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RESEARCH ON ECONOMIC INEQUALITY Series Editors: John Bishop and Yoram Amiel

RESEARCH ON ECONOMIC INEQUALITY

VOLUME 16

INEQUALITY AND OPPORTUNITY: PAPERS FROM THE SECOND ECINEQ SOCIETY MEETING EDITED BY

JOHN BISHOP East Carolina University, Greenville, NC

BUHONG ZHENG University of Colorado, Denver, CO

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

JAI Press is an imprint of Emerald Group Publishing Limited Howard House, Wagon Lane, Bingley BD16 1WA, UK First edition 2008 Copyright r 2008 Emerald Group Publishing Limited Reprints and permission service Contact: [email protected] No part of this book may be reproduced, stored in a retrieval system, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without either the prior written permission of the publisher or a licence permitting restricted copying issued in the UK by The Copyright Licensing Agency and in the USA by The Copyright Clearance Center. No responsibility is accepted for the accuracy of information contained in the text, illustrations or advertisements. The opinions expressed in these chapters are not necessarily those of the Editor or the publisher. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-1-84855-134-3 ISSN: 1049-2585 (Series)

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

LIST OF CONTRIBUTORS Ingvild Alma˚s

Norwegian School of Economics and Business Administration and University of Oslo (ESOP), Oslo, Norway

Olga Alonso-Villar

University of Vigo, Vigo, Spain

Elena Ba´rcena

University of Ma´laga, Ma´laga, Spain

Luis Beccaria

General Sarmiento National University, Buenos Aires, Argentina

Denis Cogneau

Institut de Recherche pour le De´veloppement (IRD), Paris, France

Frank Cowell

London School of Economics, London, UK

Liema Davidovitz

Ruppin Academic Center, Emek Hefer, Israel

Coral del Rı´o

University of Vigo, Vigo, Spain

Joseph Deutsch

Bar Ilan University, Ramat Gan, Israel

Henar Dı´ez

University of the Basque Country, Bilbao, Spain

Veronika V. Eberharter

University of Innsbruck, Innsbruck, Austria

Udo Ebert

University of Oldenburg, Oldenburg, Germany

Yves Flu¨ckiger

University of Geneva, Geneva, Switzerland

Fernando Groisman

General Sarmiento National University, Buenos Aires, Argentina vii

viii

LIST OF CONTRIBUTORS

Luis J. Imedio

University of Ma´laga, Ma´laga, Spain

Serge-Christophe Kolm

Ecole des Hautes Etudes en Sciences Sociales, Paris, France

Ma Casilda Lasso de la Vega

University of the Basque Country, Bilbao, Spain

Sandrine Mesple´-Somps

Institut de Recherche pour le De´veloppement (IRD), Paris, France

Juan D. Moreno-Ternero

University of Ma´laga, Ma´laga, Spain; CORE, Universite´ catolique de Louvain, Louvain-La-Neuve, Belgium

Vito Peragine

University of Bari, Bari, Italy

Laura Serlenga

University of Bari, Bari, Italy

Jacques Silber

Bar Ilan University, Ramat Gan, Israel

Ana Urrutia

University of the Basque Country, Bilbao, Spain

Buhong Zheng

University of Colorado, Denver, CO, USA

INTRODUCTION Volume 16 of Research on Economic Inequality contains a selection of papers from the Second Biannual Meeting of the Society for the Study of Economic Inequality, Berlin, July 2007. The volume opens with an essay on equal liberties by Serge-Christophe Kolm and is followed by papers on the equality of opportunity, inequality measurement issues, and an applications section. In the lead paper, Kolm characterizes two levels of equal liberties: equal social liberty and equal economic liberty (equal total freedom) with the former representing the basic human rights and the latter extending the notion of equality to the allocation of productive capacities such as natural resources. Through the mechanism of ‘‘equal-labor income equalization,’’ Kolm demonstrates that the equal liberty principles necessarily yield the same results implied by other ethical considerations. In the second paper, Cowell and Ebert address the relationship between inequality and envy. If an individual feels that he’s not ‘keeping up with the Joneses;’ falling farther behind some upward looking reference income, then is his idea of inequality increasing? Cowell and Ebert model this behavior and provide absolute and relative inequality indices that are envy-sensitive. The next four papers address issues in the equality of opportunity. Roemer’s theory of equal opportunity provides a ﬁrst workable mechanism to design equal-opportunity policies. But his theory is built upon the assumption of independent preference, i.e., agents are all self-interested. The ﬁrst of these papers by Juan D. Moreno-Ternero extends Roemer’s mechanism of equal opportunity to allow for interdependent preferences, i.e., agents also care about their peers’ well being. Moreno-Ternero illustrates his extended model with a health care example. Peragine and Serlenga consider the inequality of opportunity in higher education in Italy, comparing the less developed South to the more developed North-Central region. Measurement of the inequality of opportunity can be problematic as ex ante opportunities available to individuals are not observable. These immutable ‘circumstances’ are proxied using controls for family background. Stochastic dominance methods, combined with statistical inference procedures, are used to rank the two ix

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INTRODUCTION

regions by educational opportunities. The inequality of educational opportunities is found to be greater in Southern Italy. Denis Cogneau and Sandrine Mesple´-Somps investigate a largely unstudied problem, economic inequalities in Sub-Saharan Africa. Using ﬁve rich data sets they provide the ﬁrst ever study of the intergenerational transfer of resources in Africa. Their decompositions reveal that two former British colonies (Ghana and Uganda) share a higher degree of intergenerational education and occupation mobility than the former French colonies of Ivory Coast, Guinea, and Madagascar. Ingvild Alma˚s investigates the equality of opportunity in Germany and the United States. She begins by applying the strict equalitarian (Germany dominates) and strict libertarian ideals (US dominates). But citizens of both countries are more likely to take an intermediate (responsibility-sensitive) position. She ﬁnds that the ranking between the two countries depends on the treatment of the unexplained variation (luck or unobserved human capital) in the income generating equation. Veronika V. Eberharter also compares Germany and the United States, two countries with quite different redistribution policies. Her interest is in the intergenerational transfer of income inequality and poverty. The empirical results suggest that Germany’s more intensive redistribution policy succeeds in reducing income inequality and poverty, but at the expense of intergenerational income persistence and dynastic poverty. However, when both individual and labor market characteristics are taken into account then the United States has a higher degree of intergenerational income persistence. Zheng’s paper leads off the measurement section of the book. Zheng examines whether stochastic (Lorenz) dominance – developed for ratio data – can be applied to rank inequality and welfare of distributions with ordinal data such as self-reported health status. He derived an impossibility result for inequality measurement and a limited possibility result for welfare rankings. Zheng points out polarization as a useful but limited approach to understanding the dispersion in health outcomes. The two papers by Diez et al. and del Rio and Alonso-Villar concern the use of different monetary units in the measurement of inequality and poverty. The paper by Diez et al. extends the unit-consistency introduced in Zheng (2007a, 2007b) to multidimensional measurement of inequality and poverty and derives unit-consistent multidimensional inequality and poverty indices. The paper by del Rio and Alonso-Villar examine speciﬁcally the issue of unit-consistency and intermediate inequality indices. They illustrate

Introduction

xi

geometrically that some popular intermediate inequality measures may violate unit-consistency. Elena Ba´rcena and Luis J. Imedio examine comprehensively two newly rediscovered inequality measured originally introduced by Bonferroni and De Vergottini, respectively, in 1930 and 1940. The authors show that both measures are similar to the Gini index in many ways: all three are members of the Mehran (1976) family; all are related to Lorenz curve and all can be interpreted as measures of deprivation. Deutsch, Flu¨ckiger, and Silber recognize that measuring unemployment, like measuring poverty, is primarily an ethical problem, not simply an exercise in statistical measurement. Welfare is inﬂuenced by the rate of unemployment, the average duration, and the inequality of unemployment spells. Using Swiss data at the canton level, they identify the contribution of each of these three components on overall welfare. The last two papers in the book provide an interesting case study in inequality, Argentina, and some experimental ﬁndings. Beccaria and Groisman’s study of income inequality and mobility in Argentina covers the period 1988–2001, allowing the authors to compare a period of hyperinﬂation against a more stable macroeconomic environment. They ﬁnd that the macroeconomic stabilization achieved in the early 1990s reduced the variability of family incomes. However, labor income instability increased from the middle 1990s causing an increase in inequality. An examination of permanent as opposed to annual incomes provides similar conclusions. Liema Davidovitz uses experimental methods to address the question: Is an individual’s degree of inequality aversion inﬂuenced by the lottery’s risk level? In a three-phased experiment, the students ﬁrst perform a task in order to obtain a gambling stake. In the second phase they must choose the method of payment, equal outcomes, or individual outcomes. In the ﬁnal phase the lottery is played. Students are found to prefer individual outcomes under low-risk scenarios, but collective outcomes under high-risk scenarios. John A. Bishop Buhong Zheng

EQUAL LIBERTIES AND THE RESULTING OPTIMUM INCOME DISTRIBUTION AND TAXATION Serge-Christophe Kolm ABSTRACT The relevant basic principle for overall distribution in macrojustice turns out to be the relevant equality of liberties. This study shows the consequence of this fact for the optimum distribution, taxation, and transfers of income. The liberties in question are social liberty (freedom from forceful interference, basic rights), and the possibilities offered by domains of choice which can provide equal liberty while being different for individuals with different productivities. The method is deductive from the basic relevant concepts. The result is that this distribution consists of an equal sharing of the proceeds of the same labour for all individuals (with their different productivities). The individuals choose freely their total labour (with no other tax). This redistributive structure is Equal-Labour Income Equalization or ELIE. It also has a number of other important meanings, such as: general balanced labour reciprocity (each yields to each other the proceeds of the same labour); equal basic universal income ﬁnanced by an equal labour of all; and uniform linear concentration to the mean of the distribution of total incomes (including the value of leisure).

Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 1–36 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16001-X

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SERGE-CHRISTOPHE KOLM

This result extends to multidimensional labour (duration, education, intensity, etc.), and to partial labour including unemployments. The practical application relies on exemption of overtime labour from the income tax, and a tax credit. This is successfully applied in some countries. This constitutes a new paradigm of optimum income distribution and taxation. The old paradigm was based on welfarism not found relevant by society for this application, and it has therefore never been applied.

MACROJUSTICE Equal freedom has to be the principle of the bulk of the distribution in society, as we shall see. Therefore, this principle has to be speciﬁed, and the policy it implies has to be derived. This is the object of this study, which obtains its result thanks to two facts: it considers an actual problem (overall distribution and the income tax) and it relies on the basic legal and philosophical properties of the concept of liberty. The ﬁrst section shows that the necessary principle of ‘‘macrojustice’’ is equal liberty, notably because general opinion – investigated in Appendix A – rejects, for answering this important but speciﬁc question, the comparisons between individuals’ welfares or their variations that result from the maximization of some a priori speciﬁed aggregate of these welfares (although the outcome is Pareto efﬁcient). The second section speciﬁes equal freedom. It ﬁrst reminds us that two kinds of freedoms have to be considered. One, social liberty, deﬁned by the nature of the constraint, is the freedom from forceful interference which is the basis of our societies. When the other individual means and rights are added, one obtains the total freedom described by the possibility set. Equal total freedom has to be for non-identical domains of choice, in order to respect social liberty and Pareto efﬁciency while utilities (welfare) are irrelevant and people have different earning capacities. Various possible deﬁnitions of equal liberty lead to the same conclusion about the structure of income transfers. The important diverse meanings of this structure, and the question of implementation of the corresponding policy, are shown in the third section. The fourth section speciﬁes the relevant solution and shows its incentive compatibility. The questions of information, of the determination of the degree of income equalization, and of the relation with the rest of public ﬁnance are considered in the ﬁfth section. Finally, the result obtained is compared

Equal Liberties and Resulting Income Distribution and Taxation

3

with the best known economic and ethical results and positions in the last section. When all the people who can inﬂuence a policy – such as voters and ofﬁcials – share a certain possible view about it, this view is implemented and any alternative proposal has no chance to be. This remark applies in particular to the role of ‘‘welfarism’’ – that is, taking individuals’ ‘‘welfare’’ as ultimate social ethical reference – for judging a distribution.1 The conclusion turns out to be that welfarism is judged relevant to evaluate a distribution when it means lower suffering, or for some distributions among people who sufﬁciently know one another and more or less ‘‘empathize’’ others’ satisfaction. A consequence, shown in Appendix A, is that welfarism is not held to be relevant for the issue of macrojustice. Macrojustice concerns the general rule of society and its application to the general overall distribution of the main resources. It opposes the concept and ﬁeld of microjustice concerned with allocations that are speciﬁc according to beneﬁciaries, reasons, circumstances, or goods.2 Macrojustice includes the general rules of economic allocation (free exchange and property rights in our societies) and the general distributive policies such as the income tax or equivalent taxes and general income transfers. We consider here distribution in a large-scale society (e.g., a nation) not in a state of emergency as a result of some general catastrophe (war, natural disaster, etc.), and where basic rights, overall distribution, speciﬁc assistance and basic insurance schemes are in place. This has two consequences. First, policies aiming at the relief of suffering are particular with respect to reasons, circumstances, scope, and beneﬁciaries, and hence refer to issues of microjustice. Second, the distribution is essentially not between people who know one another. This eliminates, from the reasons for the choices of macrojustice, the two reasons one of which – at least – is present when people hold that the comparison of individuals’ welfare is relevant to judge the distribution. If, in choice theory, utility is deleted, there remains the domain of free choice. More philosophically, man is both a sentient being feeling pleasure and pain and an agent capable of free choice and action. These are the two possible general bases of individual-oriented social ethics. Hence, when welfarism is not deemed relevant, the value has to be liberty. However, two types of liberty are considered. Social liberty is freedom from the forceful interference of other people acting individually, in groups or in institutions. This is the basic principle of ‘‘liberal’’ societies, in the form of the basic (constitutional) rights which have priority.3 Individuals are only forced not to force others. Free exchange is therefore allowed and is important – and it a priori implies freedom from the forceful interference of agents not party to

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the exchange. The respect of people’s respectful actions implies the respect of the intended consequences of these actions, such as rights acquired from action or exchange. Moreover, people have other means which, along with social freedom, determine their domain of possible choice (opportunity set). Individuals’ social liberty is non-rival: the social liberty of one does not hamper that of others. When individuals’ desires, intentions, or actions oppose one another, the limit is deﬁned by the allocation of rights, which is an aspect of the general issue of the distribution of social possibilities and resources.4 Therefore, social liberty can be full for all individuals, and it is then equal in this sense. Fiscal policy that respects social liberty has to be based on items that individuals cannot affect, i.e., on inelastic items. This is also a classical condition for Pareto efﬁciency. Pareto efﬁciency is also valuable per se for several reasons. Its failure is a kind of collective lack of freedom: something prevents society from going to a possible state that everybody prefers (with the possible indifference of some). By nature, a democracy should not be prevented from reaching a possible state preferred by all (with the possible indifference of some) – even in an electoral democracy, a contending party could choose such a program and win with the unanimity of votes. Moreover, the failure to achieve a possible increase in everybody’s welfare (with the possible indifference of some) is also a priori regretted even when comparing individuals’ welfares or their variations is not considered the relevant distributive criterion. These inelastic items are the classical ‘‘natural resources’’ (intertemporally, capital is produced). Pareto efﬁciency can then result from a distribution of these resources plus the working of an efﬁcient market (with the correction of possible ‘‘market failures’’ by the public ‘‘allocation branch’’). Natural resources divide into the human and the nonhuman ones. The contribution of labour to the economic value of the social product is very much larger than that of non-human natural resources (capital being an intermediate good, in an intertemporal view).5 In addition, some productive capacities are not put to work. Therefore, overall distributive justice in macrojustice is essentially concerned with the allocation of the value of the given productive capacities.6 Moreover, ‘‘Justice is equality, as everybody thinks it is, apart from other considerations,’’ Aristotle writes in Nicomachean Ethics and Eudemian Ethics. Justice being equality prima facie (i.e., in the absence of an overpowering reason such as impossibility or the joint relevance of some other principle) is in fact a requirement of rationality in the most standard sense of providing a reason or intending to. Indeed, if someone receives something for a reason based on certain characteristics of hers, some other

Equal Liberties and Resulting Income Distribution and Taxation

5

person having the same relevant characteristics should a priori receive the same thing (possibly the same relevant value). A non-satisfaction of this equality implies a lack of justiﬁcation, an arbitrariness, which arouses a sentiment of injustice. Therefore, macrojustice is equality of liberty. Social liberty can be full and hence equal for all. This goes with Pareto efﬁciency. With social liberty, individuals freely choose their labour using productive capacities of theirs, and they buy goods with their income. Moreover, they can be submitted to transfers of the ﬁscal policy, based on their given productive capacities.7 The problem is to determine this ﬁscal policy that achieves equality in overall individual liberty. Hence, the ﬁrst task is to deﬁne this equality in economic liberty.

EQUAL ECONOMIC LIBERTY Possibilities With (equal) social liberty to choose, exchange, and earn, the remaining equality concerns the initial given conditions. This initial equality can take four forms: 1. Equal initial allocation. The other forms describe properties of the given domains of choice: 2. Socially free individuals are susceptible to choose an equal allocation. 3. Identical domains of choice. 4. Equal overall freedom provided by different domains of choice. We will see that solutions 1, 2 and 4 give the same result, whereas solution 3 is impossible in the sense that it violates Pareto efﬁciency and social liberty if individuals’ preferences are not taken into account (from non-welfarism or ignorance) to deﬁne the domain – and it may violate them even without this qualiﬁcation.8

The Simple Case, Notations We consider to begin with the simple case of unidimensional labour and constant individual wage rates (linear wage functions), because it is an important case, it simpliﬁes the presentation a little, the concepts and results extend straightforwardly to the general case of multidimensional labour

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(duration, intensity, formation, etc.) and non-linear production (see Appendix B), and the general case can often be reduced to the simple case by deﬁning a duration of labour qualiﬁed for its other characteristics (id.). The case of involuntary unemployment will be considered in Appendix C. There are n individuals, and each is indexed by i and has labour ‘i (seen as duration), and hence leisure li=1 ‘i by normalization to 1 of the total relevant time, a given wage rate wi, and a tax or subsidy ti (tiW0 for a subsidy and o0 for a tax of ti). Her labour income is wi‘i, her disposable income used to buy freely (non-leisure) consumption is yi ¼ wi ‘i þ ti

(1)

and her total income, which adds the value of leisure at its market price wi, is vi ¼ yi þ wi li ¼ wi þ ti :

(2)

We consider now a balanced distributive budget (Musgrave’s (1959) P ‘‘distribution branch’’), and hence ti=0.

Solution 1: Social Liberty from an Equal Allocation A Solution This solution is the classical (equal) social liberty from an equal allocation.9 Social liberty implies free exchange. The allocation is that of the two goods, leisure (or labour) and income which can buy consumption (from free exchange). Free exchange is, ﬁrst of all, of labour for earnings. If this initial equal labour is k (leisure 1 k), it provides each individual i with the income kwi, and, if this income is transformed into an equal piece of disposable income with balance of the distributive P budget and no waste, where w ¼ ð1=nÞ wi is the average wage each now receives the average kw, rate. Then, individual i is taken away kwi and provided with kw instead, that is, she receives the net subsidy-tax ti ¼ k ðw wi Þ:

(3)

P We have ti=0. The described operation is ‘‘Equal-Labour Income Equalization’’ (the equal sharing of the incomes produced by a given labour equal for all) or ELIE. Labour k is the ‘‘equalization labour.’’ Individual i freely chooses her (full) actual labour ‘i and the corresponding earnings wi‘i. Equivalently, this can be described as her choosing labour ‘i k above labour k, and hence earning the corresponding wi (‘i k) in

Equal Liberties and Resulting Income Distribution and Taxation

7

addition to the given kw (we will shortly see that, for the problem of macrojustice, ‘iWk will happen to hold). At any rate, her disposable income and her total income are, respectively, yi ¼ wi ‘i þ ti ¼ kw þ ð‘i kÞwi

(4)

vi ¼ wi þ ti ¼ kw þ ð1 kÞwi :

(5)

and

First Properties Formulas (3), (4) and (5) show remarkable properties in themselves. Form (4) shows that each individual income is made of two parts, an egalitarian part in which all individuals receive the same income kw for the same labour k and a liberal-self-ownership part in which each individual i receives the full product of her extra labour (‘i k) at her wage rate wi, (‘i k)wi. The equalization labour k is the cursor making the division between these two parts. Moreover, form (4) shows that yi is close to kw if wi is small, whatever ‘i. At any rate yi kw if ‘iZk, which will turn out to be the case relevant for macrojustice: there is a minimum income kw (hence, a consensus about a minimum income implies a consensus about coefﬁcient k, given that the properties that imply the structure ELIE are generally wanted). Formula (3) shows that this distributive scheme amounts to a universal basic income kw ﬁnanced by an equal labour k of all individuals, or according to capacities (each individual i pays her earnings for this labour, kwi, which is also according to her capacities wi). The way in which the result has been obtained shows that the result amounts to each individual i yielding to each other the sum kwi/n=(k/n)wi, that is, the proceeds of the same labour k/n. This is a general equal-labour reciprocity. Formula (4) shows that an individual’s total income is the weighted with k and average between her productivity wi and average productivity w, 1 k as weights. Rawls’s Final Solution In 1974, John Rawls, at the instigation of Richard Musgrave (1974), added leisure to his list of ‘‘primary goods,’’ thus bringing to two, income (related to wealth) and leisure, the economic primary goods.10 Rawls’s solution consists of basic liberties, the best description of which is social liberty which is full and hence equal for all and maximal, and an ideal of an equal initial allocation of primary goods if this is not wasteful. The above solution consists of an initial allocation in which all individuals have the same quantity of each

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good, 1 k for leisure and kw for income, from which each individual freely trades labour for income in application of social liberty. No individual can have more of one good in her initial allocation without any other initial allocation of any good to any person being lower, and the ﬁnal outcome is Pareto efﬁcient. This result can thus be said to be Rawls’s full solution (as he posed the problem after 1974).11 The Geometry of ELIE The result is shown in Fig. 1, with axes li and yi, ‘i=1 li, budget lines with slopes wi, transfers ti, and total incomes vi. The initial equal allocation is When k varies the point common to all budget lines K (‘i ¼ k; yi ¼ kw). from 0 to 1, point K describes the segment LM from point L (‘i=yi=0) to – yet, only cases where ko‘i will turn out to be point M (li ¼ 0; yi ¼ w) relevant for macrojustice. The particular case k=0, and hence ti=0 and yi=wi‘i for all i, corresponds to the full self-ownership of ‘‘classical

v2

w M

v1

w1 y1 kw

K

t1 λ1

0

L 1 k l1

Fig. 1.

The Geometry of ELIE.

t2

Equal Liberties and Resulting Income Distribution and Taxation

9

liberalism’’ (this is, for example, the position of – among scholars – Friedrich Hayek, Milton Friedman, Robert Nozick, and John Locke). The choice of the coefﬁcient or ‘‘equalization labour’’ k will be considered below.

Solution 2: Socially Free Individuals are Susceptible to Choose an Equal Allocation Individuals who have social liberty and prefer higher income (consumption) and leisure choose an allocation on their budget line. If there is one individual allocation that they all are thus susceptible to choose, these lines pass through the same point representing this allocation.12 Equation (2) with some given ti represents this budget line for individual i, and if this common point is ‘i=k (li=1 k) and yi=Z, it entails Z þ ð1 kÞwi ¼ wi þ ti

(6)

Z ¼ kwi þ ti :

(60 )

or P For a balanced distribution ti=0, and summing Eq. (6u) for all i implies hence form (3) for ti. Z ¼ kw,

Solution 3: Identical Domains of Choice Properties If individuals’ choices include the choice of effort or labour and they have different earning capacities, and if the policy maker does not take individuals’ preferences into account, presenting identical domains of choice to all individuals violates both Pareto efﬁciency and social liberty (and hence it should be impossible in a well-functioning democracy and it violates the basic rights).13 Consider, indeed, the ﬁve conditions: (1) Individuals freely choose in identical domains of choice. (2) They do not all have the same productivity. (3) Their preferences or utilities are irrelevant or unknown to determine the domain of choice. (4) Pareto efﬁciency. (5) Social liberty.

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Then, the two following results hold: (1) Properties (1), (2), (3), and (4) or/and (5), cannot hold jointly. (2) Properties (1), (2), and (4) or/and (5), may not hold jointly. Proof of Result (1) The proof results from the conditions necessary for building such a common domain of choice. In the space of leisure or labour and disposable income (consumption), at an achieved state: (1) Pareto efﬁciency and social liberty imply that each individual’s marginal rate of substitution is equal to her marginal productivity (wi); and (2) because this individual freely chooses in the domain offered to her, this state is on the domain’s border B and the marginal rate of substitution is equal to the border’s rate of transformation. Hence, at this state, this latter rate is equal to the individual’s marginal productivity. If these productivities are identical and constant, this border can be a straight line with this slope. If not, this border should respect the following condition. Call Ei the ‘‘curve’’ (more generally, set of points) where individual i’s rate of substitution is equal to wi (an Engel curve). Then, border B should cut each Ei at a point where its slope should be wi ( wi if the variable is leisure). This condition depends on the curves Ei, which are derived from the individuals’ preference orderings or utility functions. This border, and hence the common domain, cannot be built without these preferences or utilities. Fig. 2 illustrates this condition.14 Proof of Result (2) A set of individual allocations can result from individual choices on identical domains if and only if no individual prefers another’s allocation to her own (Kolm, 1971/1998).15 Moreover, this latter property may be inconsistent with Pareto efﬁciency (Pazner & Schmeidler, 1974, whose example is a case of the present simple model). Finally, social liberty with perfect markets implies Pareto efﬁciency.

Solution 4: Equal Liberty of Unequal Domains To deﬁne equal freedom of choice for different domains of choice, consider that domains can offer more or less freedom, with ‘‘neither more nor less’’ being equal. Using these relations usually implicitly implies their transitivity, which we assume. Domains of choice are thus ranked by a (weak) ordering, the freedom ordering. This ordering will be assumed to be representable by

11

Equal Liberties and Resulting Income Distribution and Taxation yi E2 -w2

u2

B

E1 u1 D 0

-w1

λi

Fig. 2.

