0 0,0,0

0,0,1 0,0,0 0,0,0 0,0,0 0,0,0

1,1,1 1,1,1 1,1,1 1,1,1 1,1,1

0,1,0 0,0,0 0,0,0 0,0,0 0,0,0

0,0,0 0,0,0 0,0,0 0,0,0 0,0,0

1,1,1 1,1,1 1,1,1 1,1,1 1,1,1

Note: ‘‘0’’ indicates elasticity being zero; ‘‘1’’ indicates normal good at 5% significance level.

status of the household’s primary female. When we adjust for either household size or deflate by food prices, we find zero elasticities for both calories and protein. In contrast, we find that the percent fat is generally a normal good. It is the conventional wisdom that by improving the status of women, household nutrition outcome will improve. While we cannot address this issue directly, we do observe that the nutrient elasticities with respect to women’s schooling or women’s wages are also zero. How do our income elasticities compare to the previous literature? Du, Mroz, Zhai, and Popkin (2004) use the CHNS data (through 1997) and a

Rising Incomes and Nutritional Inequality in China

265

two-step random effects model to estimate income elasticities. In their model, they control for food prices, fuel prices, family size, age, education, urban status, and region. They find that flour and rice are inferior goods, and animal food and edible oil are normal but inelastic. Popkin (2007) instead finds that ‘‘in China, the poor spend a larger share of their food expenditure on vegetable oil than do the rich’’ (p. 92). Meng et al. (2004) use food expenditure data (and an approximation procedure to adjust for meals eaten out), which they convert to nutrition content. They estimate calorie demand as a function of demographic variables and a proxy for uncertainty. They find elasticities greater than 0.5 and comment that their finding is in contrast to many studies ‘‘where it is common to observe low income elasticities using data from surveys designed to monitor nutrition’’ (p. 13).

CONCLUSIONS To evaluate the changes in nutrition status that occurred during a period of rapid economic growth in China we seek to answer the following three questions: First, have average nutrition intakes risen with rising incomes? Second, how has the inequality in nutrient intakes changed? Finally, how have the least well-nourished fared during this time period of rapid growth? Our findings imply that nutrient intakes are both declining and are more concentrated among the few. Furthermore, we conclude that the average change in nutrient intakes underestimates the impact that rising food prices have had on the most nutritionally disadvantaged persons. In sum, rising incomes over time did not improve nutritional outcomes. The failure of a doubling of Chinese incomes between 1991 and 2004 to improve nutritional outcomes invites a more direct question: Is nutritional intake a ‘‘normal good’’? We find that the income elasticity for calorie and protein intakes are generally zero, while we find some evidence that the percentage of fat in the calorie source is a normal good. Finally, we note that nutrient elasticities with respect to women’s schooling or women’s wages are also zero.

NOTES 1. Minoiu and Reddy (2008) provide a well-researched survey of current poverty in China.

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2. Du et al. (2004) address the relationship between rapid income growth and diet quality during the period 1989–1997 using the CHNS data. Our findings are consistent with their conclusion that ‘‘the structure of the Chinese diet is shifting away from high-carbohydrate foods toward high-fat, high-energy density foods’’ (p. 1505). 3. Meng, Gregory, and Wang (2005) report the urban food CPI for 1986–2000 in Fig. 7. 4. Testing headcount below the RDA at multiple cutoffs is equivalent to truncated first-order stochastic dominance, see Ravallion (1991). 5. We use the familiar ‘‘square-root rule’’ to adjust for differences in household size.

REFERENCES Alderman, H., Behrman, J., & Hoddinott, J. (2005). Nutrition, malnutrition, and economic growth. In: G. Lopez-Casasnovas, B. Rivera & L. Currais (Eds), Health and economic growth: Findings and policy implications. Cambridge, MA: MIT Press. Bishop, J. A., Chow, K., & Formby, J. P. (1994). Testing for marginal changes in income distributions with Lorenz and concentration curves. International Economic Review, 35, 479–488. Chinese Nutrition Society. (2000). Dietary reference intakes. Beijing: Chinese Light Industry Press. Du, S., Mroz, T., Zhai, F., & Popkin, B. (2004). Rapid income growth adversely affects diet quality in China. Social Science & Medicine, 59, 1505–1515. Kakwani, N. C. (1977). Applications of Lorenz curves in economic analysis. Econometrica, 45, 719–725. Meng, X., Gong, X., Wang, Y. (2004). Impact of income growth and economic reform on nutrition intake in urban China: 1986–2000. IZA DP No. 1128. Meng, X., Gregory, R., & Wang, Y. (2005). Poverty, inequality, and growth in urban China, 1986–2000. Journal of Comparative Economics, 33(4), 710–729. Minoiu, C., & Reddy, S. G. (2008). Chinese poverty: Assessing the impact of alternative assumptions. Review of Income and Wealth, 54(4), 572–596. Osmani, S. R. (1992). Introduction. In: S. R. Osmani (Ed.), Nutrition and poverty. Oxford: Clarendon Press. Popkin, B. (2007). The world is fat. Scientific American, 297, 88–95. Ravallion, M. (1991). Does undernutrition respond to incomes and prices? Stochastic dominance tests for Indonesia. World Bank Review, 29, 140–146. Scholl, T. O., Hediger, M. L., Schall, J. I., Khoo, C. S., & Fischer, R. L. (1994). Maternal growth during pregnancy and the competition for nutrients. American Journal of Clinical Nutrition, 60, 183–188.