1

Efﬁcient Identical Domains.

an ordinal function, the ‘‘freedom function,’’ since this will sufﬁce here. If D is a domain of choice (a set of possible choices), the freedom function F(D) is such that, if Du is another domain, F(D)=F(Du) if D and Du offer equal freedoms, and F(Du)WF(D) if Du provides more freedom than D. (In particular, if the domains D and Du are identical, F(D)=F(Du)). Let us apply this to the budget sets considered here. A generic individual can provide labour ‘Z0, hence enjoy leisure l=1 ‘Z0, and consume consumption goods in amount yZ0. Let us choose an arbitrary but given and ﬁxed unit of account, for which the price of consumption goods is PW0 (P=1 if they are taken as this nume´raire), and the generic individual’s wage rate and total income are WZ0 and VZ0, respectively. For a speciﬁc individual i, ‘, l, y, W, and V take the values ‘i, li, yi, Wi, and Vi. An individual freely chooses her leisure l=[0, 1] (and hence her labour ‘=1 l), and her consumption yZ0, subject to her budget constraint Py þ Wl V

(7)

which deﬁnes her budget set, which is her possibility set or domain of choice in the space of y and l. This set is classically characterized by the (total) income V and the prices P and W. The freedom function can be written,

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therefore, as FðV; P; WÞ:

(8)

If V, P, and W are all multiplied by the same positive number, the budget set deﬁned by condition (7) does not change. Hence, level F does not change. That is, function F is homogeneous of degree zero in its three variables V, P, and W. Moreover, to describe market possibilities when incomes and prices can vary, the prices are usually summarized by a price index which is always taken as linear (as with the classical indexes of Paasche and Laspeyre and those derived from them). Write this index as p ¼ aP þ bW

(9)

where aW0 and bZ0 are constant numbers. One has FðV; P; WÞ fðV; pÞ ¼ fðV; aP þ bWÞ:

(10)

Function f is homogeneous of degree zero in its two variables V and p since multiplying V, P, and W by the same positive number does not change the level F=f and multiplies the index p by this number. Hence, dividing both arguments of function f by p (when pW0) gives V V ;1 ¼ j (11) F ¼ fðV; pÞ ¼ f p p by deﬁnition of function j. Since functions F, f, and j are ordinal and are increasing functions of V, V/p is a speciﬁcation of function j (this is real (total) income, ﬁttingly usually called purchasing power). Therefore, the V, P, and W that provide equal freedom are such that V ¼g p

(12)

V ¼ agP þ bgW:

(120 )

for some given g, or Hence, individuals i with possibly different wage rates Wi have the same freedom if their total incomes Vi are V i ¼ agP þ bgW i ;

(13)

respectively. Hence, with real (i.e., in terms of consumption goods) wage rates Wi/P=wi and total incomes Vi/P=vi vi ¼ ag þ bgwi

(14)

13

Equal Liberties and Resulting Income Distribution and Taxation

for all i. This implies that individual i receives the net real transfer

However,

P

ti ¼ vi wi ¼ ag þ ðbg 1Þwi :

(15)

ti=0 entails ð1 bgÞw ¼ ag:

(16)

ti ¼ k ðw wi Þ:

(3)

Then, denoting 1 bg=k,

This is the same result as that of solutions 1 and 2. Moreover, individual i’s budget line in space (li, yi) is wi li þ yi ¼ vi ;

(2)

since and it contains the point (‘i ¼ k; yi ¼ kw) ð1 kÞwi þ kw ¼ wi þ ti ¼ vi : This ‘‘equalization point’’ K, independent of i, is common to all budget lines (which, therefore, constitute a ‘‘pencil’’ of lines).

EQUIVALENT PROPERTIES AND NORMATIVE MEANINGS Judging something can, and a priori should, be done according to its various properties. The obtained distributive scheme has in particular a number of characteristic (necessary and sufﬁcient) properties or sets of properties, which have (more or less) different meanings (the key issue). Each can be taken as the scheme’s deﬁnition, and as its justiﬁcation (or it can participate in it). Looking at the result from these different angles is necessary for fully ‘‘understanding’’ and ﬁnally evaluating it.16 There are more than 20 such different (although logically equivalent) meanings, which regroup into several types of issues. Equal Liberty The previous remarks have shown the following properties of the result: 1. Social liberty from an equal allocation. 2. Susceptibility to choose some equal allocation with social liberty.

14

SERGE-CHRISTOPHE KOLM

3. Equal freedom of choice (for possibly non-identical domains). 4. Rawls’s solution with leisure (post-1974). ELIE A few other notable aspects are straightforward: 5. ELIE: Redistribute equally the product of the same labour k of all individuals. k is the ‘‘equalization labour.’’ 6. Equal pay for equal work, for labour k (the rate is the average wage This is one of the most widespread claims of justice. However, it rate w). refers here to differences in productivities. 7. From each according to her capacities, to each equally (where ‘‘according to’’ is taken to mean, as it most commonly does, in proportion to): take This associates two of the kwi proportional to wi and give the same kw. most widespread claims of justice. 8. Everyone works for everyone for the same labour (k) and for herself for the rest. Deserts and Merit, Equality and Classical Liberalism, Work and Works Writing yi ¼ kw þ wi ð‘i kÞ

(4)

has shown a decomposition of income into two parts induced by two different and opposed ethics, which can be seen in various ways. 9. Equality and classical liberalism. The two parts are an equal income kw and the market remuneration wi (‘i k) of labour ‘i k. These are the two basic and opposed principles of overall distributive justice in our world. The level of coefﬁcient k favours one or the other and delimitates their respective scopes. 10. Each earns according to deserts for labour k and to merit for the rest. Merit Deserts are according to labour or effort, here k for the share kw. means according to labour or effort and to capacities. This is the second part with individual labour ‘i k and capacities wi. 11. To each according to her work (effort, input) and to her works (product, output). This classical distinction refers here, respectively, to kw in proportion to work k and to the individual’s product wi (‘i k).

Equal Liberties and Resulting Income Distribution and Taxation

15

Financed Universal Basic Income 12. Equal universal basic income ﬁnanced by equal labour (equal sacriﬁce): The result ti ¼ kw wi k can be seen as providing the same basic income kw to each individual, and ﬁnancing it by the same labour k from each (individual i pays the proceeds kwi). 13. Equal universal basic income ﬁnanced according to capacities (i.e., in proportion kwi of wi for individual i). A universal, unconditional and equal basic income has often been proposed by scholars and political ﬁgures. Yet, Achilles’s heel of such schemes is the speciﬁcation of their ﬁnancing which should be sufﬁcient and fair, and should not induce Pareto inefﬁciency. ELIE satisﬁes these conditions. The fairness cannot be an equality in money terms since this would cancel out the distributive effect. Hence, with the relevant items, it has to be equality in labour provided. Reciprocity A basic principle of fairness is reciprocity (in the framework of macrojustice, this is emphasized by Rawls). 14. General equal labour reciprocity: Each individual hands out to each other the proceeds of the same labour (r=k/n). Indeed, the ELIE operation amounts to equally sharing the proceeds kwi of each individual i ’s labour k, hence to yield to each individual the proceeds (k/n)wi of the labour k/n of each individual i (and what an individual yields to herself can be discarded). That is, ti ¼ k ðw wi Þ ¼ r

X

wj nrwi ¼

X

rwj ðn 1Þrwi :

(17)

jai

This property has an aspect of fairness which is bound to be favourable to the acceptance of this scheme from sentiments of reciprocity.17 15. Each owns the rent of the same amount of each other’s capacities (r). Progressive Transfers, Total Concentration ELIE belongs to the question of reducing inequalities, in a particularly meaningful and straightforward way (see also Note 22).

16

SERGE-CHRISTOPHE KOLM

16. Equal partial compensation of productivity differences: Each individual yields to each less productive individual the same fraction of the difference in their productivities, r (wi wj) from i to j if wiWwj. It sufﬁces to consolidate the two transfers of the general equal reciprocity in each pair of individuals. Hence, ELIE amounts to a set of ‘‘progressive transfers’’ for total incomes. This set is, in fact, quite speciﬁc (Property 18). 17. Each individual’s total income is the weighted average between average productivity and this individual’s productivity, with weights k and 1 k, since vi ¼ kw þ ð1 kÞwi :

(5)

18. A concentration of total incomes: This formula also says that the set {vi} is a uniform linear concentration towards the mean of the set {wi}, with degree k. This structure of transformation of a distribution is that which can be said to be the most inequality-reducing.18

Tax Structure and Reform The ﬁscal structure and reform that realize ELIE are very simple, clear, natural, easy to implement, and made of a few elements each of which is classical. 19. An equal tax credit or rebate, and an exemption of overtime labour over some given labour, from a ﬂat tax. Indeed, the transfer can be written as the net tax k (18) ti ¼ 0 wi ‘0 kw ‘ for some given labour ‘0 chosen such that ‘0r‘i for the chosen labours ‘i relevant for macrojustice (see below). The ﬁrst, positive, term is the ﬂat tax with rate k/‘0 on the earnings wi‘0 of labour ‘0, hence with a tax exemption of the corresponding overtime earnings of labour ‘i ‘0. The second term is the tax credit or rebate kw equal for all. This tax structure is simple, clear, with two gratiﬁcations – an exemption and a rebate. For example, the tax exemption of overtime labour over a low duration is the present law in France, with also the equivalent of a universal equal rebate (resulting from an income tax credit).

Equal Liberties and Resulting Income Distribution and Taxation

17

20. Tax reform. The ELIE distributive structure can be obtained from common income taxation by a series of a few simple and rather classical tax reforms: A negative income tax or income tax credit for low incomes, which exists in many countries. Replace actual labour by a given labour in the tax schedule, which is obtainable by exempting earnings over a given labour not exceeding actual (full-time) labours. Flatten the tax schedule, which is often advocated for a reason of simplicity (and incentive)19 – an ELIE scheme can a priori be made as redistributive as one wants by choosing a sufﬁciently high coefﬁcient k. If the scheme concerns the ‘‘distribution branch’’ in ‘‘functional ﬁnance,’’ balance the budget. Formally, from the income tax on labour income f(wi‘i), one thus 0 successively obtains, with constants aW0, bW0, P c, and ‘ W0:0 f(wi‘i)o0 if 0 0 þ c ¼ 0 and f (wi‘i)=0, bw‘ wi‘ioa; f(wi‘ ) or bwi‘i+c; bwi‘ +c; and, if ¼ ti . hence, noting b‘0=k, k ðwi wÞ

Other Meanings 21. Bi-nume´raire equal sharing of the value of productive capacities. An amount of a productive capacity (with a given productivity) can be measured by the labour that can use it (or time of use), or by the output it can produce. In an equal sharing, the choice of this measure makes a difference because individual productivities differ. If an amount of an individual’s productive capacities is measured by the labour input that can use it, each individual has initially 1 and the given allocation without any transfer is equal. If this amount is measured by the output it can produce, however, the total initial endowment of individual i is wi. Both goods – income-consumption and leisure-labour-lifetime – can be taken as nume´raire. Amounts of both are classically compared across individuals. The general solution consists in measuring a fraction of the capacities, say k, in income-value, and the rest, 1 k, in labour-value. For individual i, the and the second equalization of the ﬁrst share transforms income kwi into kw, share is already equal for all in labour value, 1 k. The result is the net income transfer ti ¼ k ðw wi Þ. One can also directly write the total income of individual i from the two parts, vi ¼ kw þ ð1 kÞwi .20,21

18

SERGE-CHRISTOPHE KOLM

REAL GAINS, INCENTIVE COMPATIBILITY Irrelevance of Non-Realized Advantages As we have noted, a concentration transformation of a distribution is, in a sense, the most inequality-reducing transfer structure. Hence, the inequalityreducing effect of a redistribution is meaningfully measured by the coefﬁcient of the concentration which produces the same effect on some measure of inequality. For a redistribution and an inequality index, the ‘‘equivalent ELIE’’ produces the same ‘‘decrease’’ in inequality in total income: its k is the degree of inequality reduction or equalization of this redistribution.22 Consider now the three following facts and judgements: (1) Present redistributions in nations amount to equally redistributing the incomes of 1 to 2 days per week (from the USA to Scandinavia). Hence, de facto – even for the most redistributive policy a country could actually achieve –, for normal full-time labour one has ‘iWk (the cases of total or partial unemployment are the object of Appendix B). (2) Moreover, people commonly understand that individuals who beneﬁt from a high wage rate be taxed to help people who are not as lucky, but only when this advantage provides an actual gain, not when it remains a mere possibility of income. Precisely, people do not agree with a tax on earning capacities that entail no earning because they are not used, that is, with a tax on leisure measuring its value by the earnings this time could provide were it used at labour (taxing to induce work is something else and has to be justiﬁed). ELIE with kW‘i would so imply, when demanding the amount kwi, demanding the value of leisure (k ‘i), (k ‘i)wi, in addition to the value of the whole product wi‘i (for equally redistributing the proceeds). If the redistribution of kw is jointly on taken into account, this would imply demanding ðk ‘i Þðwi wÞ in addition to ðwi wÞ‘ i . If wi is quite low, the leisure (k ‘i) for wi 4w, whatever ‘i. tax kwi is negligible and ti and yi are both about equal to kw, If wi ow remains substantial, and ‘iok, people would again not agree with taxing leisure (k ‘i) at unit value wi for the share (k ‘i)wi of the tax kwi (then equally redistributed). If the subsidy kw is taken into account, people would similarly not agree to subsidize the unused and inactive productive capacities in leisure (k ‘i) because they have a by the part ðk ‘i Þðw wi Þ of the relatively low productivity wi ow,

Equal Liberties and Resulting Income Distribution and Taxation

19

subsidy k ðw wi Þ. Hence, this opinion implies that people who pay an actual distributive tax kwi and receive kw as counterpart are people who choose to work ‘iWk. This common view has to be obeyed in a democracy. (3) The very few productive individuals who choose to work very little mostly choose not to beneﬁt from society’s supply of a favourable wage, and hence arguably do not have to be taxed for this advantage. They choose to drop out of the cooperative venture of collective production (and division of labour), from its advantages, and, hence, from its liabilities. People who choose not to contribute to this joint venture while they could may not be entitled to a reciprocal share of the product. These fugitives from production are not, as Rawls (1982) puts it, ‘‘fully cooperating members of the society engaged in social cooperation over a complete lifetime for mutual advantage,’’ and hence are not party in the sharing of beneﬁts.

These last two points mean that what is at stake concerns actual advantages that people actually derive from their productive capacities and society’s demand for them, rather than these capacities and demand per se – hence as potential earnings. The cases in which the chosen ‘i is lower than k are particular cases: partial or full unemployment, the few eccentric productive people who drop out of cooperative social production, victims of particular handicaps, parttime jobs which are often second wages in families, etc. These particular cases deserve particular criteria and treatments. They are, therefore, out of the scope of overall distributive justice in macrojustice. However, some can also be more or less brought back into the general case, as with involuntary unemployment (Appendix C), the case of people with capacities without market value (wi=0), or the notional equal sharing of the labour of a household among its adults. The case of the tiny fraction of people – if any – who could earn high wages for a moderate effort but decide to live ‘‘on welfare’’ if they can is not a concern for macrojustice for three sets of reasons: the noted ethical reasons and opinions; this is a particular situation (out of the deﬁnition of macrojustice); and its rarity (not an issue for overall justice). These work evaders are the object of classical other proposals and discussions.23 Finally, for all these related reasons, distributive macrojustice is only concerned with normal full-time labour and ‘iWk (the cases of unemployment will be added).

20

SERGE-CHRISTOPHE KOLM

Therefore, for macrojustice, w: yi ¼ wi ‘i þ k ðw wi Þ ¼ wi ð‘i kÞ þ kw4k

(19)

24 That is, there is a minimum income of kw. As noted, the case k=0 is full self-ownership. A case of k=2.5 days a week for a nation would correspond to a very high redistribution (there can, in addition, be various policies of more speciﬁc microjustice).

Incentive Compatibility and Information If wi denotes the highest wage rate individual i can obtain, this individual can also generally earn various rates w0i owi in not using her best (most highly paid) skills at work.25 She may make such a choice if she thinks that the ﬁscal authority bases her taxes and subsidies on this actual and observed or to w0i , in order to diminish the tax or transform it into a subsidy if wi 4w, augment the subsidy if wi ow (hence, she would beneﬁt whatever w if kW0, and therefore she need not know w to behave this way). The individual may think that the government would take the observed w0i as base either because it deems the actual wage rate to be the appropriate basis for the reasons presented in the previous section (not taxing or subsidizing unused capacities of value ðwi w0i Þ) or because it mistakes it for the value of capacities wi, or for any mixture of these reasons. Individual i thus chooses both labour ‘i and skills that earn w0i wi , that maximize some increasing ordinal utility function ui ½1 ‘i ; ð‘i kÞw0i þ kw 0

(20)

P where w 0 ¼ ð1=nÞ w0j .26 Variables ‘i and w0i are independent. The derivative @ui =@w0i has the sign of ‘i k þ k=n if individual i takes the w0j for j 6¼ i as given (no collusion), but whatever they are. Therefore, individual i chooses w0i ¼ wi if ‘iWk[1 (1/n)]. This is the case for macrojustice in which ‘iWk (see the previous section). Hence, the individuals choose to work with their best skills and thus to ‘‘reveal’’ their capacities and to exhibit their economic value. The government can understand this (it does not need to know individuals’ utilities, but only that individuals prefer higher disposable incomes for given labour). Hence, it does not need to raise questions about basing its taxes and subsidies on the actual values of capacities wi or on the

Equal Liberties and Resulting Income Distribution and Taxation

21

observed wage rates w0i since using the latter as base makes them be the wi. And the individuals can in the end know this conclusion.27

INFORMATION, DEGREE OF REDISTRIBUTION, PUBLIC FINANCE The income tax is based on individuals’ wage rates in one country (France), in the form of an exemption of overtime labour from the income tax, over a limited ofﬁcial labour duration.28 This is, therefore, possible. Note that 9/10 of labour is wage labour, as in all developed countries. Cheating is very limited because falsifying the programs of pay sheets is too complicated and could not be done without the tax administration being aware of it or informed about it (very small ﬁrms may be the only exception). By contrast, when full earned income was taxed, not declaring overtime labour was easy and amounted to about half this labour. This former evasion is now lawful and helps taxation. For the intensity and formation dimensions of labour, productivity premia and premia for previous formation – when they exist – are also exempted. All wage labour has a pay sheet, an ofﬁcial legal document for which false report is punished. A pay sheet presents all the needed information: wage rate, total pay, labour duration, overtime work and pay, type of work which often implies formation and intensity, sometimes previous formation, premia, etc. As a general rule, the tax administration uses its usual procedures for information: declarations from employer and employee, checking and cross-checking, with random deeper inspections and important penalties in case of fraud. For all labours, wage rates can be estimated directly or from earnings and labour duration. The relative easiness or difﬁculty to obtain information about these three variables depends on the activity. Earnings are not better known on average (in all countries where they are the base of the tax, about 30% evade the tax).29 Labour duration is well deﬁned, observable and contractual in many jobs, but, of course, not in all. Direct observation or estimate of wage rates provides the base without the need of knowing earnings and labour duration. Estimates often use type of occupation, qualiﬁcation, educational level, sales and proﬁts, and other information.30 Note that, contrary to what is made of his writings, Mirrlees (1971) is in fact quite perceptive concerning information: ‘‘[We] couldyintroduce a tax schedule that depends upon time worked as well as upon labour-income: with such a schedule, one can obtain the full optimumy.We also have other means of estimating a man’s

22

SERGE-CHRISTOPHE KOLM

skill-level.’’ Remark also, ﬁnally, that welfarist optimum taxation raises informational difﬁculties of a higher order of magnitude when demanding information about individuals’ different tastes and utility functions, the cardinality of the latter, comparisons of their variations or levels across individuals, and their aggregation. The equal-liberty optimum distribution is fully determined when coefﬁcient k is. This equalization labour is the fraction of the rent of productive capacities that is collectively owned (with equal sharing for lack of a reason for another distribution in macrojustice). It measures a degree of community of resources, redistribution, equalization, and solidarity. Its absence, k=0, is classical liberalism. The ELIE structure of the distribution results from structural properties practically unanimously supported (social liberty, Pareto efﬁciency, the irrelevance of welfare for macrojustice). Moreover, in a given society, it is not rare to ﬁnd some kind of more or less approximate consensus about what a minimum decent disposable income is. Since ELIE implies the minimum income kw and the average productivity w is given, this is a general opinion about the level k. More generally, in most peaceful societies, the overall level of income redistribution is more or less accepted or approved of, or most opinions in this respect vary in a relatively limited range, although this level changes in time. These opinions are inﬂuenced by the social and political discussions, dialogues, and debates. The revelation of people’s social–ethical opinions is often hampered by their self-interest, but there are a number of ways to deal with this problem (note that opinions about the level of k of people who have an average wage rate wi ¼ w provide a sample of the social–ethical views unaffected by selfinterest since, for them, ti=0 whatever k). Various methods are available to determine the level of coefﬁcient k ‘‘desired by the society;’’ they are the object of Part 4 of the volume Kolm 2004 and, hence, will not be repeated here. This degree of equalization depends on the society in question, notably on the extent to which it constitutes a community. ELIE concerns only the distribution branch or function of the public sector. If the distribution is optimum, the other public expenditures should be ﬁnanced by the method that is neutral in this respect, beneﬁt taxation. Beneﬁts should at any rate be estimated when appraising the need for this expenditure. However, this is sometimes more or less difﬁcult, and classical public ﬁnance also proposes two other principles of ﬁnancing, equal sacriﬁce and according to capacity. If the former is not equally in income (and since macrojustice is non-welfarist), it is equal sacriﬁce in labour (effort). Moreover, for earned income, according to capacity is according to capacity to earn. Then, these two principles amount to the same: each

Equal Liberties and Resulting Income Distribution and Taxation

23

individual i pays cwi, where c is both the equal labourP and the coefﬁcient This is of proportionality to capacity wi. The total amount is c wi ¼ ncw. Different principles can how ELIE ﬁnances the universal basic income kw. be used for different expenditures.

COMPARISON WITH THE OTHER ECONOMIC AND ETHICAL THEORIES Finally, the obtained equal-liberty optimum distribution can be situated in social thought and compared with other economic and social ethical ideas. This equal liberty is an equality of opportunity with two characteristics: it is an equality of opportunity which is not an identity of opportunities (which, with different individual earning capacities, would de facto violate both Pareto efﬁciency and social liberty as noted above); and it applies to the overall distribution in macrojustice. The ‘‘classical liberalism’’ of, for instance, Friedrich Hayek and Milton Friedman is full self-ownership, that is, ELIE with k=0, but they justify it by social liberty, whereas both can be separated (Nozick (1974) probably emphasizes more directly self-ownership). The freedom emphasized by James Buchanan and the school of Public Choice is not moral but is the opposition of self-interested forces; moreover, these scholars rightly emphasize that policies actually result from people’s preferences, but they probably underestimate people’s desires for justice and fairness. Michael Walzer (1983) rightly remarks that justice is considered as equality in separated ‘‘spheres,’’ but one ‘‘sphere,’’ that of overall distribution in macrojustice, is much more important than others in volume (especially since a number of services can be more or less integrated in the market system). Ronald Dworkin (1981), after E. Pazner and David Schmeidler (1978), and Hal Varian (1976), considers a simple ELIE structure with k=1, but rejects it because of the large labour it demands from the very productive people (the ‘‘slavery of the talented’’); however, this level of k, as the case k=0 with a reverse effect, cannot be equality of liberty because the domains of choice of income and leisure are related by inclusion. ELIE is a case of the often proposed universal basic income, with a speciﬁc solution for the problem of its ﬁnancing. We have proposed that the obtained equal-liberty distribution represents what the philosopher John Rawls intends to mean (after amendments, by himself or otherwise, of weaknesses of his initial presentation). At any rate, his starting point in the objection to the relevance of welfarism for macrojustice seems largely endorsed by society at large (see Appendix A). If, as Kenneth Arrow (1963)

24

SERGE-CHRISTOPHE KOLM

proposes, ‘‘The fundamental function of any theory of social welfare is to supply criteria for income distribution,’’ the ELIE tax-subsidy scheme constitutes a solution to this general problem too; the issue is that if ‘‘social choice’’ is derived from ‘‘individual values’’ (Arrow’s title) and these values are not welfarist for this problem, this social choice is not either. Finally, the important literature about axiomatic measures and comparisons of liberty should be introduced here, but this would require new technical developments.31 The most important difference with the approach retained here is that this approach considers a speciﬁc actual social problem, although an important one, macrojustice, and hence it can rest on the concepts and facts provided by society for this issue, such as social liberty (negative freedom and basic rights), types of rights (rights to act and rent-rights), the various types of resources and their relative importance, and people’s opinions about the distribution (spheres of justice, non-welfarism for macrojustice, degree of redistribution, various conceptions of fairness, etc.); a consequence is that the result can be directly applied, and, indeed, its various aspects have more or less been introduced (minimum or basic income, tax exemption of overtime labour, uniform rate, etc.).

NOTES 1. The term welfarism has been coined by Hicks (1959) to criticize the use of this principle when individual liberty is the proper value. 2. It is also sometimes fruitful to distinguish a ﬁeld of ‘‘mesojustice’’ for goods that are particular but particularly important and concern everybody, such as education or health. 3. The theory of social liberty is the full theory of classical concepts such as ‘‘negative freedom’’ (Kant, Berlin), ‘‘social or civic liberty’’ (J.S. Mill), ‘‘basic rights or liberties,’’ etc. It is developed in legal theory and the relevant philosophy. 4. Another classical conception wants to associate to each basic right – which is social liberty for a broad kind of application – material means that make it ‘‘real,’’ and it wants the resulting freedom to be ‘‘equal for all and maximal’’ (Rousseau, Condorcet, the 1789 Declaration, J.S. Mill, Rawls). However, since there is no a priori limit to these associated means (to the size of the cathedral for the freedom of cult, of the various means of communication for the freedom of expression, of private planes and airports for the freedom to move, etc.), this would determine the totality of the allocation of goods, with no rule for choosing among the various goods. 5. See Kolm 1985. 6. This includes capacities to learn in education. 7. The issue of information is considered below. 8. There are other solutions that extend solution 3 into Pareto-efﬁcient solutions, but they use individuals’ preferences even more and have other intrinsic handicaps.

Equal Liberties and Resulting Income Distribution and Taxation

25

One considers individuals’ allocations that are equivalent, for each individual, to her best choice in the common possibility set (a case of ‘‘equivalence theory’’ – see Kolm 2004, Chapter 25). Another rests on the property that individuals can choose their allocations on identical domains of choice if and only if no individual prefers any other’s allocation to her own (Kolm 1971/1998) and extends it to efﬁcient maximins based on comparisons of potential freedom by inclusion of domains (Kolm 1999b). 9. See Kolm 1971, and 1996b for this application. 10. The expression ‘‘free time,’’ rather than ‘‘leisure,’’ would probably suggest better what seems to be valid in this addition, and would better ﬁt Rawls’s conception of primary goods as means. 11. Coefﬁcient k reﬂects the relative moral/social value attached to these two primary goods, and the choice of such a weight is a classical Rawlsian problem. 12. This form is a crucial axiom in Maniquet (1998). 13. This is, for instance, done by proposals of equality of opportunity understood as identity of possibility sets. 14. More precisely, in the space (li (or ‘i), yi), call D such a common possibility set, B its border limiting it towards larger li and yi, and t(li, yi) the set of slopes of the tangents to B at point (li, yi)AB (|t|=1 if B is smooth). Call ui(li, yi) individual i’s utility function assumed to be increasing and differentiable, ui1 and ui2 its two ﬁrst derivatives, and si ðli ; yi Þ ¼ ui1 ðli ; yi Þ=ui2 ðli ; yi Þ the corresponding rate of substitution at point (li, yi). Denote ðl i ; y i Þ for all i the realized state. Pareto efﬁciency and social freedom imply si ðl i ; y i Þ ¼ wi . Individual i’s free choice on D implies ðl i ; y i Þ 2 B and si ðl i ; y i Þ 2 tðl i ; y i Þ. Hence, wi 2 tðl i ; y i Þ. Call Ei={(li, yi): si(li, yi)=wi} individual i’s relevant Engel curve. Therefore, B must satisfy the condition that, at its intersection with Ei, (li, yi)AB-Ei, one has wiAt(li, yi). If all wi were equal, any straight line with slope wi can be such a B, whatever the Ei. However, if the wi are not all equal, the construction of B and D, to satisfy the condition, must take curves Ei into account, and, therefore, must take individuals’ utility functions ui into account. Therefore, if B is built without consideration of the ui and the wi are not all equal, the result violates Pareto efﬁciency and social liberty, except fortuitously. Note that the various solutions correspond to various distributions. 15. Choices in identical domains clearly imply the absence of preferences for another person’s allocation (which the former individual could also have chosen), and when this property of preferences holds, the set of individual allocations constitutes a domain of choice in which each individual’s allocation is one that this person prefers (one can add, to this set, any individual allocation that no individual prefers to her own). 16. The requirement that a principle should be evaluated from all its angles and possible meanings is a classical and basic meta-principle of social ethics, related, for instance, to Plato’s ‘‘dialectics’’ in Republic and to Rawls’s ‘‘reﬂective equilibrium.’’ 17. cf. Kolm 1984, 2008. 18. cf. Kolm 1966a, 1999a. 19. A ﬂat tax is, for instance, implemented in all Eastern European countries including the nine fastest growing countries of the European Union. 20. With ELIE as the solution of Rawls’s full problem, k thus measures the relative importance attached to the two economic primary goods: income relative

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to leisure-labour. With the measure in labour value only, equality is satisﬁed by full self-ownership which is classical liberalism, but is also Marx’s view (he deﬁnes ‘‘exploitation’’ by theft of this property by low wages). 21. ELIE has other interesting and meaningful properties. For instance, Maniquet (1998) derives, from a number of basic axioms, a state which is about the one chosen by the individuals submitted to such a distributive scheme. Moreover, it is noteworthy that ELIE can be derived from the most famous general presentation of principles of justice, that of Plato (Laws) and Aristotle (Nicomachean Ethics), with each person receiving the fruit of her labour wi‘i in ‘‘commutative justice,’’ and an equal share (with the appropriate measure) of what is given to society in ‘‘distributive justice,’’ achieved by compensatory transfers since their capacities are attached to the individuals (‘‘diorthic justice’’) – see Kolm 2004, pp. 248–249. 22. This degree of inequality reduction of a redistribution is equal to the relative decrease in the absolute form of any synthetic index of inequality (Kolm, 1966b). Indeed, for any distribution of incomes (or other quantity) xi whose set is x and P average x ¼ ð1=nÞ xi , one can, for an index of inequality, distinguish the absolute A synthetic inequality index is by form I a(x) and the relative form I r ðxÞ ¼ I a ðxÞ=x. deﬁnition such that I a(x) is equal-invariant (invariant under any equal variation of all the xi) and I r(x) is intensive (invariant under any multiplication of all the xi by the same number). Then, the absolute form is also extensive (linearly homogeneous). A concentration of coefﬁcient k of the distribution amounts to an equiproportional decrease of all xi in proportion k, which similarly decreases the absolute index, and an equal increase that restores the total sum or the mean, which P does not affect this index. P Hence the noted property. Examples of such indexes are |xi xj| (absolute and the standard deviation. Gini), jxi xj, 23. These are, for example, people who can earn 10 times the average income for some standard labour but would prefer to stop working and live on – for instance – 1/5 to 1/3 of average income. For the very few able people who choose to work very little, there are three classical proposals: (1) They should earn their sandwich, ‘‘he who does not work does not eat’’ (Saint Paul), the solution endorsed by Rawls. (2) They should have a ‘‘right to laziness’’ (Paul Laffargue) and perhaps receive a basic income (utilitarianism may support this position, which is eloquently defended by van Parijs (1995)). (3) We may try to persuade them that they should make other people somewhat beneﬁt from the talents endowed to them by nature, providence, or their parents by working a little (at a high wage rate). If their productive capacities are due to subsidized public education which they accepted, they might be asked to refund this cost to the rest of society. If they had to pay for their possible for which they advantage in earning capacity, they would pay ti ¼ k ðwi wÞ, i Þok; however, if they still choose ‘iok, we will see that should work k ½1 ðw=w they may have an interest in hiding their skills and their value wi (yet, diplomas, previous jobs, etc., often make some estimate possible and Ooghe and Schokkaert (2008) have shown that, at any rate, the resulting waste would be very small). Finally, sheer coercion might be restricted to the limited (and possibly highly remunerated) draft of exceptional talents indispensable to society or other people’s life. Note that freedom of choice should a priori refer to the full domain of possible choice in the space of income and leisure rather than to a subset of it only – such as the case ‘i=0 put forward by solution (2). Moreover, there are other distributive units than

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nations; for instance, transfers are intense in a family, but they are gifts rather than taxes (each likes the others’ enjoyment and consumption). 24. One consequence is that, in a society, since w is given, choosing a minimum income and choosing a level of equalization labour k amounts to the same – given that the structural properties that lead to ELIE happen to be largely wanted (social liberty, Pareto efﬁciency, and non-welfarist macrojustice). The relatively frequent rough consensus about a minimum income in a given society implies the same convergence of views about coefﬁcient k. This relation is more valid the more the minimum income refers to a norm of income (and consumption and lifestyle) rather than to the alleviation of physical suffering (which may elicit relief provided by microjustice policies). The general determination of coefﬁcient k will be noted shortly and is the object of Part 4 of the volume Kolm 2004. 25. See Dasgupta and Hammond (1980). 26. Choosing a more remunerated but more painful or disagreeable activity, or the contrary, is considered as working more or less, and a corresponding full analysis has to consider, in a framework of multidimensional labour (see Appendix B), the relevant dimension(s) that affect both the productivity and the painfulness or intrinsic attractiveness of labour. 27. If the government used the wi if it could know them, with ti ¼ k ðw wi Þ, and each individual i could choose her skills used and w0i wi , her income would be ‘i w0i þ k ðw wi Þ, and she would also choose w0i ¼ wi if she chooses to work at all (‘iW0) and therefore when ‘iWk. 28. Thirty-ﬁve hours a week or 1,607 hours a year or, for executives and others whose daily hours of work are unclear, 218 days per year. Similarly, for part-time labour, the tax exemption concerns the so-called ‘‘complementary hours.’’ This tax reform was adopted from a presentation of the result of the present study. There was also previously a tax that demanded each person to pay the proceeds of the same labour time (for subsidizing dependent people). 29. See, for instance, Slemrod (2002) for the US. 30. In the theoretical literature, the incentive effects of ELIE are analysed by Ooghe and Schokkaert (2008), Fleurbaey and Maniquet (2008), Trannoy and Simula (2008), and various contributions in the volumes edited by Gamel and Lubrano (2008) and Fleurbaey, Salles and Weymark (2008). 31. The reasons for ELIE presented here constitute de facto a set of axioms. Moreover, the set of axioms provided by Maniquet (1998) leads to allocations that are practically those chosen by the individuals in an ELIE tax-subsidy regime. 32. Any more than, for instance, physical beauty. 33. Mirrlees (1971, 1986), and the ensuing literature. 34. His view on this point is shared by a large number of scholars in the various disciplines (among others Dworkin, 1981, but also ‘‘classical liberals’’). Yet the rest of their conception, as that of Rawls, raises problems. 35. The leximin in interpersonally comparable utility is the eudemonistic ‘‘practical justice’’ in Kolm 1971, discussed by Rawls, but not proposed for any speciﬁed application. 36. Beyond these general conclusions, however, most of Rawls’ more speciﬁc proposals are logically problematic for speciﬁc reasons. (1) His maximin in ‘‘primary goods’’ (the ‘‘difference principle’’) omits that the bases of transfers and taxation can

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be much less elastic (hence waste inducing) than they commonly are – the issues of deﬁning an index of these goods and of relating this to Pareto efﬁciency are much more secondary matters. (2) The theories of the ‘‘original position’’ and of the ‘‘veil of ignorance,’’ both in Rawls’s version and in Harsanyi’s (which gives a kind of utilitarianism or, at least, separable welfarism), are problematic because a selﬁsh individual choice in uncertainty does not have the same structure (and objects) as a choice of justice (see Kolm 1996a, pp. 191–194, 2004, pp. 358–360). (3) The classical theory of equal and maximal real basic liberties does not hold (see Note 4). 37. The 1789 Declaration of Rights and the American Declaration of Independence. 38. For macrojustice, the effects of other persons’ labour on an individual’s earnings pass through the prices. 39. The educational input can also be taken into account by ‘‘spreading’’ the formation time on later labour (that uses its beneﬁts) (see details in Kolm 2004, Chapter 8). 40. A reﬁnement of the analysis can ﬁnd ways of taking account of some individually chosen effort at the end of the educational period. 41. There is even a ground for compensating sociological differences more than those due to intrinsic individual capacities which belong to the person’s self, but this issue is not pursued in this simple presentation. 42. For other levels of wi, the case of individuals who choose to work very little (‘iok) is treated as indicated previously. 43. Low wi at a given time only is normally the object of an insurance (health, unemployment – see also below – etc.). 44. Computations of the effects are provided in Kolm 2004, Chapter 7. 45. A particular case can be pi(‘i)=wi‘i.

REFERENCES Arrow, K. J. (1963). Social choice and individual values. New York: Wiley; New Haven: Yale University Press. Dasgupta, P., & Hammond, P. J. (1980). Fully progressive taxation. Journal of Public Economics, 13, 141–154. Dworkin, R. (1981). ‘‘What is equality? Part I: Equality of welfare’’, ‘‘Part II: Equality of resources’’. Philosophy and Public Affairs, 10, 185–246; 283–345. Fleurbaey, M., & Maniquet, F. (2008). ELIE and incentives. In: M. Fleurbaey, M. Salles, & J. Weymark (Eds), Social ethics and normative economics, in press. Fleurbaey, M., Salles, M., & Weymark, J. (Eds). (2008). Social ethics and normative economics. Heidelberg: Springer Verlag. Gamel, C., & Lubrano, M. (Eds). (2008). Macrojustice. Heidelberg: Springer Verlag. Hicks, J. (1959). Essays in world economy. Oxford: Basil Blackwell. Kolm, S.-Ch. (1966a). Les Choix Financiers et mone´taires. Paris: Dunod. Kolm, S.-Ch. (1966b). The optimal production of social justice. In: H. Guitton & J. Margolis (Eds), Proceedings of the International Economic Association conference on public economics, Biarritz. Economie publique, 1968, Paris: CNRS, pp. 109–177. Public

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economics, 1969, London: Macmillan, pp. 145–200. Reprinted in The foundations of 20th century economics, landmark papers in general equilibrium theory, social choice and welfare, selected by K. J. Arrow and G. Debreu, 2001, Cheltenham: Edward Elgar, pp. 606–644. Kolm, S.-Ch. (1971) Justice et e´quite´, Paris: Cepremap. Reprint (1972), Paris: CNRS. English translation, (1998), Justice and equity. Cambridge MA: MIT Press. Kolm, S.-Ch. (1984). La bonne economie: La re´ciprocite´ ge´ne´rale. Paris: Presses Universitaires de France. Kolm, S.-Ch. (1985). Le contrat social libe´ral. Paris: Presses Universitaires de France. Kolm, S.-Ch. (1996a). Modern theories of justice. Cambridge, MA: MIT Press. Kolm, S.-Ch. (1996b). The theory of justice. Social Choice and Welfare, 13, 151–182. Kolm, S.-Ch. (1999a). Rational foundations of income inequality measurement. In: J. Silber (Ed.), Handbook of income inequality measurement (pp. 19–94). Dordrecht: Kluwer. Kolm, S.-Ch. (1999b). Freedom justice. Working paper no. 99-5. CREME, Universite´ de Caen. Kolm, S.-Ch. (2004). Macrojustice, the political economy of fairness. Cambridge, MA: Cambridge University Press. Kolm, S.-Ch. (2008). Reciprocily. Cambridge: Cambridge University Press. Maniquet, F. (1998). An equal right solution to the compensation-responsibility dilemma. Mathematical Social Sciences, 35, 185–202. Mirrlees, J. (1971). An exploration in the theory of optimum income taxation. Review of Economic Studies, 38, 175–208. Mirrlees, J. (1986). The theory of optimal taxation. In: K. J. Arrow & M. D. Intriligator (Eds), Handbook of mathematical economics (Vol. 3). Amsterdam: North-Holland. Musgrave, R. (1959). The theory of public ﬁnance. New York: McGraw Hill. Musgrave, R. (1974). Maximin, uncertainty, and the leisure trade-off. Quarterly Journal of Economics, 88, 625–632. Nozick, R. (1974). Anarchy, state and utopia. New York: Basic Books. Ooghe, E., and Schokkaert, E. (2008). Would full ELIE be a wasteful scheme? In: C. Gamel & M. Lubrano (Eds), Macrojustice, in press. Pazner, E., & Schmeidler, D. (1974). A difﬁculty in the concept of fairness. The Review of Economic Studies, 41(3), 441–443. Pazner, E., & Schmeidler, D. (1978). Decentralization and income distribution in socialist economies. Economic Inquiry, XVI, 257–264. Rawls, J. (1971). A theory of justice, Revised edition 1999. Cambridge, MA: Harvard University Press. Rawls, J. (1974). Reply to Alexander and Musgrave. Quarterly Journal of Economics, 88, 633–655. Rawls, J. (1982). Social unity and primary goods. In: A. Sen & B. Williams (Eds), Utilitarianism and beyond (pp. 159–185). Cambridge, MA: Cambridge University Press. Slemrod, J. (2002, Summer). Tax systems. NBER Reporter, pp. 8–13. Trannoy, A., & Simula, L. (2008). When Kolm meets Mirrlees: ELIE. In: M. Fleurbaey, M. Salles, & J. Weymark (Eds), Social ethics and normative economics, in press. Van Parijs, P. (1995). Real freedom for all. Oxford: Oxford University Press. Varian, H. (1976). Two problems in the theory of fairness. Journal of Public Economics, 5, 249–260. Walzer, M. (1983). Spheres of justice. Oxford: Blackwell.

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APPENDIX A. TESTS OF WELFARISM FOR MACROJUSTICE A normative study can be applied only if people who actually inﬂuence its implementation sufﬁciently adhere to its normative criterion (they may be voters, people at large, politicians, tax ofﬁcials, etc.). A model of optimum income taxation is probably proposed for application. Therefore, if it is based on welfare, it rests on the hypothesis that welfarism is an accepted principle for macrojustice. Scientiﬁc methodology leads one to ask: does any test falsify this hypothesis, or not? Here are a few tests among many possible ones.

The European Union Test If, as it is said, the people of Northern Europe are better at producing and those of Southern Europe more skilful at enjoying consumption, should the European Union set up a vast program of intra-European North-South income transfers? Should it tax the industrious Swedes for subsidizing the Napolitans who make a feast from a meal? This would be the injunction of utilitarianism. Or perhaps, on the contrary, should this tax subsidize the Portuguese reputedly afﬂicted by a kind of mild sadness, in order to soothe their saudade? This would be required by a maximin in utility. However, everybody should help the victims of uninsured occurrences causing insufferable pain, but these are cases of speciﬁc microjustice aiming at the relief of suffering.

The Earned Income and Legitimate Ownership Test ‘‘I take the 10 euros you just earned because I like them more than you do.’’ Is this a good reason? Or perhaps, on the contrary, ‘‘I take your earnings because you like your remaining euros more than I like mine.’’ Is this a better reason? Am I entitled to (or should I) take your money because it pleases me more than it pleases you? Or perhaps, on the contrary, because you enjoy your money left more than I am able to enjoy my own? These two opposite ways of comparing our tastes for income are, respectively, utilitarianism and maximin in utility, the two polar cases of welfarism. If, however, your 10 euros enable me to buy the drug that saves my life, most people will excuse the theft, but this is a case of speciﬁc microjustice for the alleviation of suffering.

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The Taste, Preference, or Desire Tests Should you ﬁnance somebody’s beverage because her special taste for cheap beer permits her drinking to produce utility at low cost (as utilitarianism requires)? Or because she only likes expensive wine (as egalitarian maximin or another welfarist principle may demand)? Nevertheless, you should probably give water to your thirsty neighbour, to relieve her pain cheaply. Rawls (1982) points out yet another aspect, for ‘‘social justice’’: ‘‘Desires and wants, however intense, are not by themselves reasons in matters of justice. The fact that we have a compelling desire does not argue for its satisfaction any more than the strength of a conviction argues for its truth.’’

The Income Tax Test Should you pay a higher income tax than someone else because you like the euros taken away less than she does or, on the contrary, because you like the remaining euros more than she does – as utilitarianism and maximin in utility tend to require, respectively? Are, in fact, these considerations relevant for this issue? To begin with, do these comparisons of enjoyment make sense, are they possible? At any rate, should you pay more or less because you have a cheerful character, or because the other has a cheerful character (which may lead one to enjoy a euro more or to regret its absence less – opposite effects again)? In fact, has the Internal Revenue Service ever thought about sending questionnaires to inquire about these relative propensities or capacities to enjoy? Or does it think that this would be irrelevant and, perhaps, abusively intrusive; that these psychological characteristics are private matters and not the concern of overall and general public policy and the income tax; that, for this question, people are accountable for their own tastes, entitled to their beneﬁcial effects and having to endure non-pathologically less favourable ones; and that such normal differences in tastes could not give rise to compensating claims on others’ incomes or liabilities towards them?32

The Implementability Test The welfarist theory of the optimum income tax is about a very important topic.33 It is very well known (and justly admired) by economists who want their work to be useful and seek application. Some eminent contributors to

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it have even had major economic responsibilities at world and national levels. Why, then, is this remarkable theory still waiting for the beginning of an application after nearly four decades? Can it be applied, at least in a democracy? To begin with, would ofﬁcials and voters endorse its welfarist ethic? Or in fact do they discard it – for this application – when it is explained to them?

The Distributive Opinion Test The opinions about overall distribution that exist in society have two polar positions; policies apply some mix of them or compromise between them, and individuals also often endorse more or less some mix. One polar position is income egalitarianism. It sees equality in incomes as the ideal. Since individuals have different utility functions, this cannot result from any kind of welfarism. The other polar position holds that earned income should belong to the earner (‘‘classical liberalism’’). It is not welfarist either. Hence, welfarism seems absent from actual moral positions about the overall distribution in macrojustice.

The Rawls (and Many Other Scholars) Test John Rawls is the most famous of contemporary philosophers. His basic work, A Theory of Justice, is an indictment of welfarism for macrojustice (his ‘‘social justice’’ – he once uses the term ‘‘macro’’ and says ‘‘not micro’’).34 He says he presents his own theory because a critique is fully convincing only if an alternative is proposed. Some economists hide this fact by calling ‘‘Rawlsian’’ a maximin in utility. But Rawls’ maximin (his ‘‘difference principle’’) is in ‘‘primary goods,’’ not in utility. This most basic point is unambiguous: ‘‘To interpret the difference principle as the principle of maximin utility (the principle to maximize the well-being of the least advantaged person) is a serious misunderstanding from a philosophical standpoint’’ (1982).35 Hence, his remarks that ‘‘Justice as fairness rejects the idea of comparing and maximizing satisfaction’’ and ‘‘The question of attaining the greatest net balance of satisfaction never arises in justice; this maximum is not used at all’’ (1971) intend to point out a commonsense and moral inappropriateness of welfarism. Therefore, Rawls naturally acknowledges: ‘‘A principle of equal liberty.’’ ‘‘A just social system deﬁnes the scope within which individuals must develop their aims, and it provides a

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framework of rights and opportunities and the means of satisfaction within and by the use of which these ends may be equitably pursued’’ (id.).36

The Mirrlees (1971) Test In this article, which is considered the basis of the theory of welfarist income taxation and followed by almost all the subsequent literature, Mirrlees states ‘‘Differences in tastes y raise rather different kinds of problems,’’ and uses this argument for attributing the same utility function to all individuals. However, an individual’s satisfaction, happiness, utility, or welfare depends on her consumption and her tastes. Individuals do not have the same utility functions. Hence, this unique utility function describes neither individuals’ welfares nor their actual choices as assumed by the theory (except, possibly, for one individual). In particular, the outcome is not Pareto efﬁcient. This is not actually welfarism. We do not know, moreover, what this function is and it determines the income tax. Since individuals’ tastes actually differ and their differences affect the result (as shown in Mirrlees, 1986), discarding the effects of these differences implies deleting tastes and utility functions altogether: this is Rawls’s solution. If we add the second of Mirrlees’s moral statement, ‘‘The great desirability of y offsetting the unmerited favours that some of us receive from our genes and family advantages,’’ there results that Mirrlees’s (1971) ethical view is exactly that of Rawls. Nevertheless, Mirrlees (1986) chose the other solution and considered different utility functions, thus raising the corresponding informational problem and, more deeply, opposing common views for this application.

The Constitution Test The basic principle of our societies, the transgression of which is unlawful and punished, is given by our constitutions and founding declarations. It consists of liberty and rights rather than welfare. Happiness is essential but private. ‘‘Men are free and equal in rights.’’ They should be secured the liberty and means to ‘‘pursue happiness’’ as they see ﬁt, rather than some level of happiness.37 Property rights are basic, and the legitimacy of someone’s property of something is provided not by some beneﬁcial consequence but by the condition of its acquisition, notably free actions and exchanges.

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APPENDIX B. MULTIDIMENSIONAL LABOUR, NON-LINEAR PRODUCTION Labour has a priori various dimensions, such as duration, individual effort and costs in previous education and training, intensity (strength, concentration), speed, etc. Moreover, the output may not be a linear function of labour. Let ‘i denote a multidimensional labour of individual i, and pi(‘i) the corresponding earnings.38 All the reasonings, results, and meanings presented for the simple case can be repeated for this general case practically identically. The equalization labour k is now multidimensional. The taxsubsidy is pi ðkÞ ti ¼ pðkÞ ¼ ð1=nÞ where pð‘Þ

P

(B.1)

pi ð‘Þ, and individual i’s disposable income is yi ¼ pi ð‘i Þ pi ðkÞ þ pðkÞ:

(B.2)

This multidimensional case can often practically be reduced to a onedimensional case with labour duration adjusted for the other characteristics of labour. Indeed, labour can generally be considered as a ﬂow, and as steady in some given period (which can be taken as short as one wants). Then, if ‘0i denotes the duration of labour ‘i and ‘00i the set of its other parameters, function pi can be written as pi ð‘i Þ ¼ ‘0i qi ð‘00i Þ. If individuals’ particular productivities are of the classical ‘‘output augmenting’’ type qi ð‘i00 Þ ¼ ai f ð‘i00 Þ, then pi(‘i)=wiLi where Li ¼ ‘i0 f ð‘i00 Þ is individual i ’s ‘‘labour duration augmented for the other characteristics of labour’’, and wi=ai is the corresponding competitive wage rate.39 In the expression of earnings from labour ‘i, pi(‘i), labour ‘i represents items chosen by individual i, and the function pi( ) the other items, that is, individual i’s productivity and the labour market. Formation, education, and training (as health care) increase later productivity. They depend on the persons’ given capacities for learning. They also involve acts of the individual and possibly various costs for her (time, effort, direct costs, foregone earnings, etc.). However, the bulk of the formation and education received in the ﬁrst period of life is provided by the family, or determined by it through choice, support, information, and induced motivation. Globally, at a macrolevel and apart from exceptions, individuals’ level of education is essentially a sociological phenomenon. Hence, for macrojustice and as a ﬁrst approximation, its effects on earnings have to be incorporated in the productivity pi( ) or the wage rate wi under consideration. By contrast,

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training and formation undertaken later a priori constitute a dimension of labour.40 Note that the effects of different pi( ) or wi are equalized only for labour k and not for the rest of labour. This effect of the family should also be considered with the issue of bequest – its cost can be seen as a part of it.41 Family-induced education could be sensitive to future taxation, but this is much attenuated by the fact that taxes decades later are very uncertain and by the non-pecuniary values of education as providing larger occupational opportunities and freedom of choice, jobs that are less painful and more interesting and gratifying, the status of educational level and occupations, culture, and the pursuit of family traditions.

APPENDIX C. UNEMPLOYMENT Situations of unemployment raise particular speciﬁc issues, but, given their importance, they should be related to the general results for macrojustice. If wi=0, individual i’s labour is neither supplied for income nor demanded, the minimum or basic and the formula ti=k (w wi ) gives yi ¼ ti ¼ kw, whatever ‘i. These people’s income. If wi is low, ti and yi are close to kw, actual labour level makes little ﬁnancial difference.42 Hence, the general principle can be applied to these cases (apart from the other policies of formation, education, taking care of handicaps, etc.).43 In involuntary unemployment, the individual faces a constraint ‘i ‘i0 . This unemployment may be partial or total (labor duration of zero), and for duration or for other dimensions of labor (for instance, as underqualiﬁcation for formation). Reasons for discarding cases ‘iok from macrojustice may not hold any longer for this case: these people do not abstain voluntarily from participation in social production, and their number may not be small. Of course, good macroeconomic policy in the ﬁrst place, unemployment insurance, and speciﬁc policies about the labour market and formation are in order. However, the obtained distributive policy can have three important positive effects on employment. By basing taxes and subsidies on items less elastic than actual labour, it generally induces higher labour. The other two effects concern involuntary unemployment in the strict sense. First, the income support to people with low wage rates provided by the obtained scheme can supersede, to everybody’s beneﬁt, a number of wage rigidities of public or private nature which are important causes of unemployment (minimum wages, collusions, etc.).44 Second, the general results for macrojustice can also apply to the case of involuntary unemployment, by using the logical device of considering someone who

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cannot work more as someone who cannot earn more by working more (and works to earn). What the market presents to the individual is then described solely in terms of the remuneration of each labour (yet, for partial unemployment it cannot be a linear function of labour). Considering one-dimensional labour for simplicity in presentation, the outcome is that someone involuntarily unemployed at ‘0i k (in particular ~ ¼ totallyPunemployed) has income pðkÞ which derives from the average pðkÞ ð1=nÞ pi ðkÞ by replacing the pi(k) of such individuals by pi ð‘i0 Þ (0 for full unemployment). This results from the application of the noted device by replacing, for each i with a constraint ‘i ‘i0 , the function pi(‘i) by its truncation at ‘i0 :45 Pi(‘i)=pi(‘i) if ‘i ‘i0 and Pi(‘i)=pi(‘i0 ) if ‘i ‘0i , with pi(0)=0 for full unemployment. Then, applying the ELIE scheme to functions Pi gives ti ¼ Pi ðkÞ and yi=Pi(‘i)+ti=Pi(‘i) Pi(k)+PðkÞ. PðkÞ If ‘i ¼ ‘0i and ‘0i k, 0 0 ~ ¼ pðkÞ. This is in Pi(k)=pi(‘i )=Pi(‘i )=Pi(‘i), and therefore yi ¼ PðkÞ particular the case for full unemployment, ‘0i ¼ 0. Moreover, if, when ‘0i 40, person i chooses to work less than ‘0i , her income is reduced by the corresponding loss in output.

INEQUALITY AND ENVY Frank Cowell and Udo Ebert ABSTRACT Our purpose is to examine the ‘‘envy’’ within the context of income inequality measurement. We use a simple axiomatic structure that takes into account ‘‘envy’’ in the income distribution. The concept of envy incorporated here concerns the distance of each person’s income from his or her immediately richer neighbour. We derive two classes of inequality indices – absolute and relative. The envy concept is shown to be similar to justice concepts based on income relativities. This is the ﬁrst time a complete characterisation has been provided for envy-related inequality.

1. INTRODUCTION There is considerable interest in a possible relationship between inequality and envy. However, there has been little attempt to incorporate the concept of individual envy directly into the formal analysis of inequality measurement. Of course, there is a substantial economics literature that models envy in terms of individual utility – see for example Arnsperger (1994) – but our focus here is different in that we concentrate directly on incomes rather than on utility and commodities. We seek an alternative way of characterising Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 37–47 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16002-1

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envy broadly within the literature that has formalised related concepts such as deprivation and individual complaints about income distribution. Indeed, there is an aspect of envy that can be considered as akin to the notion of a ‘‘complaint’’ that has been used as a basic building block of inequality analysis (Temkin, 1993).1 A further motivation for our analytical approach can be found in the work of social scientists who have sought to characterise issues of distributive justice in terms of relative rewards. This is sometimes based on a model of individual utility that has, as arguments, not only one’s own income, consumption or performance, but also that of others in the community. A recent example of this approach is the model in Falk and Knell (2004) where a person’s utility is increasing in his or her own income and decreasing in some reference income; Falk and Knell (2004) raise the key question as to what constitutes reference income. Should it be the same for all or relative to each person’s income? Should it be upward looking, as in the case of envy? Here we address these issues without explicitly introducing individual utility. Our approach has a connection with the seminal contribution of Merton (1957) who focuses on a proportionate relationship between an individual’s income and a reference income characterised in terms of justice: we will show that there is a close relationship between some of the inequality measures developed below and Merton’s work. The paper is organised as follows: Section 2 outlines the basic framework within which we develop our analysis; Section 3 characterises and examines the properties of a class of absolute inequality measures; and Section 4 analyses the corresponding class of relative indices.

2. THE APPROACH We assume that the problem is one of evaluating and comparing income distributions in a ﬁnite ﬁxed-sized population of at least two members, where individual ‘‘income’’ has been deﬁned as a real number, not necessarily positive. Throughout the following, we will work with vectors of ordered incomes.

2.1. Notation and Deﬁnitions Let D be the set of all logically possible values of income. For different parts of the analysis, we will need two different versions of this set, namely,

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D ¼ R and D ¼ Rþþ . An income distribution is a vector x :¼ ðx1 ; x2 ; . . . ; xn Þ 2 Dn where the components are arranged in ascending order and nZ2. So, the space of all possible income distributions is given by XðDÞ :¼ fxjx 2 Dn ; x1 x2 xn g Write 1k for the k-vector (1,1,y,1) and let x(k, d) denote the vector x modiﬁed by increasing the kth component by d and decreasing the component k þ 1 by d: xðk; dÞ :¼ ðx1 ; x2 ; . . . ; xk þ d; xkþ1 d; . . . ; xn Þ where 1okon and 1 0od ½xkþ1 xk 2

(1)

Deﬁnition 1. An inequality measure is a function J : XðDÞ ! Rþ . Deﬁnition 2. For any k such that 1okon and any x 2 XðDÞ such that xk oxkþ1 , a progressive transfer at position k is a transformation x 7! xðk; dÞ such that Eq. (1) is satisﬁed. Note that Deﬁnition 2 applies the concept of Dalton (1920) to transfers between neighbours. It should also be noted that we describe our envyrelated index everywhere as an inequality measure even where we do not insist on the application of the principle of progressive transfers. As is common in the inequality literature, we will deal with both absolute and relative approaches to inequality measurement.

2.2. Basic Axioms Our main ethical principle is captured in the following two axioms. Axiom 1. (Decomposability: non-overlapping sub-groups). For all x 2 XðDÞ and 1 k n 1: JðxÞ ¼ Jðx1 ; . . . ; xk ; xk 1nk Þ þ Jðxkþ1 1k ; xkþ1 ; . . . ; xn Þ þ Jðxk 1k ; xkþ1 1nk Þ (2)

40

FRANK COWELL AND UDO EBERT

Axiom 2. (Monotonicity). For all x; y 2 D such that xry and for all 1 k n 1 inequality Jðx1k ; y1nk Þ is increasing in y. Axiom 1 is fundamental in that it captures the aspect of the income distribution that matters in terms of envy at any position k. It might be seen as analogous to a standard decomposition – total inequality [on the left of Eq. (2)] equals the sum of inequality in the lower and upper sub-groups deﬁned by position k [the ﬁrst two terms on the right of Eq. (2)] and a between-group component (last term on the right). However, the analogy with conventional decomposition by sub-groups is not exact – note for example that the ﬁrst two terms on the right are not true sub-groupinequality expressions (which would have to have population sizes k and n – k respectively), but are instead modiﬁed forms of the whole distribution. Indeed, a better analogy is with the focus axiom in poverty analysis: in the breakdown depicted in Eq. (2), we have ﬁrst the information in the rightcensored distribution than that in the left-censored distribution then the information about pure envy at position k. Axiom 2 has the interpretation that an increase in the pure envy component in Eq. (2) must always increase inequality and that this increase is independent of the rest of the income distribution. To make progress, we also need some assumptions that impose further structure on comparisons of income distributions. We will ﬁrst consider the following two axioms: Axiom 3. (Translatability). For all x 2 XðDÞ and 2 R: Jðx þ 1n Þ ¼ JðxÞ. Axiom 4. (Linear homogeneity). For all x 2 XðDÞ and l 2 Rþþ : JðlxÞ ¼ lJðxÞ. Axioms 3 and 4 are standard in the literature; however, in Section 4 we will examine the possibility of replacing these with an alternative structure.

3. ABSOLUTE MEASURES We begin with results for the most general deﬁnition of the space of incomes. Here incomes can have any value, positive, zero or negative; that is D ¼ R. We will ﬁrst characterise the class of measures that is implied by the parsimonious axiomatic structure set out in Section 2, and then we will examine this class in the light of the conventional properties with which inequality measures are conventionally endowed.

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3.1. Characterisation To start with, let us note that the decomposability assumption implies that J has a convenient property for a distribution displaying perfect equality: Proposition 1. Axiom 1 implies that for all x 2 D: Jðx1n Þ ¼ 0

(3)

Proof. For an arbitrary integer k such that 1 k n 1, Axiom 1 implies Jðx1n Þ ¼ 3Jðx1k ; x1nk Þ ¼ 3Jðx1n Þ But this is only true if Eq. (3) holds. We use this property in the proof of the main result, Proposition 3 below. Proposition 2. Axioms 1 and 2 imply that 1. for all x 2 XðDÞ such that not all components of x are equal: (4)

JðxÞ40 2. for all x 2 XðDÞ JðxÞ ¼

n1 X

K i ðxi ; xiþ1 Þ

(5)

i¼1

where each Ki satisﬁes the property K i ðxi ; xiþ1 Þ40 if xiþ1 4xi . Proof. Applying Axiom 1 in the case k ¼ 1, we have JðxÞ ¼ Jðx1 ; x1 1n1 Þ þ Jðx2 ; x2 ; x3 ; . . . ; xn Þ þ Jðx1 ; x2 1n1 Þ So, by Proposition 1, we have JðxÞ ¼ Jðx2 ; x2 ; x3 ; . . . ; xn Þ þ Jðx1 ; x2 1n1 Þ

(6)

Applying Axiom 1 again to the ﬁrst term in Eq. (6), we obtain JðxÞ ¼ ½Jðx3 ; x3 ; x3 ; x4 ; . . . ; xn Þ þ Jðx2 12 ; x3 1n2 Þ þ Jðx1 ; x2 1n1 Þ Repeated application of the same argument gives us JðxÞ ¼

n1 X i¼1

Jðxi 1i ; xiþ1 1ni Þ ¼

n1 X i¼1

K i ðxi ; xiþ1 Þ

(7)

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FRANK COWELL AND UDO EBERT

where K i ðÞ is deﬁned such that K i ðx; yÞ ¼ Jðx1i ; y1ni Þ. Axiom 2 and Proposition 1 together imply that K i ðx; yÞ40 if y>x. Part 1 of Proposition 2 shows that J satisﬁes a minimal inequality property; part 2 demonstrates that Axioms 1 and 2 are sufﬁcient to induce an appealing decomposability property. Proposition 3. For D ¼ R, the inequality measure J satisﬁes Axioms 1–42 if and only if there exist weights a1 ; . . . ; an1 2 Rþþ such that, for all x 2 XðDÞ: JðxÞ ¼

n1 X

ai ½xiþ1 xi

(8)

i¼1

Proof. The ‘‘if ’’ part is immediate. From the proof of Proposition 2, we know that J must have the form of Eq. (7). Using Axiom 3 for the distribution ðx1i ; y1ni Þ, we have K i ðx; yÞ ¼ Jðx1i ; y1ni Þ ¼ Jð0 1i ; ½ y x1ni Þ ¼ K i ð0; y xÞ

(9)

Putting x ¼ 0 in Eq. (9) and using Axiom 4, it is clear that K i ð0; yÞ ¼ Jð0 1i ; y1ni Þ ¼ yJð0 1i ; 1ni Þ ¼ ai y where ai :¼ K i ð0; 1Þ. Applying Axiom 2 to Eq. (7), we have ai 40.

(10)

Let us note that, by rearrangement, of Eq. (8) we have the convenient form JðxÞ ¼ an1 xn þ

n1 X

½ai1 ai xi a1 x1

(11)

i¼2

This weighted-additive structure is useful for clarifying the distributive properties of the index J.

3.2. Properties of the J-Class To give shape to the class of measures found in Proposition 3, we need to introduce some extra distributive principles. The following axiom may be stated in weak or strict form for each position k where 1 k n 1: Axiom 5. (Position-k monotonicity). For all x 2 XðDÞ such that xk oxkþ1 inequality J(x) is decreasing or constant in xk.

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Inequality and Envy

However, the progressive-transfers axiom requires a slightly tighter choice of k: Axiom 6. (Progressive transfers). For any k such that 1okon and any x 2 XðDÞ such that xk oxkþ1 , a progressive transfer at position k implies Jðxðk; dÞÞ JðxÞ

(12)

These two axioms have some interesting implications for the structure of the inequality measure J. However, their properties are independent and in each case there is an argument for considering the axiom in a strong or a weak form. In the light of this, there are a number of special cases that may appear to be ethically attractive, including the following: Strong position-k monotonicity only – inequality is strictly decreasing in xk for all positions k. Strong progressive transfers – the ‘‘o’’ part in Eq. (12) is true for all positions k. Indifference across positions – inequality is constant in xk for all positions k in the statement of Axiom 5. Indifference clearly implies that the ‘‘=’’ part in Eq. (12) is true. Imposition of one or other form of Axioms 5 and 6 will have implications for the structure of the positional weights fak g. First, it is clear from Eq. (11) that Axiom 5 implies 0oa1 ak akþ1 an1

(13)

since we have a1 40 in view of Proposition 3. So, increasing the poorest person’s income always reduces J-inequality. Second, if we adopt the position of indifference across positions in Axiom 5, then 0oa1 ¼ ¼ ak ¼ akþ1 ¼ ¼ an1 In this case, it is clear from Eq. (11) that the inequality measure becomes just a multiple of the range.3 Third, an important property follows directly from Axiom 6 alone: Proposition 4. Given the conditions of Proposition 3, imposition of the principle of progressive transfers at each position k, k ¼ 2; . . . ; n 1 implies that the weights ak in Eq. (8) can be written as ak ¼ jðkÞ where j is a concave function.

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FRANK COWELL AND UDO EBERT

Proof. Applying Axiom 6 to Eq. (11), we get JðxÞ Jðxðk; dÞÞ ¼ ½ak1 ak d ½ak akþ1 d 0 Given that d>0 implies ak which establishes concavity.

ak1 þ akþ1 2

4. RELATIVE MEASURES The discussion in Section 3 is essentially ‘‘absolutist’’ in nature – the translatability property (Axiom 3) ensures this. Here, we look at the possibility of a ‘‘relativist’’ approach to characterising an envy-regarding index. 4.1. Characterisation In this case, we have to deal with a restricted domain: D can consist only of positive numbers ðD ¼ Rþþ Þ and we impose the following axioms: Axiom 7. (Zero homogeneity). For all x 2 XðRþþ Þ and l 2 Rþþ : JðlxÞ ¼ JðxÞ. Axiom 8. (Transformation). For all x 2 XðRþþ Þ and 2 Rþþ : Jðx Þ ¼ JðxÞ, where x :¼ ðx1 ; x2 ; . . . ; xn Þ. Axioms 7 and 8 replace Axioms 3 and 4 now that the deﬁnition of D is changed from R to Rþþ . This enables us to introduce a modiﬁed characterisation result: Proposition 5. For D ¼ Rþþ , the inequality measure J satisﬁes Axioms 1, 2, 7 and 8 if and only if there exist weights a1 ; . . . ; an1 2 Rþþ such that, for all x 2 XðDÞ: JðxÞ ¼

n1 X

ai ½ln xiþ1 ln xi

(14)

i¼1

Proof. For any y 2 XðRÞ let x :¼ ðey1 ; ey2 ; . . . ; eyn Þ

(15)

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Inequality and Envy

^ deﬁned as JðyÞ ^ :¼ JðxÞ with x given by Eq. (15). If J and consider JðÞ satisﬁes Axioms 1, 2, 7 and 8, then J^ satisﬁes Axioms 1–4 on XðDÞ for D ¼ R. Using Proposition 3, we have therefore ^ ¼ JðyÞ

n1 X

ai ½yiþ1 yi

(16)

i¼1

with a1 ; . . . ; an1 2 Rþþ . Using the transformation (15) in Eq. (16) gives the result.

4.2. Properties The properties of J are similar to those established in Section 3.2. Clearly JðxÞ ¼ an1 ln xn þ

n1 X

½ai1 ai ln xi a1 ln x1

(17)

i¼2

and position-k monotonicity (Axiom 5) again implies condition (13) and the corollaries of this condition still apply. The counterpart of Proposition 4 is as follows. Proposition 6. Given the conditions of Proposition 5, imposition of the principle of progressive transfers at each position k, k ¼ 2; :::; n 1 implies that the weights ak in Eq. (14) can be written as ak ¼ jðkÞ where j is a concave function. Proof. If we have a progressive transfer at position k, then from Eq. (17) the reduction in inequality is given by JðxÞ Jðxðk; dÞÞ ¼ ½ak akþ1 ln xkþ1 þ ½ak1 ak ln xk ½ak akþ1 lnðxkþ1 dÞ ½ak1 ak lnðxk þ dÞ d d þ ½ak ak1 ln 1 þ ¼ ½akþ1 ak ln 1 xkþ1 xk Expanding the last line of this expression, we get " # " # d 1 d 2 d 1 d 2 ½akþ1 ak þ ½ak ak1 þ xkþ1 2 xkþ1 xk 2 xk

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which, neglecting second-order and higher terms for small d, gives JðxÞ Jðxðk; dÞÞ ’ ½ak ak1

d d ½akþ1 ak xk xkþ1

(18)

Applying Axiom 6, expression (18) must be non-negative which, given that d>0, implies ak ½xkþ1 þ xk ak1 xkþ1 þ akþ1 xk

(19)

Deﬁning y :¼

xkþ1 xkþ1 þ xk

condition (19) becomes ak yak1 þ ½1 yakþ1 where ð1=2Þ yo1, which is sufﬁcient to establish concavity.

5. DISCUSSION As we noted in the introduction, an important application of the relative indices developed here is the formalisation of Merton’s index, which is based on a sum of ‘‘justice evaluations’’. An individual’s justice evaluation is given by A (20) ln C where A is the actual amount reward and C is the just reward – see also Jasso (2000, p. 338). Since we are concerned with inequality (and its counterpart distributive injustice), it makes sense to consider the inverse of A/C. If the just reward for individual i is an immediately upward-looking concept, then A ¼ xi and C ¼ xiþ1 and we should focus on C ln (21) ¼ ln xiþ1 ln xi A which is exactly the form that we have in Eq. (14). Finally, note that the indices derived here, although based on a set of axioms that might appear similar to those used in conventional inequality analysis are fundamentally different from those associated with

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conventional non-overlapping decomposable inequality indices (Ebert, 1988). Instead, the measures (8) and (14) capture a type of ‘‘keeping up with the Joneses’’ form of envy.

NOTES 1. Our methodology is similar to that used in the analysis of poverty (Ebert & Moyes, 2002), individual deprivation (Bossert & D’Ambrosio, 2006; Yitzhaki, 1982) and complaint-inequality (Cowell & Ebert, 2004). 2. If n ¼ 2, then Axiom 1 is not required. 3. We have to say ‘‘a multiple of’’ because we have not introduced a normalisation axiom to ﬁx, say, a1 ¼ 1.

ACKNOWLEDGMENT We would like to thank STICERD for hosting Ebert in order to facilitate our collaboration.

REFERENCES Arnsperger, C. (1994). Envy-freeness and distributive justice. Journal of Economic Surveys, 8, 155–186. Bossert, W., & D’Ambrosio, C. (2006). Reference groups and individual deprivation. Economics Letters, 90, 421–426. Cowell, F. A., & Ebert, U. (2004). Complaints and inequality. Social Choice and Welfare, 23, 71–89. Dalton, H. (1920). Measurement of the inequality of incomes. The Economic Journal, 30, 348–361. Ebert, U. (1988). On the decomposition of inequality: Partitions into nonoverlapping subgroups. In: W. Eichhorn (Ed.), Measurement in Economics. Heidelberg: Physica Verlag. Ebert, U., & Moyes, P. (2002). A simple axiomatization of the Foster-Greer-Thorbecke poverty orderings. Journal of Public Economic Theory, 4, 455–473. Falk, A., & Knell, M. (2004). Choosing the Joneses: Endogenous goals and reference standards. Scandinavian Journal of Economics, 106, 417–435. Jasso, G. (2000). Some of Robert K. Merton’s contributions to justice theory. Sociological Theory, 18, 331–339. Merton, R. K. (1957). Social theory and social structure (2nd ed.). New York: Free Press. Temkin, L. S. (1993). Inequality. Oxford: Oxford University Press. Yitzhaki, S. (1982). Relative deprivation and economic welfare. European Economic Review, 17(1), 99–114.

INTERDEPENDENT PREFERENCES IN THE DESIGN OF EQUAL-OPPORTUNITY POLICIES Juan D. Moreno-Ternero ABSTRACT We study mechanisms to construct equal-opportunity policies for resource allocation. In our model, agents enjoy welfare as a function of the effort they expend and the amount of a socially provided resource they consume. Nevertheless, agents have interdependent preferences, that is, they not only care about their own allocation, but also about their peers’ allocations. As in the standard approach to equality of opportunity, the aim is to allocate the social resource so that welfare across individuals at the same relative effort level is as equal as possible. We show how pursuing this same aim while assuming that agents have interdependent preferences might crucially alter the results.

1. INTRODUCTION Distributive justice concerns the fair distribution of social welfare among the citizens of a society. Probably, the most universally supported conception of distributive justice is that of equality of opportunity. Traditionally, equality

Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 49–65 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16003-3

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JUAN D. MORENO-TERNERO

of opportunity was understood as the absence of legal bar to access to education, to all positions and jobs, and the fact that all hiring was meritocratic. From the 1970s, several authors, most notably, John Rawls (1971), Amartya Sen (1980), and Ronald Dworkin (1981a, 1981b), started calling for a more radical notion of equality of opportunity. Nowadays, it is well accepted that real equality of opportunity requires compensating individuals for aspects of their situation for which they are not responsible (and that hamper their achievement of whatever is valuable in life) but only for those differences between aspects of their situations for which they are responsible should not be a concern for justice. John Roemer (1993, 1998) has formalized a precise (and quite inﬂuential) notion of equality of opportunity in order to resolve distributive issues.1 In a pure distribution context, a policy reduces to a proposal for the allocation of some ﬁnite amount of resource, across types of individuals, where the resource is to be interpreted as the instrument to achieve a certain objective. Roemer postulates that an equal-opportunity policy, with respect to an objective, should allocate the resource so that it makes the degree to which an individual achieves the objective a function only of his or her effort (i.e., aspects that inﬂuence the individual’s status but over which he or she has at least some control), and therefore independent of his or her circumstances (i.e., aspects beyond the individual’s control that also inﬂuence his or her status). Roemer’s mechanism is constructed under the standard assumption in most economic models that all people are exclusively motivated by their material self-interest (the so-called self-interest hypothesis). That is, in his model, Roemer assumes that agents have ‘‘independent preferences,’’ hence, only caring about their own allocations. In recent years, however, there has been experimental and ﬁeld evidence systematically refuting the self-interest hypothesis and suggesting that many people are strongly motivated by others’ preferences and that concerns for fairness and reciprocity cannot be ignored in social interactions (e.g., Guth, Schmittberger, & Schwarze, 1982; Fehr, Kirchsteiger, & Riedl, 1993; Fehr & Gachter, 2000). As a result, several models that relax the assumption of individual greed, upon expanding the notion of preferences, to account for the above evidence, have been proposed (e.g., Fehr & Schmidt, 1999; Bolton & Ockenfels, 2000). A feature that most of these models share is that individuals dislike payoff inequality and that individual preferences also depend on the payoff of others. If a social planner cares about equality of opportunity, it seems reasonable to assume that (at least, some) agents do too.2 An agent caring

Interdependent Preferences in the Design of Equal-Opportunity Policies

51

about equality of opportunity is likely to care about social goals per se and not just about material self-interest. Therefore, in order to design equal-opportunity policies, it makes sense to assume that agents have interdependent preferences instead of self-interest (independent) preferences. This is indeed the aim of this paper. That is, to explore the design of equal-opportunity policies in the event in which individuals not only care about their own allocation, but also about their peers’ allocations, where peer here is interpreted as an equally deserving (in terms of relative effort) individual. We shall formalize a suitable mechanism for the design of equal-opportunity policies in this context and will highlight how the assumption of interdependent preferences can make a difference with respect to the standard approach to equality of opportunity. Our mechanism will be mostly framed as an extension to Roemer’s original mechanism. As we shall see later, it can also be easily adapted to account for the suitable extension of a different, but somewhat related, mechanism developed by Dirk Van de gaer (e.g., Van de gaer, 1993; Ooghe, Schokkaert, & Van de gaer, 2007).3 The related literature to this paper can be divided into two broad categories. First, the literature on compensation and responsibility in fair allocation rules [see Fleurbaey and Maniquet (2004) or Fleurbaey (2008) for excellent surveys]. This literature deals with two antagonistic principles (to neutralize the inﬂuence over agents’ outcomes of the characteristics that elicit compensation, and to do nothing about inequalities entailed by other characteristics) typically modeled by axioms that are logically independent, and even sometimes substantially incompatible. The main issue then is to solve the ethical dilemma of how to balance these two principles in the social allocation of resources. The mechanism presented in this paper can be considered as a step in that direction.4 Second, the literature on interdependent preferences [see Fehr and Schmidt (2003) or Sobel (2005) for excellent surveys]. As mentioned above, research on interdependent preferences mostly originated to give account of the growing empirical and experimental evidence that human behavior could not be explained only by the hypothesis of self-interested material payoff maximization. In this paper, we make use of one of the (successful) existing models accounting for this evidence so that the design of equalopportunity policies becomes more accurate. The rest of the paper is organized as follows. In Section 2, we introduce the preliminaries of the model. In Section 3, we present the standard and new mechanisms to construct equal-opportunity policies. In Section 4, we

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JUAN D. MORENO-TERNERO

provide an application to obtain equal-opportunity policies in the context of health care delivery. Section 5 concludes.

2. THE MODEL Consider a population whose members enjoy welfare as a result of the amount of a socially provided resource they consume, and the amount of effort they expend. The amount of effort an individual expends comes determined not only by his or her autonomous volition, but also by his or her circumstances. We assume there is a ﬁxed set of circumstances and let T={1,y,n} be the set of resulting types in which the population is exhaustively partitioned (i.e., two individuals in the same type share the same proﬁle of circumstances, whereas individuals in different types have different proﬁles). Each type is characterized by a function denoted ut( , ) representing the material welfare of an individual of type t, as a function of the amount of the resource he or she consumes and the effort he or she expends. We assume that these utility functions are fully interpersonally 0 comparable. That is, ut ðx; eÞ ut ðx0 ; e0 Þ means that an individual in type t, who receives an amount of resource x and expends a level of effort e, enjoys at least the same material welfare level than an individual in type tu, who receives an amount of resource xu and expends a level of effort eu.5 Suppose there exists an amount o (per capita) of the resource to allocate among individuals in the population. The issue is to determine how to allocate o properly to achieve equality of opportunity. For each tAT, let jt : Rþ 7!Rþ be the function that indicates the amount of resource that an individual of type t receives with respect to the effort he or she expends. An ntuple j=(j1,y,jn) of such functions will be called a policy and each of its components jt will be called an allocation rule. Let F be the set of available policies. Suppose the distribution of effort expended by members of type t is given by the probability measure F tjt . Let et(p, jt) be the level of effort expended by the individual at the pth quantile of that effort distribution. Formally, et(p, jt) is such that Z et ðp;jt Þ p¼ dF tjt 0

Now, we deﬁne the indirect material utility function vt(p, jt), that is, the level of material welfare enjoyed by an individual of type t who reached the

Interdependent Preferences in the Design of Equal-Opportunity Policies

53

pth degree of effort and faced the allocation rule jt, as follows: vt ðp; jt Þ ¼ ut ðjt ðet ðp; jt ÞÞ; et ðp; jt ÞÞ Let pA[0,1] and j=(j1,y,jn)AF be given, and consider vðp; jÞ ¼ ðv1 ðp; j1 Þ; v2 ðp; j2 Þ; :::; vn ðp; jn ÞÞ the vector of indirect material utilities of the individuals at the pth degree of effort of each type, after implementing policy j. Now, we also assume that individuals enjoy immaterial welfare that comes determined by their peers’ material welfare levels, where ‘‘peers’’ here refers to agents that are equally deserving, that is, agents expending a comparable level of effort.6 In other words, individuals are not necessarily purely selﬁsh subjects and they might dislike inequitable outcomes for individuals who are equally deserving. As modeled by Fehr and Schmidt (1999), we assume that, in general, individuals suffer more from inequity that is to their material disadvantage than from inequity that is to their material advantage. Formally, let Vt( , ) denote the function representing the overall (material and immaterial) welfare of an individual of type t, as a function of his or her quantile at the type’s effort distribution and the policy being implemented. Then, for pA[0, 1] and j=(j1,y,jn)AF, we have at X V t ðp; jÞ ¼ vt ðp; jt Þ maxfvs ðp; js Þ vt ðp; jt Þ; 0g n 1 s2T=ftg

bt X maxfvt ðp; jt Þ vs ðp; js Þ; 0g n 1 s2T=ftg

where atZmax{0, bt}. Note that the ﬁrst term measures the utility loss from disadvantageous inequality, while the second term measures the loss from advantageous inequality. Furthermore, the assumption atZbt captures the idea that an individual suffers more from inequality that is to his or her disadvantage. Note also that we do not impose the standard assumption of btZ0, as we do not want to rule out from the outset the existence of subjects who like to be better off than their peers.

3. THE MECHANISMS The issue now is to construct a mechanism that yields for each environment a particular policy in F. For a given quantile p of effort expended, suppose

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JUAN D. MORENO-TERNERO

we are only concerned with equalizing the advantage of all individuals, across types, who expended the pth degree of effort. For this case, Roemer (1998, p. 27) proposes to select the policy that maximizes the minimum level of material advantage of these individuals. Formally, jp ¼ arg max minfvt ðp; jt Þg j2F

t2T

(1)

If, instead, the goal would be adjusted to consider immaterial advantage too, the program would become b p ¼ arg max minfV t ðp; jÞg j j2F

t2T

(2)

At the risk of stressing the obvious, note that whereas program (1) is only concerned with the material advantage achieved by the worst-off individual, out of those at the same (relative) level of effort, program (2) is concerned with the whole distribution of material advantage within the group. It is also worth noting, nonetheless, that if we assume at=bt=0, for all tAT, in program (2), then we obtain program (1), which shows that program (2) is indeed a generalization of program (1), to account for possible interdependent preferences. Now, if we wish to equalize advantage across types for every p, either using program (1) or (2), we would have in general a continuum of different jp : p 2 ½0; 1g. If, by chance, all the programs policies, fjp : p 2 ½0; 1g or fb would provide the same policy, then that would be, unambiguously, the equal-opportunity policy recommended. In general, this will not be the case and, therefore, we need to adopt a compromise solution. To do so, Roemer (1998) proposes a modiﬁcation of program (1) upon replacing the maximandum for a social objective function consisting of the average of the maximanda in each of the programs. More precisely, Z 1 minfvt ðp; jÞgdp (3) jR ¼ arg max j2F

0

t2T

The analogous extension of our proposal would generate the following program: Z 1 R b ¼ arg max minfV t ðp; jÞgdp (4) j j2F

0

t2T

As mentioned above, there is a second approach to equality of opportunity that focuses on the opportunity sets to which people have

Interdependent Preferences in the Design of Equal-Opportunity Policies

55

access (rather than their outcomes), and tries to make these sets as equal as possible (e.g., Van de gaer, 1993; Ooghe et al., 2007). Formally, for jAF, let vj be the average of the indirect (material) utilities of each type at each degree of effort, after implementing j, that is, Z 1 Z 1 1 1 n n vj ¼ v ðp; j Þ dp; :::; v ðp; j Þ dp 0

0

Each component of the representative vector vj can be interpreted as the opportunity set of each type. Then, Van de gaer’s mechanism amounts to the following program: Z 1 V t t j ¼ arg max min v ðp; j Þ dp (5) j2F

t2T

0

The counterpart mechanism for interdependent preferences would then be the following: Z 1 V t b ¼ arg max min j V ðp; jÞ dp (6) j2F

t2T

0

3.1. A Particular Case For ease of exposition, and in order to gain some insight into the design of equal-opportunity policies, let us focus now on the twotype case. Formally, let T={1,2}. For pA[0,1] and j=(j1, j2)AF, we have V 1 ðp; jÞ ¼ v1 ðp; j1 Þ a1 maxfv2 ðp; j2 Þ v1 ðp; j1 Þ; 0g b1 maxfv1 ðp; j1 Þ v2 ðp; j2 Þ; 0g where a1Zmax{0, b1}, and, V 2 ðp; jÞ ¼ v2 ðp; j2 Þ a2 maxfv1 ðp; j1 Þ v2 ðp; j2 Þ; 0g b2 maxfv2 ðp; j2 Þ v1 ðp; j1 Þ; 0g where a2Zmax{0, b2}. Let us now assume that type 1 is handicapped with respect to type 2. Formally, let us assume that, for j=(j1, j2)AF given, v1(p, j1)rv2(p, j2) for all pA[0, 1]. Then, the above expressions become V 1 ðp; jÞ ¼ v1 ðp; j1 Þ a1 ðv2 ðp; j2 Þ v1 ðp; j1 ÞÞ

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and V 2 ðp; jÞ ¼ v2 ðp; j2 Þ b2 ðv2 ðp; j2 Þ v1 ðp; j1 ÞÞ Then, it is straightforward to show that, for j=(j1, j2)AF and pA[0, 1] given, V1(p, j)rV2(p, j) if and only if a1 Z b2 1. Therefore, program (4) would become Z 1 Z 1 v1 ðp; jÞdp a1 v2 ðp; jÞdp (7) maxð1 þ a1 Þ j2F

0 1

0

2

for the case in which a Zb 1, and Z 1 Z v1 ðp; jÞdp þ ð1 b2 Þ max b2 j2F

0

1

v2 ðp; jÞdp

(8)

0

for the case in which a1rb21. Program (7), however, would become Z 1 max v1 ðp; jÞdp (9) j2F

0

which clearly highlights the differences in policy recommendations between the case of interdependent and independent preferences. For instance, it is not difﬁcult to showR that if the average (material) 1 advantage of the handicapped group [i.e., 0 v1 ðp; jÞdp] is a single-peaked function with respect R 1 to j, whereas the average (material) advantage of the other group [i.e., 0 v2 ðp; jÞdp] is an increasing function with respect to j (as it will be the case for the illustration in the next section), then we have that the resulting equal-opportunity policy upon assuming interdependent preferences [i.e., the solution to program (7), or to program (8)] gives more priority to the handicapped group than the resulting equal-opportunity policy upon assuming independent preferences [i.e., the solution to program (9)].7

4. AN ILLUSTRATION: EQUAL-OPPORTUNITY POLICIES FOR HEALTH CARE We show in this section, by means of a stylized example, that when it comes to design equal-opportunity policies, considering interdependent preferences can make a difference. This example comes from Roemer (2002) and it is presented here with some slight modiﬁcations.8 It consists of a framework to select equal-opportunity policies for the delivery of health care resources. Assume a society with two types of individuals, the rich and the poor, where we suppose that a person is not to be held accountable for his or her

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socioeconomic status in regard to his or her health. Let us say that one-half of the population is poor while the other half is rich. The rich have, on average, more healthy life styles than the poor. This is formalized by assuming that the poor have life-style qualities uniformly distributed on the interval [0,1], while the rich have life-style qualities that are uniformly distributed on the interval [0.5, 1.5]. We suppose that members of the population die from cancer or tuberculosis. The probability of contracting cancer, as a function of lifestyle quality (q), is the same for both types, and given by C rC R ðqÞ ¼ rP ðqÞ ¼ 1

2q 3

whereas the probability of contracting tuberculosis is only positive for the poor people and given by q rTP ðqÞ ¼ 1 3 In particular, the rich do not contract tuberculosis at all. Suppose that life expectancy for a rich individual has the following expression: 8 70 if cancer is not contracted; > > < xc 1; 000 LE R ¼ 60 þ 10 xc þ 1; 000 > > : if cancer is contracted and xc is spent on its treatment Thus, if the disease is contracted, life expectancy will lie between 50 and 70, depending on how much is spent on treatment (from zero to an inﬁnite amount). This is a simple way of modeling the fact that nobody dies of cancer before age 50, and that life expectancy increases as resources spent increase and approaches 70 if resources spent become inﬁnite. Suppose that life expectancy for a poor individual is 8 70 if neither disease is contracted; > > > > xc 1; 000 > > 60 þ 10 > > > xc þ 1; 000 > > > > if cancer is contracted and xc is spent on its treatment; > > > < xt 10; 000 LE P ¼ 50 þ 20 xt þ 10; 000 > > > > > if tuberculosis is contracted and xt is spent on its treatment; > > > > > xc 1; 000 xt 1; 000 > > ; 50 þ 20 min 60 þ 10 > > xc þ 1; 000 xt þ 1; 000 > > : if both diseases are contracted

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Thus, the poor can die at age 30 if they contract tuberculosis and it is not treated. With large expenditures, an individual who contracts tuberculosis can live to age 70. We also assume that if a poor individual contracts both cancer and tuberculosis, then his or her life expectancy will be the minimum of the above two numbers. Finally, assume that national health care budget is $4,000 per capita. The instrument is (xc, xt), the schedule of how much will be spent on treating an occurrence of each disease. The objective is to equalize opportunities, for the rich and the poor, for life expectancy. With the data mentioned above, one can easily compute that 1/3 of the rich will contract cancer, 1/9 of the poor will contract only cancer, 5/18 of the poor will contract only tuberculosis, and 5/9 of the poor will contract both tuberculosis and cancer. Hence, the budget constraint can be expressed as 1 1 1 2 1 5 þ xc þ xt ¼ 4; 000 2 3 2 3 2 6 or equivalently, 6xc+5xt=48,000. It is also straightforward to see that the probability that individuals at quantile p of their effort distribution contract a disease is

Rich Poor

Cancer

Tuberculosis

2 1 ðp þ 0:5Þ 3 2 1 p 3

0 1

Thus, life expectancies are

p 3

2 2 xc 1; 000 vR ðp; xc Þ ¼ ðp þ 0:5Þ 70 þ ð1 pÞ 60 þ 10 3 3 xc þ 1; 000

and vP ðp; xc ; xt Þ ¼

2p2 p 2p xt 10; 000 70 þ 1 50 þ 20 9 3 3 xt þ 10; 000 2p p xc 1; 000 60 þ 10 þ 1 3 3 xc þ 1; 000 p 2p xc 1; 000 ; 50 min 60 þ 10 þ 1 1 3 3 xc þ 1; 000 xt 10; 000 þ 20 xt þ 10; 000

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The solution that Roemer’s mechanism would propose for this example is obtained by solving the problem: Z 1 maxfxc ;xt g minfvR ðp; xc Þ; vP ðp; xc ; xt Þg dp 0

s:t: 6xc þ 5xt ¼ 48; 000 It can be shown that, for (xc, xt) given, vR ðp; xc Þ vP ðp; xc ; xt Þ for all pA[0, 1]. Thus, the above program becomes Z 1 vP ðp; xc ; xt Þ dp maxfxc ;xt g 0

s:t: 6xc þ 5xt ¼ 48; 000 whose solution turns out to be R R xc ; xt ¼ ð$310; $9;230Þ that is, $310 spent in the treatment of cancer and $9,230 in the treatment of tuberculosis.9 Let us now assume that individuals have interdependent preferences. That is, they not only care about their life expectancies, but also about their peers’ life expectancy, here ‘‘peers’’ refers to agents at the same quantile of their corresponding life-style distributions. Formally, let V P ðp; xc ; xt Þ ¼ vP ðp; xc ; xt Þ aP maxfvR ðp; xc Þ vP ðp; xc ; xt Þ; 0g bP maxfvP ðp; xc ; xt Þ vR ðp; xc Þ; 0g where aPZmax{0, bP}Z0, and V R ðp; xc ; xt Þ ¼ vR ðp; xc Þ aR maxfvP ðp; xc ; xt Þ vR ðp; xc Þ; 0g bR maxfvR ðp; xc Þ vP ðp; xc ; xt Þ; 0g where aRZmax{bR, 0}. Since, for (xc, xt) given, vR ðp; xc Þ vP ðp; xc ; xt Þ for all pA[0, 1], we have the following: V P ðp; xc ; xt Þ ¼ vP ðp; xc ; xt Þ aP ðvR ðp; xc Þ vP ðp; xc ; xt ÞÞ and V R ðp; xc ; xt Þ ¼ vR ðp; xc Þ bR ðvR ðp; xc Þ vP ðp; xc ; xt ÞÞ

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The mechanism we propose in this paper would amount to solving the following program: Z 1 minfV P ðp; xc ; xt Þ; V R ðp; xc Þg dp maxfxc ;xt g 0

s:t: 6xc þ 5xt ¼ 48; 000 Let us assume ﬁrst that aPZbR1. Then, it follows that, for (xc, xt) given, V R ðp; xc ; xt Þ V P ðp; xc ; xt Þ for all pA[0, 1]. Thus, the above program translates into Z 1 V P ðp; xc ; xt Þ dp maxfxc ;xt g 0

s:t: 6xc þ 5xt ¼ 48; 000 Equivalently,

Z

Z

1

vP ðp; xc ; xt Þ aP

maxfxc ;xt g ð1 þ aP Þ 0

1

vR ðp; xc Þ dp 0

s:t: 6xc þ 5xt ¼ 48; 000 Obviously, for aP=0 this program becomes Roemer’s original program. For aP>0, however, we have different programs and different solutions. a the More precisely, let b a ¼ 0:13842. Then, it turns out that for aP b solution is given by ðxc ; xt Þ ¼ ð$0; $9;600Þ that is, everything is spent in the treatment of tuberculosis. For 0oaP ob a, the corresponding programs have aP-speciﬁc solutions, ðxc ; xt Þ ¼ ðxac P ; xat P Þ, moving from Roemer’s solution to the above solution. If we now assume that aPrbR1, then results do not change. More precisely, under this assumption, V R ðp; xc ; xt Þ V P ðp; xc ; xt Þ for all pA[0, 1], which implies solving the following program: Z 1 Z 1 vP ðp; xc ; xt Þ þ ð1 bR Þ vR ðp; xc Þ dp maxfxc ;xt g bR 0

0

s:t: 6xc þ 5xt ¼ 48; 000 Then, for bR=1, which is the lowest possible value for bR (and equivalent to the case in which aP=0) under the assumption aPrbR1, this a ¼ 1:13842, program becomes Roemer’s original program. For bR 1 þ b

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the solution is given by ðxc ; xt Þ ¼ ð$0; $9;600Þ that is, everything is spent in the treatment of tuberculosis. For 1obRo1.13842, the corresponding programs have bR-speciﬁc solutions that coincide with the aP-speciﬁc solutions described above.10 Thus, no new solutions emerge under the assumption aPrbR1. In summary, we obtain that all possible solutions under the assumption of interdependent preferences are more prioritarian than under the assumption of selﬁsh preferences, as they involve a higher expenditure on the disease aP R that is speciﬁc to the poor type. Formally, xac P oxR c and xt 4xt for all 11 aP>0. To conclude, it is interesting to note that the policy in which everything is spent in the treatment of tuberculosis is precisely the so-called Rawlsian policy (i.e., the policy that maximizes the condition of the worst-off individual) for this example.12 Therefore, the above condition shows that a small degree of inequity aversion in individuals’ preferences is enough to guarantee a broad consensus on the Rawlsian solution, at least for this problem.

5. DISCUSSION Equality of opportunity amounts to combine the idea of compensation with the concept of responsibility for the design of policies. Roemer’s theory of equality of opportunity is a very important contribution in this direction, providing perhaps the ﬁrst workable proposal (in an economic model) to design equal-opportunity policies, by means of a precise balance between the ideas of compensation and responsibility.13 The theory assumes the socalled self-interest hypothesis, by which all individuals are assumed to be exclusively pursuing their material self-interest without caring about ‘‘social’’ goals per se. In this paper, we have argued that interdependent preferences (which have proven to be useful in explaining several puzzles arising under the self-interest hypothesis) might be more consonant with the idea of equality of opportunity. We have also shown that the counterpart mechanism to Roemer’s original mechanism, extended to incorporate interdependent preferences, might produce signiﬁcantly different recommendations. It is worth remarking that the introduction of interdependent preferences in the model we have considered does not preclude the existence of agents

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obeying the standard economic assumptions of rationality and individual greed. One of the insights of some of the newly developed theoretical models with interdependent preferences is that the interaction between fair and selﬁsh individuals is key to understand the observed behavior in strategic settings (e.g., Fehr & Schmidt, 1999; Bolton & Ockenfels, 2000). These models explain why in some strategic settings almost all people behave as if they are completely selﬁsh, while in others the same people will behave as if they are driven by fairness. In this respect, our mechanism is a generalization of Roemer’s (and also Van de gaer’s) mechanism that can be seen as an extreme case in which all agents in the model are selﬁsh. It does allow, nonetheless, for more realistic situations in which individuals do care about social goals, such as fairness in the allocation process, relative deprivation, and status seeking. A somewhat related modiﬁcation to Roemer’s (and also Van de gaer’s) approach to the design of equal-opportunity policies has also been recently proposed (e.g., Moreno-Ternero, 2007). In this case, the idea is to recommend the policy that minimizes the inequality (according to a certain inequality index) of welfare across individuals that are equally deserving. In doing so, a concern for relative deprivation and status seeking is captured, albeit without imposing interdependent preferences in the model. Interestingly enough, the case in which the maximin inequality index is considered gives rise to a mechanism that can be derived from Roemer’s (and also Van de gaer’s) mechanism upon assuming interdependent preferences, provided there is a sufﬁciently high concern for inequality aversion [see Moreno-Ternero (2007) for further details]. We have also provided in this paper an application of these mechanisms to the case of designing equal-opportunity policies for the ﬁnance of health care in a stylized example. We have shown in this example that a small concern for equity, among equally deserving agents, is enough to recommend more radical solutions than the ones advocated by Roemer’s (and Van de gaer’s) mechanism, even leading to the Rawlsian recommendation for this setting. Rawlsian policies have often been criticized for being too extreme as a result of not invoking any concern for individual responsibility. Our results, however, show that Rawlsian recommendations can actually be supported by a responsibility-sensitive theory of egalitarianism, such as the one proposed in this paper. This work leaves two main routes to be explored for further research. On the one hand, it would be desirable to provide general (analytical) results extending the features highlighted in the application presented in this paper. For instance, to characterize the domains in which the equal-opportunity

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policies obtained under the assumption of interdependent preferences are more prioritarian than in the case of selﬁsh preferences, or even the domains in which the equal-opportunity policies with interdependent preferences lead to the Rawlsian recommendation. On the other hand, it would be interesting to explore an alternative interpretation of the model in which agents would care about their peers’ allocation, but interpreting peers as individuals at the same type (i.e., with the same set of circumstances) rather than at the same level (or degree) of effort.

NOTES 1. It is worth noting that Roemer’s theory has a broader range of applications, although we shall restrict our attention here to the pure distribution context, for ease of exposition. 2. Except, perhaps, if the social planner endorses the enlightened despotism maxim ‘‘everything for the people, nothing by the people.’’ 3. Van de gaer’s mechanism focuses on the opportunity sets (i.e., sets of available outcomes) of equally-deserving individuals in different types, rather than focusing on their actual outcomes (as Roemer’s mechanism does). 4. To be more precise, the mechanism in this paper refers to the so-called utilitarian-reward approach to equality of opportunity (also adopted by Roemer and Van de gaer), which postulates that the social objective is to maximize the sum of individual outcomes, once the undue inﬂuence of characteristics calling for compensation has properly been taken into account. There is an opposite approach based on the libertarian principle that if all agents were identical in the characteristics that elicit compensation, there would be no reason to make any transfer between the agents (the so-called natural-reward principle). See Fleurbaey and Maniquet (2004) or Fleurbaey (2008) for further scrutiny of both reward principles, as well as their connections with the principle of compensation. 5. For instance, think of the case of future earning power as a function of the (per capita) expenditure in education and years of schooling. Also, think of life expectancy, or the number of quality-adjusted life years (QALYs), as a function of the (per capita) health care expenditure and the life style (e.g., smoking behavior, physical exercise). 6. Here, and following Roemer (1998), comparable effort will refer to the same relative effort level, that is, the same quantile at the corresponding effort distribution. 7. For an axiomatic study of the ethics of priority, see Moreno-Ternero and Roemer (2006). 8. See also Moreno-Ternero (2007). 9. Note that this would also be the solution that Van de gaer’s mechanism would propose here, as it would amount to solve the same problem given that, in this example, opportunity sets do not cross (e.g., Ooghe et al., 2007). b b P P 10. More precisely, for 1obRo1.13842, ðxc R ; xt R Þ ¼ ðx1þa ; x1þa Þ. t c

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11. It is worth stressing again that this feature is also obtained under more general conditions than the ones supporting this example, as mentioned in Section 3.1. 12. Formally, the Rawlsian policy is given by the program jRW ¼ arg maxj2F minðt;pÞ2T½0;1 fvt ðp; jÞg: It is not difﬁcult to show that the solution of this program, for the example of this section, is obtained by solving the problem: maxfxc ;xt g fvP ð0; xc ; xt Þg such that 6xc+5xt=48,000, whose solution turns out to RW Þ ¼ ð$0; $9;600Þ. be ðxRW c ; xt 13. Other important contributions balancing the ideas of compensation and responsibility in a general framework, as well as in applications, have recently appeared in the literature (e.g., Fleurbaey & Maniquet, 2004, 2006; Ooghe et al., 2007; Fleurbaey, 2008).

ACKNOWLEDGMENTS I owe my gratitude to John Roemer for his inspiring work and stimulating conversations, which are the main reason why I became interested in the formal study of equality of opportunity. I retain, however, the responsibility for any shortcomings in the outcomes of my study. I am also most grateful to Dirk Van de gaer and an anonymous referee for helpful comments and suggestions. Thanks are also due to the audience at the ECINEQ’s Second Biannual Conference (Berlin, 2007). Financial support from the Spanish Ministry of Education and Science (SEJ 2005-04805) and Junta de Andaluca (P06-SEJ-01645) is gratefully acknowledged.

REFERENCES Bolton, G., & Ockenfels, A. (2000). ERC: A theory of equity, reciprocity and competition. American Economic Review, 90, 166–193. Dworkin, R. (1981a). What is equality? Part 1: Equality of welfare. Philosophy & Public Affairs, 10, 185–246. Dworkin, R. (1981b). What is equality? Part 2: Equality of resources. Philosophy & Public Affairs, 10, 283–345. Fehr, E., & Gachter, S. (2000). Cooperation and punishment in public goods experiments. American Economic Review, 90, 980–994. Fehr, E., Kirchsteiger, G., & Riedl, A. (1993). Does fairness prevent market clearing? An experimental investigation. Quarterly Journal of Economics, 108, 437–460. Fehr, E., & Schmidt, K. (1999). A theory of fairness, competition and cooperation. Quarterly Journal of Economics, 114, 817–868. Fehr, E., & Schmidt, K. (2003). Theories of fairness and reciprocity. Evidence and economic applications. In: M. Dewatripont, L. P. Hansen & S. J. Turnovsky (Eds), Advances in Economics and Econometrics: 8th World Congress. Cambridge: Cambridge University Press.

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Fleurbaey, M. (2008). Fairness, responsibility, and welfare. New York, NY: Oxford University Press. Fleurbaey, M., & Maniquet, F. (2004). Compensation and responsibility. In: K. Arrow, A. Sen & K. Suzumura (Eds.), The Handbook for Social Choice and Welfare. (Vol. 2, Forthcoming). Fleurbaey, M., & Maniquet, F. (2006). Fair income tax. Review of Economic Studies, 73, 55–83. Guth, W., Schmittberger, R., & Schwarze, B. (1982). An experimental analysis of ultimatum bargaining. Journal of Economic Behavior and Organization, 3, 367–388. Moreno-Ternero, J. D. (2007). On the design of equal opportunity policies. Investigaciones Econo´micas, 31, 351–374. Moreno-Ternero, J. D., & Roemer, J. (2006). Impartiality, priority and solidarity in the theory of justice. Econometrica, 74, 1419–1427. Ooghe, E., Schokkaert, E., & Van de gaer, D. (2007). Equality of opportunity versus equality of opportunity sets. Social Choice and Welfare, 28, 209–230. Rawls, J. (1971). A theory of justice. Cambridge, MA: Harvard University Press. Roemer, J. E. (1993). A pragmatic theory of responsibility for the egalitarian planner. Philosophy & Public Affairs, 22, 146–166. Roemer, J. E. (1998). Equality of opportunity. Cambridge, MA: Harvard University Press. Roemer, J. E. (2002). Equity in health care delivery. Yale University: Mimeo. Sen, A. (1980). Equality of what? In: S. M. McMurrin (Ed.), Tanner Lectures on Human Values (Vol. I). Cambridge: Cambridge University Press. Sobel, J. (2005). Interdependent preferences and reciprocity. Journal of Economic Literature, 43, 392–436. Van de gaer, D. (1993). Equality of opportunity and investment in human capital, Ph.D. thesis, K.U. Leuven.

HIGHER EDUCATION AND EQUALITY OF OPPORTUNITY IN ITALY Vito Peragine and Laura Serlenga ABSTRACT Purpose: This paper aims at studying the degree of equality of educational opportunity in the Italian university system. Methodology: We build on the approaches developed by Peragine (2004, 2005) and Lefranc et al. (2006a, 2006b) and focus on the equality of educational opportunities for individuals of different social background. We propose different deﬁnitions of equality of opportunity in education. Then, we provide testable conditions with the aim of (i) testing for the existence of equality of opportunity (EOp) in a given distribution and (ii) ranking distributions on the basis of EOp. Deﬁnitions and conditions resort to standard stochastic conditions that are tested by using nonparametric tests developed by Beach and Davidson (1983) and Davidson and Duclos (2000). Findings: Our empirical results show a strong family effect on the performances of students in the higher education and on the transition of graduates in the labor market. Moreover the inequality of opportunity turns out to be more severe in the South than in the regions of the NorthCenter. Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 67–97 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16004-5

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Originality: This work contributes to the literature in three ways: ﬁrst, it proposes a deﬁnition of equality of educational opportunities. Second, the paper develops a methodology in order to test for the existence of equality of opportunity in a given distribution and to rank distributions according to equality of opportunity. Third, we present empirical evidence on the degree of equality of educational opportunity in the Italian university system.

1. INTRODUCTION Equality of opportunity (EOp) is a widely accepted principle of distributive justice in western liberal societies and the leading idea of most political platforms in several countries. The crucial role played by the educational system in determining the extent of equality of opportunities in a society is also broadly recognized. It is, therefore, of prime policy interest to evaluate the effects of education policies from the point of view of EOp. However, in addition to data limitation and empirical constraints, such evaluation is by no means straightforward from a theoretical point of view. It is sometimes thought that opportunity equalization, in the dimension of education, is implemented by the provision of equal educational resources to all young citizens; alternatively, by the provision of equality in the educational attainments of all individuals. Arrow, Bowles, and Durlauf (2000), for instance, argue that ‘‘even so basic a concept as equality of educational opportunity eludes deﬁnition, with proposals ranging from securing the absence of overt discrimination based on race or gender to the far more ambitious goal of eliminating race, gender, and class differences in educational outcomes’’ (pp. ix). This state of affairs is the starting point of our paper, which contributes to the literature in three ways: First, building on the literature on EOp that has recently ﬂourished in the area of normative economics, it proposes a deﬁnition of equality of educational opportunities. Second, the paper develops a methodology in order to test the existence of EOp in a given distribution and to rank distributions according to EOp. Third, we present empirical evidence on the degree of equality of educational opportunity in the Italian university system. After the inﬂuential contributions by Arneson (1989), Cohen (1989), Dworkin (1981), Roemer (1998), and Sen (1980), a large body of literature has explored the conception of EOp as ‘‘leveling the playing ﬁeld,’’ according to which society should equally split the means to reach a

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valuable outcome among its members; once the set of opportunities have been equalized, which particular opportunity the individual chooses from those open to him/her is outside the scope of justice. Translated in terms of inequality measurement, this means that ex ante inequalities (i.e., inequalities in the set of opportunities open to individuals) are inequitable while ex post inequalities (i.e., inequalities in the ﬁnal achievements) are not necessarily inequitable. There is an extensive literature concerned with the measurement of inequality of opportunity, with both a theoretical and an empirical ﬂavor. For empirically oriented contributions, see, among others, Bourguignon, Ferreira, and Menendez (2003), Goux and Maurin (2003), Checchi and Peragine (2005), Dardanoni, Fields, Roemer, and Sanchez Puerta (2005), Lefranc, Pistolesi, and Trannoy (2006a, 2006b), O’Neill, Sweetman, and Van De Gaer (1999), Peragine (2002, 2004, 2005), Ruiz-Castillo (2003), and Villar (2006). In this paper, we build on the approaches developed by Peragine (2004, 2005) and Lefranc et al. (2006a, 2006b), but the focus is on the equality of educational opportunities for individuals of different social background. The application of the opportunity inequality framework to the educational opportunities is problematic. A ﬁrst question is related to the partition in circumstances and effort and concerns the status of innate abilities. Innate talents and abilities are exogenous variables, chosen by nature, not by individuals. Thus, according to the opportunity egalitarian ethics, their effects on the ﬁnal achievements should be compensated by society. However, such a prescription seems in contrast with a role generally attributed to the educational system in a society that seeks to be meritocratic: the role of selecting talents and ‘‘signaling’’ those talents, together with the acquired competencies, to the labor market. In general, efﬁciency considerations suggest much caution in designing measures intended to neutralize the effects of different abilities. In our empirical speciﬁcation, the individual circumstances are represented only by family background; hence, we implicitly assume that talents are part of the individual’s sphere of responsibility. A second question refers to the deﬁnition of individual achievement. A simple application of the EOp scheme to the school system would imply evaluating the effects of circumstances on the individual educational outcomes (years of schooling, test scores, graduation marks, etc.). However, even without denying that education has a value per se, one could also argue that education has an indirect or instrumental value and that the ﬁnal achievements of education should be expressed by some indicators of the

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value assigned to education in the labor market. Education can be seen as an important determinant of future earning capacity of individuals and, thereby, of their future well-being. To defend such a consequentialist view of education, consider that this kind of reasoning is perfectly in tune with the economic role recognized to education in mainstream economic theory, namely, the one that sees education essentially as an investment in human capital (Becker, 1993). Hence, it seems coherent with such an approach to evaluate different education systems, also in equity terms, by looking at their effects on the earnings of individuals, for these are the ﬁnal achievements of an investment in education. Consequently, in our empirical application, we ﬁrst study the extent of EOp with respect to academic achievements, as measured by the probability of graduation and the actual graduation marks; second, we study the transition of university graduates to the labor market and analyze the EOp with respect to actual earnings. Let us now summarize the strategy we propose. Consider a given population and a distribution of a particular form of individual outcomes (income, educational achievements, etc.) that is assumed to be determined by two classes of variables: circumstances and effort. Now partition the population into types, a type being a group of people endowed with the same circumstances. If we assume that the individual outcome is determined only by circumstances and effort, and that the distribution of effort is independent from circumstances, then all the variation of individual outcomes within a given type would be assumed to be caused by differential personal effort. That is, the outcome distribution conditional to circumstances can be interpreted as the set of outcomes open to individuals with the same circumstances: the opportunity set – expressed in outcome terms – open to any individual in that type. Hence, comparing the opportunity sets of two individuals endowed with different circumstances amounts to comparing their type-conditional distributions. Roughly speaking, inequality of opportunities in this scenario is revealed by inequality between type distributions. Exploiting this idea, we ﬁrst propose different deﬁnitions of EOp in education. Then, we provide testable conditions with the aim of (i) testing for the existence of EOp in a given distribution and (ii) ranking distributions on the basis of EOp. Deﬁnitions and conditions resort to standard stochastic conditions, which are tested by using nonparametric tests developed by Beach and Davidson (1983) and Davidson and Duclos (2000). We then propose an empirical analysis of EOp in the Italian university system, using different surveys over the period 2000–2004, and compare two Italian macro-regions, South and North-Center. Our empirical results show

Higher Education and Equality of Opportunity in Italy

71

that the strong family effect detected by previous studies is also preserved both in tertiary education and in the transition of graduates to the labor market. It also reveals that the inequality of opportunity is stronger when looking at the effects of family background on graduation marks and dropout rates than when examining graduates’ incomes. Moreover, it turns out that the inequality of opportunity is more severe in the South than in the regions of the North-Center, particularly in the case of income distributions. The rest of the paper is organized as follows. Section 2 discusses our characterization of equality of educational opportunity. We ﬁrst propose a deﬁnition of equality of educational opportunities; we then develop a comprehensive model that allows to test for EOp and to rank distributions according to educational EOp. In Section 3, we provide an empirical analysis of EOp for higher education in Italy. Some concluding remarks appear in Section 4.

2. EQUALITY OF EDUCATIONAL OPPORTUNITIES 2.1. The Analytical Framework We have a society of individuals where each individual is completely described by a list of traits partitioned into two different classes: traits beyond the individual responsibility, represented by a person’s set of circumstances O, belonging to a ﬁnite set O={O1,y,On}, with |O|=n; and factors for which the individual is fully responsible, effort for short, represented by a variable eAY. Different partitions of the individual traits into circumstances and effort correspond to different notions of EOp. The value of e actually chosen by each individual is unobservable. Individual outcome is generated by a function g : O Y ! <þ , so that x=g(e, O), with x 2 ½0; z <þ , where z is the maximal income level. For instance, g can be interpreted as the income function or the education production function. We do not know the form of the function g and do not make any assumption about the degree of substitutability or complementarity between effort and circumstances; this issue, which is indeed important at an empirical level, is not speciﬁed here in order to keep the approach as general as possible. We assume, however, that the function g is ﬁxed and the same for all individuals. A societal outcome distribution is represented by a cumulative distribution function (c.d.f.) F : <þ ! ½0; 1, belonging to the set C. We can partition any given population into n subpopulations, each representing a

72

VITO PERAGINE AND LAURA SERLENGA

class identiﬁed by the variable O. For Oi A O, we call ‘‘type i’’ the set of individuals whose set of circumstances is Oi. Within type i, there will be a distribution of outcomes, with density fi(x), c.d.f. Fi(x), population share qFi , and average mFi . Hence, for all Oi A O, Fi(x) is the outcome distribution conditional to circumstances Oi. The distributions of income will differ across types; however, note that the distribution function is a characteristic of the type, not of any individual. The distribution F i(x) represents the set of outcome levels that can be achieved by exerting different degrees of effort – starting from the circumstances Oi. That is to say, the distribution F i(x) is a representation of the opportunity set – expressed in outcome terms – open to any individual endowed with circumstances Oi. Hence, comparing the opportunity sets of two individuals endowed with circumstances Oi and Oj amounts to comparing their type-relevant outcome distributions Fi (x) and Fj (x). Moreover, evaluating the distribution of opportunity sets among individuals in a society amounts to evaluating the set of distributions F ¼ fF 1 ðxÞ; . . . ; F n ðxÞg. In the following sections, we exploit this idea.

2.2. Deﬁning Equality of Opportunity We ﬁrst introduce a general deﬁnition of EOp. Deﬁnition 1. Given a set of distributions F ¼ fF 1 ðxÞ; . . . ; F n ðxÞg, there is EOp if and only if, for any pair of distributions F i ; F j 2 F, neither Fi is preferred to Fj nor Fj preferred to Fi. To give content to the above deﬁnition, one needs to deﬁne a preference relation on the set of type-conditional distributions. We assume that such a preference relation can be represented by an evaluation function V : F ! <þ and impose some conditions on such function. A ﬁrst assumption concerns the aggregation issue that, in this case, is a within-type aggregation. We propose the following additive evaluation function1 V for a given type i: Z z VðF i Þ ¼ U i ðxÞf i ðxÞdx (1) 0

where U i : ½0; z ! <þ is the evaluation function of an individual in type i; it is assumed to be twice differentiable (almost everywhere) in x. Next, we introduce a common monotonicity assumption, which guarantees that social welfare does not decrease as a result of an outcome

73

Higher Education and Equality of Opportunity in Italy

increment, whatever be the type: ðC:1Þ 8i 2 f1; . . . ; ng;

dU i ðxÞ 0; 8x 2 ½0; z dx

Next, we assume that our evaluation function is inequality averse. We require within-type strict inequality aversion: ðC:2Þ 8i 2 f1; . . . ; ng;

d 2 U i ðxÞ o0; 8x 2 ½0; z dx2

Alternatively, we could require our function V to be indifferent to outcome inequality within the same type, therefore assuming within-type inequality neutrality: ðC:3Þ 8i 2 f1; . . . ; ng;

d 2 U i ðxÞ ¼ 0; 8x 2 ½0; z dx2

This condition says that a reduction in outcome inequality within a type that leaves the mean of the type unchanged has no welfare effects. Note that this welfare condition implies that the function Ui is afﬁne. Conditions (C.1), (C.2), and (C.3) identify several classes of individual utility functions U that implicitly deﬁne classes of evaluation functions V. Now we deﬁne three such classes: the class of type evaluation functions V constructed as in Eq. (1) and with utility functions satisfying conditions (C.1) is denoted by V1; the class of type evaluation functions constructed as in Eq. (1) and with utility functions satisfying conditions (C.1) and (C.2) is denoted by V12; the class of type evaluation functions constructed as in Eq. (1) and with utility functions satisfying conditions (C.1) and (C.3) is denoted by V13. The next step consists deriving suitable criteria for choosing opportunity sets by requiring unanimous agreement among these classes. Hence, we have the following deﬁnitions of a preference relation over the set F of type distribution functions. Deﬁnition 2. For all F i ; F j 2 F, F i V 1 F j if and only if VðF i Þ VðF j Þ

for all V 2 V1

F i V 12 F j if and only if VðF i Þ VðF j Þ

for all V 2 V12

F i V 13 F j if and only if VðF i Þ VðF j Þ

for all V 2 V13

74

VITO PERAGINE AND LAURA SERLENGA

Standard results in inequality theory allow to identify the distributional conditions corresponding to the above welfare criteria. The ranking V 1 is equivalent to ﬁrst-order stochastic dominance ðFSD Þ: Remark 3. For all F i ; F j 2 F, F i V 1 F j if and only if F i FSD F j 3F j ðxÞ F i ðxÞ

for all x 2 ½0; z

with strict inequality for some x. The ranking V 12 is equivalent to second-order stochastic dominance ðSSD Þ: Remark 4. For all F i ; F j 2 F, F i V 12 F j if and only if Z t Z t F j ðxÞdx F i ðxÞdx for all t 2 ½0; z F i SSD F j 3 0

0

with strict inequality for some x. Finally, the ranking V 13 is equivalent to higher expected value. Remark 5. For all F i ; F j 2 F, F i W13 F j if and only if mi 4mj . Given the distributional conditions discussed above, we can now introduce some criteria to test for the existence of EOp.

2.3. Testing for the Existence of EOp In this section, we make use of the deﬁnition of EOp and the criteria derived in the previous section in order to identify empirical tests for the existence of EOp in a distribution of opportunity sets.2 We start with the strongest deﬁnition of EOp, requiring that individuals face identical prospects of outcome, regardless of their circumstances. Deﬁnition 6. Strong EOp. There is EOp if and only if 8F i ; F j 2 F, F i ðxÞ ¼ F j ðxÞ; 8x 2 ½0; z

Since the above condition is extremely demanding and will be violated in most cases, we turn to less-demanding conditions.

75

Higher Education and Equality of Opportunity in Italy

The ﬁrst test is based on the preference relation V 13 : Deﬁnition 7. Weak EOp. There is EOp if and only if 8F i ; F j 2 F, Fi - V 13 F j

and

F j - V 13 F i 3mi ¼ mj

A second test is based on the preference relation V 1 : Deﬁnition 8. EOp1 (EOp of the ﬁrst order). There is EOp if and only if 8F i ; F j 2 F, F i - V1 F j

and

F j - V 1 F i 3F i - FSD F j

and

F j - FSD F i

The next test is based on the preference relation V 12 : Deﬁnition 9. EOp2 (EOp of the second order). There is EOp if and only if 8F i ; F j 2 F, F i - V 12 F j

and

F j - V 12 F i 3F i - SSD F j

and

F j - SSD F i

These tests allow us to conclude whether, in a given distribution, there is EOp or not according to the different deﬁnitions introduced. In the next section, we address the problem of ranking distributions of opportunity set on the basis of EOp. 2.4. Ranking Distributions of Opportunity Sets Our aim is to derive welfare criteria and dominance condition in analogy with the analysis conducted in the previous section. However, here the criteria have to be deﬁned over the set of distributions C. Again, we assume that a preference relation over C can be represented by a social evaluation function W : C ! <þ . A generalization of the evaluation function V, discussed in the previous section, to the case of income distributions that can be decomposed across homogeneous subgroups, is obtained by aggregating the welfare of each type, weighted by the relevant population share, and using type-speciﬁc utility functions. In this case, we assume that there exist an ordering over O, assumed to be antisymmetric, so that, in general, Oiþ1 Oi for i 2 f1; . . . ; n 1g. Hence, we are able to rank individuals according to their circumstances. An example would be a ranking based on

76

VITO PERAGINE AND LAURA SERLENGA

parental education (as it will be the case in our empirical application) or parental status. As for the aggregation of the type welfare, we opt for an additive social evaluation function: WðFÞ ¼

n X i¼1

qFi

Z

z

U i ðxÞf i ðxÞdx

(2)

0

We now try to capture the basic intuition beyond the opportunity egalitarian ethics, by selecting different classes of utility functions oU 1 ðxÞ; . . . ; U n ðxÞ4. First, we could impose on the type-speciﬁc functions Ui properties (C.1), (C.2), and (C.3) already introduced in the previous section, which are not type speciﬁc. In addition, we now formulate some type-dependent properties. First, we deﬁne a condition expressing inequality aversion between the opportunity sets. The condition stating between-type inequality aversion is the following: ðC:4Þ

dU i ðxÞ dU iþ1 ðxÞ ; 8i 2 f1; . . . ; n 1g; 8x 2 ½0; z dx dx

which says that the marginal increase in welfare due to an increment of income is a decreasing function of circumstances.3 To the properties already introduced, we now add the following condition:4 ðC:5Þ

8i; j 2 f1; . . . ; ng; U i ðzÞ ¼ U j ðzÞ

where z is the maximum possible income. By introducing condition (C.5), any afﬁne transformation, such as U i ! ai þ bU i , is supposed to be able to affect the results of social comparisons.5 This requirement is necessary in a context with different type populations. We now deﬁne two classes of social evaluation functions: the class of social evaluation functions constructed as in Eq. (2) and with utility functions satisfying conditions (C.1), (C.3), (C.4), and (C.5), denoted by WEOp1; the class of social evaluation functions constructed as in Eq. (2) and with utility functions satisfying conditions (C.1), (C.4), and (C.5), denoted by WEOp2. The next step consists deriving suitable welfare and distributional conditions by requiring unanimous agreement among these classes.

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Higher Education and Equality of Opportunity in Italy

Deﬁnition 10. For all F; G 2 C, F k EOp1 G if and only if WðFÞ WðGÞ

for all W 2 WEOp1

F k EOp2 G if and only if WðFÞ WðGÞ

for all W 2 WEOp2

Thus, we turn to identify a range of tests that, if successful, will ensure welfare dominance for appropriate classes of SEFs. The aim of the analysis is the following: given a class of utility functions Ui(x) expressing our ethical concerns, we seek conditions, expressed in terms of distribution functions Fi(x) and Gi(x) and population shares qFi and qG i , that are necessary and sufﬁcient for welfare dominance according to the criteria deﬁned above. We ﬁrst propose the following distributional condition, which was obtained by Peragine (2004): Theorem 1. For all F; G 2 C, FkIOP1 G if and only if k X

qiF miF

i¼1

k X

qiG miG ; 8k 2 f1; . . . ; ng

i¼1

This test can be interpreted as a second order stochastic dominance (generalized Lorenz dominance) applied to the distribution of the type means weighted by the relevant population shares: ðqF1 mF1 ; . . . ; qFn mFn Þ. Note that by applying such a test, we are implicitly making the following operations: we evaluate (i) the opportunity set of each type by the weighted mean qFi mFi and (ii) the distribution of opportunity sets by the generalized Lorenz criterion. The second distributional condition we obtain is the following: Theorem 2. For all F; G 2 C, F IOP2 G if and only if k X

qFi F i ðxÞ

k X

i¼1

qG i Gi ðxÞ; 8x 2 ½0; z; 8k 2 ð1; . . . ; nÞ

i¼1

Proof. See the appendix. This theorem characterizes a sequential ﬁrst-order stochastic dominance condition, where each type distribution is weighted by the relevant population share. This condition dictates the following procedure: take

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VITO PERAGINE AND LAURA SERLENGA

the ﬁrst lowest type of the two distributions and check for dominance; then we add the second lowest type, then the third lowest type, and so on, until all the population is included, performing the dominance check at every stage. We have to perform n different tests, starting from the lowest type, until all types are merged. If these tests are always positive, then we have welfare dominance for the family WEOp2, and the converse is also true. We therefore implicitly make the following operations: we evaluate (i) the opportunity set of each type by the weighted c.d.f. qFi F i ðxÞ and (ii) the distribution of opportunity sets by the generalized Lorenz criterion. Hence, the difference with the condition obtained in Theorem 1 lies in the evaluation of the individual opportunity set: in the criterion ZIOP1, it is evaluated by the weighted expected value, while in the criterion ZIOP2, each opportunity set is evaluated by looking at the entire weighted distribution. As for the ranking of proﬁles of opportunity sets, the distributive criterion remains the same. A ﬁnal remark is in order. We have focused on unanimous preference orderings for classes of opportunity egalitarian social decision makers rather than on pure (opportunity) inequality criteria. Consequently, the distributional conditions obtained are expressed in terms of means, c.d.f., and generalized Lorenz dominance, rather than simple Lorenz dominance.

3. THE EMPIRICAL ANALYSIS: EQUALITY OF OPPORTUNITY IN THE ITALIAN HIGHER EDUCATION SYSTEM In this section, we apply the theoretical framework proposed in the previous section with the aim of analyzing EOp in the Italian higher education system. We examine whether ﬁnal graduate student outcomes and their salary distributions are characterized by EOp. Since we strongly believe that placement in the labor market should also be considered in order to fully evaluate individual tertiary education outcomes, we analyze both the income distribution after three years from graduation and the income distribution of those who have held a degree for more than three years. The choice of three years as a threshold is related to the speciﬁc design of the survey of graduates we use in the empirical application (individuals are interviewed after three years from the completion of their studies). In our analysis, individual circumstances are represented by parental education. Moreover, since our analysis also extends to consider the existence of regional

Higher Education and Equality of Opportunity in Italy

79

disparities in Italy, the conditional distributions of two Italian macroregions, the North-Center and the South, are compared and ranked according to different notions of EOp. In what follows, we present the data and the empirical methodology; ﬁnally, we discuss the results.

3.1. Data Description In this application, we use three outcome variables: graduation marks,6 net monthly income after three years from graduation, and annual disposal income earned after more than three years from graduation, which we simply call income. On the other hand, parental education is measured by the highest educational attainment of the parents and is divided into four classes. We therefore allocate the individuals in four types, according to parental education, as follows: the ﬁrst type corresponds to primary school degree, the second type to lower secondary school degree, the third type to upper secondary degree, and ﬁnally, the fourth type to graduates who have at least one of the parents with a bachelor (or higher) degree. Furthermore, the Italian regions are divided into two macro-regions: the North-Center and the South.7 Information on graduation marks and net monthly income after three years from graduation are taken from ‘‘Indagine sull’Inserimento Professionale dei Laureati’’ (IIPL, hereafter), a survey on the transition from college to work of a representative sample of Italian graduates conducted by the National Statistical Ofﬁce (Istat) in 2004, whereas data on annual disposal income of individuals who have held a degree for longer than three years are drawn from the Bank of Italy ‘‘Survey on Household Income and Wealth’’ (SHIW, hereafter). The IIPL contains information on individuals who graduated in 2001 and covers a large share of the population, up to 17%. The SHIW has been conducted regularly by the Bank of Italy since 1965. However, since we need information on the year of college completion, we consider only the last-three waves (2000, 2002, and 2004) available.8 Notice that while, in the case of the IIPL, the sample is divided into North-Center and South with respect to the geographical location of the university attended, in the case of SHIW, the macro-regions are deﬁned on the basis of the individuals’ region of residence. In fact, because of internal mobility, the university site would not be a good proxy for geographical location after a long period from graduation. Lastly, we acknowledge that, in order to investigate the academic performance, we must consider not only the students who succeeded

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VITO PERAGINE AND LAURA SERLENGA

and graduated but also those who failed their attempts to complete tertiary education. In order to do so, we use a parallel survey conducted by Istat on the transition from high school to college, ‘‘Indagine sull’Inserimento Professionale dei Diplomati’’ (this information is indeed not available in the IIPL, where only graduates are interviewed). The survey conducted in 2001 collects information on students who completed high school in 1998. In order to take into account the dropout rate, we calculate the probability of dropping out after three years from matriculation for each type and macro-region. Hence, we add to the sample of graduates of the IIPL, in each type and region, a number of students with their ﬁnal marks equal to zero proportionally to the dropout rate. Notice also that, in this analysis, we ignore the fact that the data contained in those surveys do not concern the same individuals, and the information is matched on the basis of circumstances. Summary statistics follow in the appendix.

3.2. Statistical Analysis and Methodology Our samples allow to build outcome (i.e., ﬁnal marks and income) distributions conditional on circumstances and perform a simple twofold analysis. We assess equality of distribution as developed in Beach and Davidson (1983) and perform ﬁrst- and second-order stochastic dominance tests using the Davidson and Duclos (2000) methodology. The details of the tests implemented are illustrated in the appendix. In order to draw our conclusion, we carry out the following empirical procedure, as described in Lefranc et al. (2006a, 2006b). We conduct a separate analysis for the two macro-regions, and for all the possible pairs of circumstances i and j within the same region, we perform four tests independently: - Test (1) (Weak EOp) tests the null of equality of the means of the distribution of types i and j; - Test (2) (Strong EOp) tests the null of equality of the distributions of types i and j; - Test (3) (EOp1) tests the null of ﬁrst-order stochastic dominance of the distribution of type i over j and vice versa; - Test (4) (EOp2) tests the null of second-order dominance of the distribution of type i over j and vice versa.

Higher Education and Equality of Opportunity in Italy

81

Then we pursue the following strategy: If the null of Test (1) or (2) is not rejected, we conclude that Weak or Strong EOp is satisﬁed. If Test (3) or (4) accepts dominance of one distribution over the other but not the other way around, we say that EOp is violated. If Test (3) rejects dominance of each distribution over the other, we say that EOp of the ﬁrst order is supported. If Tests (3) and (4) conclude that the two distributions dominate each other, we give priority to the results of Test (2). Accordingly, we proceed by comparing the results obtained for the two macro-regions. The drawback of such approach is that it does not allow us to rank different situations in which we would reject EOp. Hence, in case we ﬁnd evidence of inequality of opportunity, we move to the second step of our analysis, that is, we look for partial ranking of the distributions of opportunity sets in the two macro-regions. In order to do so, we rely on the dominance conditions characterized in Theorems 1 and 2. We ﬁrst verify the existence of the partial ranking ZIOP1 by numerical comparison of the distributions of the type (weighted) means of the two regions [Test (5)]. Next, we apply the second criterion ðF IOP2 GÞ by sequentially testing the following null hypotheses P2 F of ﬁrst-order P2 G stochastic P3 dominance: F G F F ðxÞ q G ðxÞ; (ii) q F ðxÞ q G ðxÞ; (iii) (i) q i i i¼1 i P i¼1 i i¼1 qi F i ðxÞ 1 1 P P3 1 G1 4 4 F G q G ðxÞ; (iv) q F ðxÞ q G ðxÞ, where F and G are the i i i¼1 i i¼1 i i i¼1 i conditional outcome distributions of the North-Center and the South, respectively [Test (6)]. In these cases, the same strategy of Test (3) is implemented.

3.3. Empirical Results In this section, we report the results of the tests of equality and stochastic dominance for the outcome distributions conditional on four classes of parental education in the two macro-regions. Figs. 1, 2, and 3 show the c.d.fs. conditional on parental education of marks, income after three years from graduation, and income. As far as the graduation ﬁnal marks are concerned, a clear ranking of types emerges both in the North-Center and in the South. The distribution of the fourth type dominates over the third, the third over the second, and the second over the ﬁrst. This visual ranking is strongly conﬁrmed by the

VITO PERAGINE AND LAURA SERLENGA

.6 0

0

.2

.2

.4

.4

.6

.8

.8

1

1

82

0

50 100 Graduation Mark South class1 class3

Fig. 1.

0

50 100 Graduation Mark North-Centre

class2

class1

class2

class4

class3

class4

Graduate Final Marks c.d.f.

results of the tests of equality and stochastic dominance (Table 1). The tests clearly indicate evidence of strong inequality of opportunity among individuals belonging to different types, in both the South and the NorthCenter. We have also repeated the same analysis, conditioning the graduation marks on the type of upper secondary school attended, ‘‘liceo’’ and ‘‘istituto’’ school type).9 The results obtained in those cases conﬁrm what described in the general unconditioned one (Fig. 1bis, i.e., the c.d.f. of ﬁnal graduation marks for graduates who attended the ‘‘istituto’’ school type). This result is quite surprising. Indeed, it is generally recognized that, in Italy, students are streamed in different tracks more in consequence of their background than of their ability and that, after enrolling in a track, the probability of entering in higher education still depends on their family background (Checchi & Flabbi, 2007). Interestingly, our analysis shows that the effects of family background goes even further: after sorting the students according to their family backgrounds in different tracks, we ﬁnd that the family of origin still matters for future academic performances within each track.

83

1 .2

.2

.4

.4

.6

.6

.8

.8

1

Higher Education and Equality of Opportunity in Italy

0

50 100 Cond Grad Mark South Istituto

Fig. 1bis.

0 50 100 Cond Grad Mark North-Centre Istituto

class1

class2

class1

class2

class3

class4

class3

class4

Graduate Final Marks c.d.f. Conditional to School Type (‘‘Istituto’’).

Turning to the dominance conditions, we notice that marks are higher in the South than in the North-Center, and this is true for each type (Table 2). However, results based on either of the criterion ZIOP1 or ZIOP2 show a mixed pattern: the South dominates the North-Center in all cases but the third. Hence, we cannot conclude for any dominance according to the ZIOP criterion in the case of mark distribution. Notice that our main conclusion on ﬁnal graduation marks does not change even when we condition the distribution on the area of academic specialization (humanities and social science separately from medicine, science, and engineering). Hence, it is a quite robust result. While analyzing the income variables, we notice that we are not able to make an explicit assessment by just observing the cumulative distributions of income (Figs. 2 and 3): in both cases, the visual ranking is not very clear. Similarly, the results of the statistical tests are not so deﬁnite as in the case of the graduation marks. In particular, in the case of the IIPL income, the hypothesis of the equivalence of means is not rejected in more cases in the North-Center than in the South. From the tests of stochastic dominance (Table 1), we generally notice that there is more evidence of equality of

.8 .6 .4 .2 0

0

.2

.4

.6

.8

1

VITO PERAGINE AND LAURA SERLENGA

1

84

0

1000 2000 3000 4000 Income South class1 class2 class3

class4

1000 2000 3000 4000 Income North-Centre class1 class2 class3

class4

.8 .6 .4 .2 0

0

.2

.4

.6

.8

1

Income after Three Years from Graduation c.d.f.

1

Fig. 2.

0

0

1000 2000 3000 4000 Income|Marks 105-max South

Fig. 2bis.

0 1000 2000 3000 4000 Income|Marks 105-max North-Centre

class1

class2

class1

class2

class3

class4

class3

class4

Income after Three Years from Graduation Conditional to High-Marks c.d.f.

85

1 .8 .6 .4 .2 0

0

.2

.4

.6

.8

1

Higher Education and Equality of Opportunity in Italy

0

20000 40000 60000 80000100000 Income South class1 class2 class3 class4

Fig. 3.

0

100000 200000 300000 400000 Income North-Centre class1 class2 class3 class4

Income c.d.f.

Test (2) – (4) EOp1 and EOp2.

Table 1.

First-Order Dominance North-Center

First-Order Dominance

South

North-Center

Graduation Mark 1

2

3

4

o

o o

o o o

1

South

Graduation Mark

2

3

4

o

o o

o o o

1 – – 2 – – 3 – – 4 – Income after three years from graduation 1 – o o o – 6¼ o 2 – 6¼ 6¼ – o 3 – 6¼ – 4 – Income 1 – 6¼ o o – 6¼ 6¼ 2 – 6¼ o – 6¼ 3 – o – 4 –

1

2

3

4

–

o

o o

o o o

–

2

3

4

–

o

o o

o o o –

–

– – Income after three years from graduation o – o o o – 6¼ o o – 6 ¼ 6¼ – o o – 6¼ – – – Income o – 6¼ o o – 6¼ 6¼ o – 6¼ o – 6¼ o – o – – – –

–

1

o o o – o o o –

Notes: W the row dominates the column; o the column dominates the row; ¼ the curves are equal; 6¼ the curves are different and cannot be ranked. Denotes 5% level of signiﬁcance. Denotes 10% level of signiﬁcance.

86

VITO PERAGINE AND LAURA SERLENGA

Table 2.

Test (5) IOP1. Graduation Mark

North-Center 1 1þ2 1þ2þ3 1þ2þ3þ4 Income after three years from graduation 1 1þ2 1þ2þ3 1þ2þ3þ4 Income 1 1þ2 1þ2þ3 1þ2þ3þ4

South o o W o W W W W W W W W

Note: See notes to Table 1.

opportunities in the North-Center than in the South, that is, we do not reach a clear-cut conclusion on dominance in three cases in the North-Center and in only one case in the South. On the other hand, the results from inequality of opportunity comparisons allow us to reach a more deﬁnite result. Since (i) the weighted means are in all cases higher in the North-Center than in the South (Table 2) and (ii) the sequential ﬁrst-order stochastic dominance condition is satisﬁed (Table 3), we can conclude that there is more EOp in the North-Center than in the South when looking at levels of income three years after graduation. These ﬁgures are consistent with the general view of less intergenerational mobility in the South than in the North of Italy. Also in this case, we have repeated the exercise for conditioned income distribution. In particular, here we have separately studied the earnings of those who got medium-low graduation marks and those who got mediumhigh graduation marks. Overall, the results do not change when analyzing those two groups with respect to the general unconditioned case (Fig. 2bis, i.e., the c.d.f. of income after three years from graduation of individuals who received a medium-high ﬁnal mark). As far as the SHIW income is concerned, the hypothesis of equivalence of means cannot be rejected in one more case in the South than in the NorthCenter (Table 4). The same evidence is shown in the tests for ﬁrst and second

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Table 3.

Test (6) IOP2.

Graduation Mark South

North-Center 1

1 W 1þ2 1þ2þ3 1þ2þ3þ4 Income after 3 years from graduation 1 o 1þ2 1þ2þ3 1þ2þ3þ4 Income 1 6¼ 1þ2 1þ2þ3 1þ2þ3þ4

1þ2

1þ2þ3

1þ2þ3þ4

W o

o

o

o

o

W

o

o

Note: See notes to Table 1. Denotes 5% level of signiﬁcance.

dominance (Table 1). Turning to the EOp ranking, the ﬁgures obtained for income show evidence of the dominance of the North-Center over the South only according to the ZIOP1 criterion, whereas the distributions are not comparable according to ZIOP2. Indeed, the North-Center dominates the South in all the steps of the sequential procedure except the ﬁrst. Summarizing, although some of our dominance conditions are not fully satisﬁed – and this is quite normal when using partial ranking – a general picture seems to emerge from the analysis of the opportunity inequality for income: the southern regions have lower per capita income accompanied by greater overall income inequality and higher degree of opportunity inequality. Also note that in the interpretation of the results, it should be reminded that our dominance criteria ZIOP1 and ZIOP2 reﬂect both distributive and aggregate aspects. It is possible that the dominance of the North-Center over the South in income levels is driven by an average effect rather than a pure inequality effect. Therefore, we have also performed pure inequality comparisons by Lorenz dominance tests. In general, the results exhibit the same evidence found for the ZIOP1 and ZIOP2 dominance tests for graduation marks. On the other hand, in the case of income

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VITO PERAGINE AND LAURA SERLENGA

Table 4.

Test (1) Weak EOp. Graduation Mark

North-Center 1

2

3

o 1 – o 2 – o 3 – 4 Income after three years from graduation o 1 – = 2 – = 3 – 4 Income o 1 – = 2 – = 3 – 4

South 4

1

2

3

4

o o o –

–

o –

o o –

o o o –

o = = –

–

o –

= o –

o o = –

o o o –

–

= –

= = –

o o o –

Notes: o the mean of the distribution in the colum is greater than the mean of the distribution in the row; ¼ means are equal. Denotes 5% level of signiﬁcance. Denotes 10% level of signiﬁcance.

distributions, Lorenz dominance tests show dominance of the South over the North-Center in the ﬁrst step of the sequential strategy and the NorthCenter over the South in the remaining steps. In conclusion, our analysis shows that, while most of the parental background exerts its effect through favoring the educational attainment of the students, it keeps on playing a role in the labor market, independently from education. This could represent the impact that family networking plays a role in ﬁnding good jobs. Our evidence shows that this effect is stronger in the South than in the North-Center.

4. CONCLUDING REMARKS Building on the existing literature on EOp, in this paper, we have proposed a deﬁnition of equality of educational opportunities and a methodology to test for the existence of EOp in a given distribution and to rank distributions

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89

according to EOp. Moreover, we have provided an empirical application by studying the degree of equality of educational opportunities in the Italian university system. We have compared two Italian macro-regions, the South and the NorthCenter, according to EOp. In the ﬁrst application, we have focused on individual graduation scores, while in the second, we have considered the distribution of incomes among Italian graduates; in both cases, we have studied how these different individual achievements vary according to the family background, as measured by the level of parental education. Our empirical results show a strong family effect on the performances of students in the university and the transition of graduates in the labor market. In addition, our analysis reveals that the degree of opportunity inequality is stronger when looking at the effects of family background on graduation marks and dropout rates than when examining graduates’ incomes. Moreover, the inequality of opportunity turns out to be more severe in the South than in the regions of the North-Center, especially for income distributions. One wonders whether these effects may be addressed by appropriate policies. To begin with, our results point to the role of higher education policies. In recent years, the Italian university system has been involved in a deep process of reform, which has reduced the years of enrollment and signiﬁcantly enlarged the educational supply, in terms of possible curricula from which the students may choose. So far, the existing data show that this reform has increased the number of students enrolled in the university. Indeed, it would be extremely interesting to study the effect of such a reform from the EOp viewpoint: has the incidence of social background on the academic performance of students increased or decreased as an effect of the reform? Unfortunately, the available data do not allow yet to draw conclusions on the effect of the reform. However, as soon as relevant data will become available, this will certainly be an object of further investigation. The same type of difﬁculty seems to emerge in the labor market. The effect of social origin plays a role in the earnings distributions, even among graduates with the same ﬁnal marks, and again, this effect is stronger in the South than in the North. This could represent the impact of family networking in ﬁnding good jobs as well as a reduced availability of good jobs in less technologically advanced areas. These greater obstacles and/or lack of adequate incentives in local labor markets can be linked to existing evidence of internal migration ﬂows, which speaks of a sort of ‘‘brain drain,’’ that is, strong migration of highly skilled workers from the South

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toward the northern regions. While part of this migration is certainly explained by the different unemployment rates, existing studies show that the choice to migrate is specially concentrated among individuals with poor family background (Coniglio & Peragine, 2007). Finally, we would suggest some possible connection between what is observed in the distributions of graduation scores of individuals coming from different social origins and what is seen in the income distribution of the same social groups. Inequality of opportunities in the labor market may stem not only from the opaque Italian labor market but also from some features of the university system. The evidence of generally higher marks in southern regions and strong effects of family background on those marks speaks of a university system that is hardly able to properly signal abilities and competencies of the students. But if the school system fails to be fully meritocratic and select according to abilities, then it is easier that other allocation mechanisms might prevail also in the labor market. Unfortunately, this does not come as a surprise in a country where more than 50% of the working population declares to have obtained the current job through recommendations of their relatives or friends.

NOTES 1. For non-utilitarian aggregation, both in the within-type and between-type stages, see for instance Roemer (1998) and Peragine (2002). 2. Here, we follow the approach proposed by Lefranc et al. (2006a, 2006b). 3. Conditions (C.1), (C.3), and (C.4) entail cardinal unit comparability (Sen, 1970). 4. An analog condition is introduced by Jenkins and Lambert (1993) in the context of income inequality in the presence of differences in needs and in order to extend the ‘‘sequential generalized Lorenz dominance’’ to the case of distributions with different type partitions. 5. By adding condition (C.5), we pass from cardinal unit comparability to cardinal full comparability (Sen, 1970). 6. In Italy, the ﬁnal graduation mark ranges from 66 to 110 cum laude; in this analysis, the 110 cum laude was simply transformed into 111. 7. The North-Center comprehends Piemonte, Lombardia, Veneto, Liguria, Trentino, Friuli, Emilia Romagna, Toscana, Umbria, and Marche while the South includes Lazio, Abbruzzo, Molise, Campania, Puglia, Basilicata, Calabria, Sicilia, and Sardegna. Note that we include Lazio among the southern regions in order to balance the number of observations between the two macro-regions. However, considering Lazio as a southern or northern region does not signiﬁcantly change the results that are available from the authors upon request. 8. We drop the panel component of those three waves and express income in terms of euro at 2000.

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9. The type of secondary school attended is a crucial variable when studying the family background effect on Italian students’ performances. In fact, the Italian upper secondary school is a tracked system, with three different paths that can be freely chosen by the pupils at the age of 13 (i.e., by their parents): an academically oriented generalist education provided by high schools (ﬁve years, called licei), a technically oriented education provided by technical schools (ﬁve years, called istituti tecnici), and a vocational training offered by local schools organized at regional levels (three years, called istituti di formazione professionale). After a debated reform in 1969, students from any track are entitled to enrol in colleges and universities, conditional on having successfully completed ﬁve years of upper secondary schooling. However, each of these tracks predicts very different outcomes in terms of additional education acquired and labor market performance. As a matter of fact, more than 88% of students who graduate from licei enrol in a university as opposed to 17.8% of the students coming from the vocational track. 10. We determine the weights w numerically. We draw 1,000 multivariate standard normal vectors and pre-multiply it by the Cholesky decomposition of a consistent estimate of S. Hence, we compute the proportion of vectors with l positive elements. This proportion is an estimate of the weight w(k, l, S).

ACKNOWLEDGMENTS An earlier version of the paper was presented at seminars in Milan and Siena, at the Second Meeting of the ECINEQ Society in Berlin, at the Second Winter School on Inequality and Collective Welfare Theory in Canazei, at the XX SIEP Conference in Pavia, and at the EEEPE meeting in London. We would like to thank the participants at these conferences and seminars for their discussion of the paper. Moreover, we are particularly grateful to Valentino Dardanoni, Carlo Fiorio, and Alain Trannoy for valuable comments and suggestions. The usual disclaimer applies.

REFERENCES Arneson, R. (1989). Equality of opportunity for welfare. Philosophical Studies, 56, 77–93. Arrow, K., Bowles, S., & Durlauf, S. N. (Eds). (2000). Meritocracy and economic inequality. Princeton: Princeton University Press. Barrett, G. F., & Donald, S. G. (2003). Consistent tests for stochastic dominance. Econometrica, 71, 71–104. Beach, C. M., & Davidson, R. (1983). Distribution-free statistical inference with Lorenz curves and income shares. The Review of Economic Studies, 723–735. Becker, G. S. (1993). Human capital: A theoretical and empirical analysis, with special reference to education (First edition, 1964). Chicago, IL: University of Chicago Press.

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Bourguignon, F., Ferreira, F. H. G., & Menendez, M. (2003). Inequality of outcomes and inequality of opportunities in Brazil. DELTA Working Paper no. 24. Checchi, D., & Flabbi, L. (2007). Intergenerational mobility and schooling decisions in Italy and Germany: The impact of secondary school track. IZA Working Paper no. 2879. Checchi, D., & Peragine, V. (2005). Regional disparities and inequality of opportunity: The case of Italy. IZA Working Paper no. 1874. Cohen, G. A. (1989). On the currency of egalitarian justice. Ethics, 99, 906–944. Coniglio, N., & Peragine, V. (2007). Giovani al Sud: Tra immobilita` sociale e mobilita` territoriale. In: N. Coniglio & G. Ferri (Eds), Primo Rapporto Banche e Mezzogiorno. Bari, Italy: Universita` degli Studi di Bari. Dardanoni, V., Fields, G., Roemer, J., & Sanchez Puerta, M. (2005). How demanding should equality of opportunity be, and how much have we achieved? In: S. L. Morgan, D. Grusky & G. Fields (Eds), Mobility and inequality: Frontiers of research in sociology and economics. Stanford, CA: Stanford University Press. Davidson, R., & Duclos, J. Y. (2000). Statistical inference for stochastic dominance and for the measurement of poverty and inequality. Econometrica, 68, 1435–1464. Dworkin, R. (1981). What is equality? Part 1: Equality of welfare. Part 2: Equality of resources. Philosophy & Public Affairs, 10, 185–246; 283–345. Goux, D., & Maurin, E. (2003). On the evaluation of equality of opportunity for income: Axioms and evidence. Paris, France: CREST, mimeo. Jenkins, S., & Lambert, P. J. (1993). Ranking income distributions when needs differ. Review of Income and Wealth, 39, 337–356. Lefranc, A., Pistolesi, N., & Trannoy, A. (2006a). Inequality of opportunities vs inequality of outcomes: Are Western Societies all alike? Working Paper no. 54, ECINEQ, Society for the Study of Economic Inequality. Lefranc, A., Pistolesi, N., & Trannoy, A. (2006b). Equality of Opportunity: deﬁnitions and testable conditions with an application to France. Working Paper no. 53, ECINEQ, Society for the Study of Economic Inequality. O’Neill, D., Sweetman, O., & Van De Gaer, D. (1999). Equality of opportunity and kernel density estimation: An application to intergenerational mobility. Economics Department Working Paper no. 950999, National University of Ireland, Maynooth. Peragine, V. (2002). Opportunity egalitarianism and income inequality. Mathematical Social Sciences, 44, 45–64. Peragine, V. (2004). Measuring and implementing equality of opportunity for income. Social Choice and Welfare, 22, 187–210. Peragine, V. (2005). Ranking income distributions according to equality of opportunity. The Journal of Economic Inequality, 2, 11–30. Roemer, J. E. (1998). Equality of opportunity. Cambridge, MA: Harvard University Press. Ruiz-Castillo, J. (2003). The measurement of inequality of opportunities. In: J. Bishop & Y. Amiel (Eds), Research in economic inequality (9, pp. 1–34). Amsterdam: Elsevier. Sen, A. (1970). Collective choice and social welfare. San Francisco, CA: Holden Day. Sen, A. (1980). Equality of what? In: The Tanner lecture on human values (Vol. 1, pp. 197–220). Cambridge: Cambridge University Press. Villar, A. (2006). On the welfare evaluation of income and opportunity. Contributions to Theoretical Economics, 5, p. 1129. Wolak, F. A. (1989). Testing inequality constraints in linear econometric models. Journal of Econometrics, 41, 205–235.

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APPENDIX Summary Statistics Table A1. Types

Summary Statistics.

North-Center a

Mean std. err.

b

South N

c

a

Mean std. err.b

Nc

Graduation mark 1 83:24

2,090

82:03

1,649

2

84:80

4,217

85:87

2,845

3

88:53

6,723

90:75

4,243

4

97:62

4,451

99:39

3,630

Tot

89:31

17,481

91:1

12,367

Income after three years from graduationd 1 1097:6 1,288

866:3

866

40:8 39:6

36:48 25:95 35:9

548:5

42:12 39:97 36:07 26:08 35:9

631:1

2

1119:9

2,610

945:01

1,529

3

1126:2

4,246

973:84

2,373

4

1133:9

2,534

996:05

1,771

Tot

1123:1

10,678

958:87

6,539

Incomee 1

27117:6

287

22031:3

177

2

29374:1

189

22132:1

100

3

31838:3

256

22134:6

124

4

38239:1

261

26571:7

162

Tot

31687:3

993

23944:5

563

a

548:3 573:6 639:5 581:1

25520:9 23496:6 31780:8 39384:2 31256:1

Sample mean. Sample standard error. c Sample number of observations. d Monthly income. e Annual income. b

635:2 612:2 663:5 635:4

14454:1 12809:4 15672:4 17524:4 15473:3

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VITO PERAGINE AND LAURA SERLENGA

Proofs Proof of Theorem 2 We ﬁrst state and prove the following lemma: Pn Lemma 1 k¼1 vk wk 0 for all sets of real P numbers {vk} such that vk vkþ1 0, 8k 2 f1; . . . ; ng, if and only if ki¼1 wi 0, 8k 2 f1; . . . ; ng. Proof of Lemma 1 Pn Pn Pk Applying Abel’s decomposition, k vkþ1 Þ k¼1 vk wk ¼ k¼1 ðv i¼1 wi . P Pk It is obvious that, if i¼1 wi 0, 8kP2 f1; . . . ; ng, then nk¼1 vk wk 0. As of numbers for the necessity part, suppose that nk¼1 vk wk 0 for all sets P j {vk} such that vk vkþ1 0, but ( j 2 f1; . . . ; ng such that i¼1 wi o0. v Þ & 0; 8kaj. We obtain Consider what happens when ðv k kþ1 Pj Pn v w ! ðv v Þ w o0, which is the desired contradiction. j jþ1 k¼1 k k i¼1 i We can now prove the theorem. By deﬁnition, DW ¼ WðFÞ WðGÞ 0, 8W 2 WEOp2 , if and only if X Z z X Z z U i ðxÞf i ðxÞ dx qG U i ðxÞgi ðxÞ dx 0 qFi i 0

0

i

for all the functions U satisfying conditions (C.1) and (C.4). Using integration by parts, we obtain that DW Z 0 if and only if X Z z dU i X X z i i F i ðxÞ dx qFi qG qiF ½U i ðxÞF i ðxÞz0 i ½U ðxÞ G ðxÞ0 0 dx X Z z dU i Gi ðxÞ dx 0 þ qG i 0 dx As F i ðzÞ ¼ Gi ðzÞ ¼ 1, the above expression reduces to X Z z dU i X i F G i F i qG qi qi U ðzÞ þ i G ðxÞ qi F ðxÞ dx 0 0 dx Pn F that, by condition (C.4), U i ðzÞ ¼ U j ðzÞ and that i¼1 qi ¼ PConsider n G q ¼ 1. Hence, we obtain that DW Z 0 if and only if i i¼1 n Z z X dU i G i ½qi G ðxÞ qFi F i ðxÞ dx 0 dx 0 i¼1 Rz or, equivalently, 0 TðxÞdx 0, where TðxÞ ¼

n X dU i G i qi G ðxÞ qFi F i ðxÞ dx i¼1

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Now considering that, by conditions (C.1) and (C.3), ðdU i ðxÞ=dx dU iþ1 ðxÞ=dxÞ 0, we can apply Lemma 1. Hence, we obtain that TðxÞ 0; 8 U satisfying (C.1) and (C.3), if and only if k X i¼1

qFi Gi ðxÞ

k X

qG i F i ðxÞ; 8x 2 ½0; z; 8k 2 ð1; . . . ; nÞ

i¼1

Rz Clearly, if TðxÞ 0 8x, then 0 TðxÞdx 0; 8x, which proves the sufﬁciency part of the theorem. As for the necessity part, suppose, for a contradiction, that DW 0; 8W 2 W EOp2 and 8F; G 2 C but 9h 2 f1; . . . ; ng and 9I ½a; b ½0; z P such that hi¼1 ðqFi Gi ðxÞ qG i F i ðxÞÞo0; 8x 2 I. Then, by Lemma Pn1, ( is ia set of functions fU i : ½0; z ! <þ ; i 2 f1; . . . ; ngg such that i¼1 ðdU =dxÞ R1 i F i ½qG 2 I. Thus, we have DW ¼ TðxÞdx, where i G ðxÞ qi F ðxÞo0 8x 0 Rb TðxÞo08x 2 I. Clearly, a TðxÞdxo0. Now we can select a function T(x) (i.e., sets of functions Ui and distributions Fi(x) and Gi(x)) R 1 such that TðxÞ & 0; 8x 2 ½0; zI. In this case, we obtain that DW ¼ 0 TðxÞdx ! Rb a TðxÞdxo0, a contradiction. Statistical Tests The testing procedures used in the empirical application of this paper have been developed by Beach and Davidson (1983) and Davidson and Duclos (2000). We consider mainly equality tests and stochastic dominance tests of ﬁrst and second order. In this appendix, as a matter of notation, we represent the two order of stochastic dominance using the integral operator, I j ð:; FÞ to be the function that integrates the function F to order j. I 1 ðx; FÞ ¼ FðxÞ Z x Z I 2 ðx; FÞ ¼ FðtÞdt ¼ 0

x

I 1 ðt; FÞdt 0

The general hypotheses for testing stochastic dominance of order j of the distribution F over G can be written as follow: H j0 : I j ðx; FÞ I j ðx; GÞ 8x 2 ½0; z H j1 : I j ðx; FÞ4I j ðx; GÞ 8x 2 ½0; z The tests are based on comparisons of the (difference in the) distribution functions (and integrals thereof) at a ﬁxed number of points, k (in the paper,

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VITO PERAGINE AND LAURA SERLENGA

we use ﬁve points in the outcome variable range). Deﬁning those differences as a (k 1) vector Dj ðxl Þ ¼ I j ðxl ; FÞ I j ðxl ; GÞ, we specify the equality and stochastic dominance tests accordingly. Equality tests. The equality of distributions test is performed by a Wald test, and apply a w2 test. The hypothesis system is the following: H 0 : Dj ðxl Þ ¼ 0 H 1 : Dj ðxl Þa0

for all l 2 f1; . . . ; kg for some l 2 f1; . . . ; kg

^ j as the (k 1) vector of estimates of Dj(xl) and R ^ j as the Deﬁning D ^ estimates of the variance covariance matrix of Dj , it can be shown that, under the null hypothesis, the (k 1) vector D^ j is asymptotically normal such that: D^ j Nð0; R^ j Þ where R^ j ¼ RjF =N F þ RjG =N G . Therefore, the statistic under the null hypothesis is ^ 0 R^ 1 D^ j w2 W^ j1 ¼ D j j k See Beach and Davidson (1983) for details. Stochastic dominance tests. In this case, we follow the methods considered in Davidson and Duclos (2000), which are designed to test the following hypotheses: H j0 : Dj ðxl Þ 0 H j1

: Dj ðxl Þ40

for all l 2 f1; . . . ; kg for some l 2 f1; . . . ; kg

then the Wald test can be obtained by W^ j2

¼ min

n

D2Rkþ

¼

k P l¼0

0 o ^ j Dj R^ 1 D ^ j Dj D j

wðk; k l; RÞPr w2j c

with the weights w denoting the probability that k1 elements of Dj are strictly positive. As shown by Wolak (1989), the Wald statistic has an asymptotic distribution that is a mixture of chi-squared random variables. In particular, as noted in Barrett and Donald (2003), we compute the

Higher Education and Equality of Opportunity in Italy

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solutions to a large number of quadratic programming problems in order to estimate the weights that appear in the chi-squared mixture limiting ^ j2 , using a Monte Carlo distribution and estimate the p-value of W simulation.10 See Wolak (1989), Davidson and Duclos (2000), and Barrett and Donald (2003) for details.

INEQUALITY OF OPPORTUNITY FOR INCOME IN FIVE COUNTRIES OF AFRICA Denis Cogneau and Sandrine Mesple´-Somps ABSTRACT Purpose: This paper examines for the ﬁrst time inequality of opportunity for income in Africa, by analyzing large-sample surveys, all providing information on individuals’ parental background, in ﬁve comparable SubSaharan countries: Ivory Coast, Ghana, Guinea, Madagascar, and Uganda. Methodology/approach: We compute inequality of opportunity indexes in keeping with the main proposals in the literature, and propose a decomposition of between-country differences that distinguishes the respective impacts of intergenerational mobility between social origins and positions, of the distribution of education and occupations, and of the earnings structure. Findings: Among our ﬁve countries, Ghana in 1988 has by far the lowest income inequality between individuals of different social origins, while Madagascar in 1993 displays the highest. Ghana in 1998, Ivory Coast in 1985–1988, Guinea in 1994, and Uganda in 1992 stand in-between. Decompositions reveal that the two former British colonies (Ghana and Uganda) share a much higher intergenerational educational and Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 99–128 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16005-7

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DENIS COGNEAU AND SANDRINE MESPLE´-SOMPS

occupational mobility than the three former French colonies. Further, Ghana distinguishes itself from the four other countries, because of the combination of widespread secondary schooling, low returns to education, and low income dualism against agriculture. Nevertheless, it displays marked regional inequality insofar as being born in the Northern part of this country produces a signiﬁcant restriction of income opportunities. Originality/value of paper: By providing the ﬁrst ﬁgures for ﬁve countries of Sub-Saharan Africa, this paper allows enlarging the sample of international comparisons in the study of inequality of opportunity. It also reveals some suggestive evidence regarding the long-term origin of intergenerational mobility differences, and in particular the colonial legacy of school extension and of dualism against agriculture.

1. INTRODUCTION The ﬁrst compilation of international income inequality statistics covering a signiﬁcant number of Sub-Saharan African countries was published 10 years ago. It showed this subcontinent to be essentially as inegalitarian as Latin America, a region long known to have a high level of inequality (Deininger & Squire, 1996).1 However, misgivings about household survey quality mean that the idea of a high-inequality Africa is still subject to caution, aside from in the speciﬁc cases of South Africa where apartheid has long made for a glaring level of inequality (Lam, 1999; Louw, Van Der Berg, & Yu, 2006; Leite, Mc Kinley, & Osorio, 2006). In the rest of Africa, economic inequalities remain largely understudied. While equity has recently been raised by the World Bank as a fundamental determinant of economic development (World Bank, 2005), the study of equality of opportunity is only at its beginning (see also, Cogneau, 2006). The paper sets out to make a detailed analysis of inequality of opportunity for income in ﬁve African countries. This essentially descriptive exercise is innovative in that it makes the ﬁrst ever comparative measurement of the extent of the intergenerational transmission of resources and its contribution to the observed income inequality in Africa. This is made possible by having large-sample surveys providing information on the social origins of the individuals interviewed: parents’ education and occupation, and place of birth. This angle on inequalities dictated the countries and surveys chosen since, to our knowledge, very few representative national surveys contain this type of information. The countries in question are Ivory

Inequality of Opportunity for Income in Africa

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Coast in 1985–1988, Ghana in 1987–1988 and in 1998, Guinea in 1994, Madagascar in 1993, and Uganda in 1992. The ﬁve African countries under review here have certain characteristics in common: they are of average size, do not have large mineral resources and derive most of their trade income from agricultural exports. When computed over arable land, population density is very much similar across the ﬁve countries. The bulk of the labor force is still working in agriculture everywhere, although there is some variation between the most urbanized country, Ivory Coast, and the most rural, Madagascar. The vast majority of agricultural workers are small landowners or shareholders. However, the ﬁve countries’ colonial and post-colonial histories are quite different. Three were colonized by the French and two by the British in the late nineteenth century. Furthermore, the three former French colonies took different roads after independence in 1960: Ivory Coast established itself as the main partner of the former colonial power in Africa, Guinea broke with the past and introduced a form of socialist government, while Madagascar displayed a succession of those two polities. Ghana and Uganda had turbulent histories with political conﬂicts and severe macroeconomic crises through to the mid-1980s. The outline and main ﬁndings of this paper are as follows. The ﬁrst section introduces to the main inequality concepts and indexes, as well as to the decomposition techniques that will be used in the remainder of the paper. The second section describes the data and the construction of variables, as well as the countries’ basic socioeconomic features, including the level of overall income inequality. The third section presents the results. It ﬁrst reveals that income inequality differentials are not wide enough to modify average income levels comparisons: 1985–1988 Ivory Coast dominates all other countries in terms of social welfare, followed by other countries with the same ranks as for GDP in PPP. It however shows that Ghana in 1987–1988 had by far the lowest income inequality between individuals of different social origins, while Madagascar in 1993 displayed the highest inequality of opportunity from the same point of view. In-between, Ghana in 1998, Ivory Coast in 1985–1988, Guinea in 1994, and Uganda in 1994 cannot be ranked without ambiguity. Inequality of opportunity with respect to the region of birth, that is most meaningful in the African context of ethnic fractionalization, also carries its weight in the case of Ghanaian and Ivorian Northerners. Among the ﬁve countries, inequality of opportunity for income seems to correlate more with overall income inequality than with average national

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income. We then introduce the sons’ education and occupation as an intermediary variable, and try to distinguish unequal access to social positions from the earnings inequality between positions. Decomposition results reveal that a signiﬁcant part of differences in inequality of opportunity for income can be attributed to differences in intergenerational mobility matrices linking fathers’ education and occupation and sons’ education and occupation. As far as this ﬁrst channel is concerned, the two former British colonies (Ghana and Uganda) share a much larger intergenerational educational and occupational mobility than the three former French colonies. A second channel corresponds to the differences in the distribution of educational levels and of occupations and in the earnings attached to them. With respect to this second channel, Madagascar also stands out as the country with both the highest share of farmers in the population and the highest income dualism against agriculture. At the other end of the spectrum, Ghana combines a more even distribution of education and, in 1987–1988, the lowest returns to education as well as the most limited income dualism. The raise in earnings differentials between education levels and occupations and regions is responsible for the decrease of equality of opportunity in this latter country during the 1990s. The fourth section concludes, by raising general equilibrium issues and historical issues that both warrant further research.

2. THE MEASUREMENT AND ANALYSIS OF INEQUALITY OF OPPORTUNITY We construct inequality of opportunity indexes in keeping with the two main proposals of literature on economic justice and equality of opportunity (Roemer, 1996, 1998; Van de Gaer, 1993). For a given outcome variable (here household consumption per capita), both proposals distinguish between what is due to ‘‘circumstances,’’ deﬁned as an individual’s characteristics that inﬂuence his/her outcome but over which he/she has no control (here father’s education and occupation and region of birth), and what is due to ‘‘effort’’ for which the individual is held responsible or more generally to all the factors considered irrelevant to the establishment of illegitimate inequality. The ﬁrst approach proposed by Roemer considers that only the relative ‘‘efforts’’ in each group of ‘‘circumstances’’ (also called types by this author) are comparable. The inequality between types are then measured by

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comparing individuals with the same relative level of effort; the inequality of opportunity is measured at different points of the distribution of relative levels of effort and these measurements are then aggregated into a single index. Roemer proposes measuring relative levels of efforts as within-types percentiles for the outcome variable. We here choose to compare deciles of income conditional on the types of social origin. We calculate the inequality indexes at each decile and aggregate them taking their average. These ‘‘Roemer’’ indexes are written: ROE ¼

10 1 X I½Y p ðoÞ; pðoÞ 10 p¼1

(1)

where o is an index for the different types of social origins, Yp(o), the mean income at decile p for type o, p(o), the observed weight of type o, and I, an index of inequality. Instead of a traditional index of inequality like Gini or Theil, Roemer favors the minimum function (I=min), in keeping with a Rawlsian maximin principle. We compute this original Roemer’s index. The second approach proposed by Van de Gaer considers that there is equality of opportunity when the distribution of expected earnings is independent of social origins. The extent of inequality of opportunity is then measured by an indicator of the inequality of income expectations obtained by individuals of different origins. These conditional income expectations can be obtained from the distribution of average income estimated by categories of origin; very simply, we can choose for instance the Gini of mean income by type of origin. In their general form, these ‘‘Van de Gaer’’ indexes are written: VdG ¼ I½EðYjoÞ; pðoÞ

(2)

where I is again an inequality index and E(Y|o) the income expectation conditional on social origin o. We compute the Van de Gaer index using three inequality indexes: the minimum function, the Gini, and the Theil-T.2 As argued by Van de Gaer, Schokkaert, and Martinez (2001), the two ‘‘Roemer’’ and ‘‘Van de Gaer’’ measurements considered here produce the same rankings when the transition matrices between origins and income deciles are ‘‘Shorrocks monotonic’’ (Shorrocks, 1978), i.e., when the most underprivileged types of origin in each decile are the same (see also Gajdos & Maurin, 2004, on the related issue of ex-ante and ex-post inequality with uncertainty). The matrices we compute come out as monotonic, so that we mainly use the Van de Gaer index that is easier to compute and to decompose. In the particular case of maximin, the Roemer is even equal to

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DENIS COGNEAU AND SANDRINE MESPLE´-SOMPS

the Van de Gaer index. We disregard inequality of opportunity with respect to income risk, i.e., differences in the within-types (social origins) income variance, mainly because its assessment raises measurement errors issues that are difﬁcult to deal with, especially in a comparative context. Whether in the Roemer or in the Van de Gaer cases, minimum indexes are akin to Rawlsian social welfare indexes: comparisons between countries include a trade-off between efﬁciency (average level) and equity (inequality of opportunity). Minimum indexes imply comparing per capita consumption levels of the worst-off type of social origin between countries. For that purpose, we use purchasing power parity (PPP) exchange rates. These minimum indexes can also be divided by the overall average income, so that they only reﬂect an equity component and disregard efﬁciency considerations, like in the cases of the Gini or Theil-T inequality indexes. We also examine generalized (welfare and equity) and simple (pure equity) Lorenz dominance in order to assess the inﬂuence of the choice of a particular social welfare or inequality index. When we introduce intermediary outcome variables like sons’ education and occupation, we propose a decomposition of the Van de Gaer inequality of opportunity index that is inspired from Cogneau and Gignoux (2008). We write: X E c ðYjo; sÞpc ðsjoÞ (3) E c ðYjoÞ ¼ s

and still, VdGc ¼ I½E c ðYjoÞ; pc ðoÞ

(4)

where c indexes the country under analysis, o (still) social origins, and s the intermediary outcome, i.e., son’s social position; pc(s|o) is the conditional probability of reaching the social position s given the social origin o, coming from the social mobility matrix crossing social origins and social positions: pc(s|o)=pc(s, o)/pc(o), where pc(s, o) are the observed joint probabilities of each matrix cell and pc(o) the row marginal probabilities. In the case of large samples, the conditional expectations, Ec(Y|o, s), can be estimated by the empirical means for each sub-population (o, s). In order to analyze the between-countries differences in the Van de Gaer index we could devise a rather straightforward decomposition, when passing from country c to country cu: ﬁrst change the conditional probabilities from pc(s|o) to pc0 ðsjoÞ in (3), holding constant the country-speciﬁc earnings

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structure Ec(Y|o, s), as well as the social origins distribution pc(o) in (4); second, change the social origins distribution from pc(o) to pc0 ðoÞ in (4); third and last, change the earnings structures from Ec(Y|o, s) to E c0 ðYjo; sÞ. However, this kind of decomposition does not clearly distinguish the impact of the strength of the link between social positions and social origins from the impact of the distribution of social origins and social positions in the population. For instance, passing from pc(s|o) to pc0 ðsjoÞ in a ﬁrst step implicitly induces a shift in the distribution of positions, as the distribution of social origins pc(o) is held constant; likewise, when passing from pc(o) to pc0 ðoÞ in a second step, the reweighing of origins induces a reweighing of positions p(s), as pc0 ðsjoÞ is left constant in the computation of counterfactual conditional expectation using (3). Besides, when general equilibrium considerations are borne in mind, such shifts in the distribution of education levels and occupations in the sons’ generation are not necessarily independent from counterfactual changes in the income conditional expectations (i.e., returns to education and earnings attached to occupations). In order to isolate the pure social mobility effect, holding constant both the marginal distribution of social origins and of social positions, we consider the odds-ratios (OR, henceforth) of the mobility matrices, as it is traditional in quantitative sociology. OR allow us to compare the strength of association between origin and destination across time and/or space, regardless of the fact that the weight of some origins and some destinations varies between countries or periods. More precisely, they express the relative probability for two individuals of different origins to reach a speciﬁc destination rather than another one. ORc ðs; o; s0 ; o0 Þ ¼

pc ðsjoÞpc ðs0 jo0 Þ q ðsjoÞ=½1 qc ðsjoÞ ¼ c 0 0 0 pc ðs joÞpc ðsjo Þ qc ðs=o Þ=½1 qc ðsjo0 Þ

(5)

where qc(s|o) is the conditional probability in the 2 rows and 2 columns submatrix crossing social origins (o; ou) and social positions ðs; s0 Þ : qc ðsjoÞ ¼ pc ðsjoÞ=½ pc ðsjoÞ þ pc ðs0 joÞ. Being ratios of conditional probabilities, the ORs are arithmetically independent of row and column marginal probabilities pc(o) and pc(s). We use the log-linear model that provides a useful parameterization of ORs. It allows us to construct a ﬁctional mobility table where row and column margins are those of country c and ORs are those of country cu. The appendix gives details about this construction.

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In the end, our preferred decomposition of the between-country differences in inequality of opportunity indexes is the following: VdGc VdGc0 ¼ I½E c ðYjoÞ; pc ðsjoÞ I½M c!c0 ðYjoÞ; pc ðsjoÞ þ I½M c!c0 ðYjoÞ; pc ðsjoÞ I½M c!c0 ðYjoÞ; pc ðsjoÞ þ I½M c!c0 ðYjoÞ; pc ðsjoÞ I½E c0 ðYjoÞ; pc0 ðsjoÞ

ð6Þ

When passing from country c to country cu, the ﬁrst term of (6) gives the impact of simply changing the ORs of the social mobility table, i.e., the country-speciﬁc features of the pure association between social origins and social positions, irrespective of those latter variables’ marginal distributions. The second term then shifts both the distributions of social origins p(o) and social positions p(s) in one step. The third and last term corresponds to the change in the earnings structure E(Y|s, o). The precise deﬁnitions of the two counterfactual income expectations M c!c0 ðYjoÞ and M c!c0 ðYjoÞ are given in the appendix. Of course, this kind of decomposition is path-dependent. One can deﬁne M c0 !c ðYjoÞ and M c0 !c ðYjoÞ (with obvious notations) and combine the three decomposition steps to devise six different paths for going from c to cu.3 As we have 6 countries-years to compare, we also have 15 pairs of countries; this makes 90 decompositions to consider in total. Instead, we chose to implement decompositions in the order of Eq. (6), as we believe the second and third terms should be better considered together or at least successively, for reasons exposed thereafter: this limits the number of decompositions to 30. We also choose as benchmarks the two countries showing, respectively, the lowest and the highest inequality of opportunity (Ghana in 1987–1988 and Madagascar in 1993): this limits the number of decompositions to consider to 18. It can be noticed that the second and third steps of our decomposition are very similar to the Oaxaca–Blinder decomposition that decomposes average wage differentials into a ﬁrst term of population differences in average characteristics (here distribution of social origins and social positions) and a second term of returns to these characteristics (Blinder, 1973; Oaxaca, 1973). We, however, do not make any parametric assumption about the function linking income and social origins and positions, like for instance a loglinear ‘‘Mincerian’’ speciﬁcation. Nevertheless, like all the Oaxaca–Blinder decomposition procedures, even the most sophisticated ones (Juhn, Murphy, & Pierce, 1993; Di Nardo, Fortin, & Lemieux, 1996; Bourguignon, Ferreira, & Leite, 2002), our decomposition assumes independence between the earnings structure (here by social origins and social positions) and the distribution of the population. This assumption implies the absence of

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general equilibrium effects: the counterfactual redistribution of the population by social origins and/or positions does not alter the structure of earnings. This kind of assumption is certainly strong when trying to disentangle the impact of the distribution of sons’ population by education levels and occupations (i.e., what we call social positions) and the impact of the returns to these positions, in keeping with the traditional Oaxaca– Blinder procedure. If this assumption does not hold, the second and third steps of the decomposition (6) cannot be separated. It could be judged that changing only the ORs of the matrix (while holding ﬁxed the supply of educations levels and occupations) should have less general equilibrium consequences, so that our ﬁrst step would really reﬂect the causal impact of a counterfactual change in ‘‘pure’’ social mobility. However, real-world changes in social mobility could also underlie composition effects resulting in ﬁne changes in the structure of labor supply, themselves having in turn an impact on earnings structures (either purely compositional or through general equilibrium resolution). Besides, the relevance of this ﬁrst step of the decomposition depends on another independence assumption: independence between ORs and marginal frequencies. While arithmetically correct, this latter assumption is an issue, as for instance improvements in educational equality of opportunity may crucially depend on the broad extension of access to higher levels of education, as it has been observed historically in many contexts (see, e.g., Cogneau & Gignoux, 2008, opus cited, on Brazil) and also as some of our African case studies will illustrate.

3. DATA AND VARIABLES CONSTRUCTION We use household surveys covering large nationally representative samples for ﬁve African countries: Ivory Coast from 1985 to 1988, Ghana in 1988 and 1998, Guinea in 1994, Madagascar in 1993, and Uganda in 1992. The Ivory Coast, Ghana, and Madagascar surveys are ‘‘integrated’’ Living Standard Measurement Surveys (LSMS) designed by the World Bank in the 1980s; the format of the two other for Guinea and Uganda is inspired from them. The appendix Table A1 gives more details on the surveys.4 To our knowledge, the surveys that we selected are the only large sample nationally representative surveys in Africa that provide information on parental background for adult respondents. We restrict the sample to men 20–69-year-old and family backgrounds to fathers’ positions. Combining information on education and main

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occupation of fathers, we deﬁne three social origins: farmers (whatever their level of education); nonfarmers with no education or primary level; and nonfarmers having reached a secondary or tertiary level of education. For the purpose of the decompositions whose methodology has been detailed at length in the previous section, individuals’ (sons’) positions are deﬁned like fathers’ except that an inactive class is added to include unemployed, students, and retired people. We also deﬁne two other circumstances to try to take into account a variable of outmost importance in the African context of State consolidation and ethnic conﬂicts: region of birth. Unfortunately, this variable is not available in the case of Uganda. First, for each country except Uganda, we are able to distinguish individuals born in the most advantaged region: the capital town district. When we interact this region of birth dichotomy with the three social origins as deﬁned above, we obtain six groups of social/regional origin. Second, we may additionally distinguish individuals born in the most peripheral and disadvantaged regions of Ivory Coast and Ghana, i.e., the Northern parts of each country; making a similar divide in the cases of Guinea and Madagascar was more difﬁcult. In the case of Ivory Coast, we aggregate foreign-born migrants born in Burkina-Faso and Mali to Northerners born in Ivory Coast, as these two populations may be confronted to the same restrictions in their income opportunity set. In the cases of Ivory Coast and Ghana, the three possible regions of birth (capital/ North/other) are only interacted with the ‘‘father farmer’’ social origin, in order to get sufﬁcient sample sizes in each cell. This generates ﬁve groups of social/regional origin. Table 1 shows the size and the breakdown of the ﬁve samples by social origins and social positions.5 Samples are quite large from 2,700 (Ghana 1987–1988) to 8,530 observations (Uganda). Most of the fathers of age 20–69 years are farmers, even if there is some variation from the Ivory Coast and Madagascar cases (more than 80%), through Guinea and Uganda (around 78%), to Ghana with only 73% in 1988 and 65% in 1998. In the generation of sons, the share of farmers is still higher than 50% everywhere, except that we observe some divergence across time due to the speed of urbanization. In Madagascar, the share of farmers among sons (74%) is still close to that among fathers (82%), whereas in Ivory Coast we observe a 30 percentage points fall, from 85% to 55%. Although it is slowing down, this structural change is still rapid in some countries, as we see that in Ghana this share fell by 6 percentage points over a decade (1988–1998).

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Inequality of Opportunity for Income in Africa

Table 1.

Sample size

Sample Description.

Ivory Coast

Ghana 1988

Ghana 1998

Guinea

4,843

2,702

4,625

5,513

Madagascar

4,427

Uganda

8,530

Fathers (%) Farmer Nonfarmer, low educationa Nonfarmer, high educationb Born in the Northc Born in the capital regiond

85.4 10.5 4.1 31.3 8.5

73.4 14.1 12.5 21.1 8.5

64.8 14.1 21.2 20.9 9.6

78.4 17.9 3.8 n.a. 6.8

82.2 10.6 7.1 n.a. 29.7

78.9 13.1 8.0 n.a. n.a.

Individuals (%) Farmer Nonfarmer, low educationa Nonfarmer, high educationb Inactive

55.7 19.3 11.7 13.3

56.4 10.6 25.9 7.1

49.8 9.0 28.9 12.3

56.7 22.3 10.6 10.4

74.4 9.3 11.6 4.6

63.5 14.5 11.2 10.9

Note: Coverage: Men 20–69 years old. Source: Ivory Coast LSMS 1985–1988; Ghana GLSS 1987–1988 and 1998; Guinea, EIBC 1994; Madagascar, EPM 1993; Uganda Integrated Household Survey 1992; calculations by the authors. a Never been at school or has achieved at most primary level. In the case of fathers in Ivory Coast, means having obtained at most a primary degree (CEPE). b Having achieved more than primary school level. In the case of fathers in Ivory Coast, means having middle school degree (BEPC). c Ivory Coast: born in de´partements of Bouna, Bondoukou, Boundiali, Dabakala, Ferkessedougou, Katiola, Korhogo, Mankono, Odienne´, Se´gue´la, Tengrela, and Touba (20.5%) and born in Burkina-Faso or Mali (10.8%); Ghana: born in regions Northern, Northern West, Northern East. d Ivory Coast: Abidjan; Ghana: Greater Accra; Guinea: Conakry; Madagascar: Antananarivo.

Regarding the two other classes of origin and the distribution of education levels, Ghana stands out as the country where secondary education is the most widespread among fathers as well as among sons. In Ghana, middle school level is in fact more like an upper primary level. For the generations concerned (the system was reformed in 1987), the Ghanaian education system offered much longer schooling than elsewhere based on the ‘‘6-4-5-2’’ format: 6 years in primary school, 4 in middle school, 5 in secondary school, and 2 pre-university years (lower sixth and upper sixth). Individuals could pass an exam to go directly from primary to secondary school, cutting out middle school. However, since primary school had no system of repeating a failed year, half of the individuals (those who had at least reached middle school) had at least completed these 6 years of

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schooling. Most of the other half had never attended school, with only a small minority having left school at primary level. Even before independence, Madagascar and Uganda experienced an early start in primary schooling, due to the policies of Merina and Buganda kingdoms and in particular the openness to European missionaries. Yet this advantage does not give rise to a high proportion of individuals completing primary school and disappears completely at the secondary level when compared with Ghana. In Madagascar and Uganda, two-thirds of individuals aged 20 and over had successfully completed 1 year of primary education, but very few had completed all 5 (Madagascar, ‘‘5-4-3’’) or 7 (Uganda, ‘‘7-4-2’’) years of this level. At the other extreme, Ivory Coast and even more so, Guinea, stand out as countries where primary education was reserved to a small minority before independence. In fact, Madagascar makes an exception among former French colonies: a continental overview conﬁrms the British colonies’ large advantage in terms of school extension before 1960 (Benavot & Riddle, 1988). While being still behind in terms of access to school in the 1970s, Ivory Coast and Guinea had caught up with Madagascar and Uganda at the middle (‘‘colle`ge’’ in the French-origin systems) and secondary levels, as is revealed by the sons’ education levels distribution. The outcome variable is the consumption per head for the household in which the individual lives. In low-income countries, it is more reliable to measure consumption (including home-produced consumption) than income (Deaton, 1997). For each country, consumption components have been meticulously reconstructed from raw survey data using a uniform methodology for comparison purposes.6 For the ease of exposition, household consumption per capita is called ‘‘income’’ in the remainder of the paper. Table 2 ﬁrst reveals a wide range of mean income levels, once consumption per capita in domestic currency is translated into international dollars using two sources for Purchasing Power Parity (PPP) exchange rates (Heston, Summers, & Aten, 2002; Maddison, 2003). The ranking of countries is fairly consistent with GDP in international dollars as estimated by the same sources. During 1985–1988 Ivory Coast comes out by far as the wealthiest country, followed by Ghana, Uganda, and Madagascar. A very large uncertainty regarding price levels in Guinea results in a wide discrepancy between the two income levels estimates for that country; our preference goes to the (much lower) Maddison’s estimate. All ﬁve African countries exhibit a very high level of income inequality, comparable with Latin American standards.7 Our own estimates are

Mean Income Levels and Overall Income Inequality Levels.

Ivory Coast

Ghana 1988

Mean per capita consumption in international $ 1,796 1,003 PPP PWTa 1,586 1,041 PPP Maddisonb Income inequality indexes Gini index 0.44 [0.43; 0.46] Theil-T index 0.37 [0.34; 0.41]

0.40 [0.39; 0.41] 0.29 [0.27; 0.32]

Ghana 1998

Guinea

Madagascar

Uganda

1,024 1,063

2,023 504

354 322

578 540

0.45 [0.43; 0.47] 0.40 [0.43; 0.47]

0.47 [0.46; 0.49] 0.42 [0.39; 0.46]

0.48 [0.45; 0.49] 0.44 [0.40; 0.46]

0.47 [0.46; 0.48] 0.43 [0.40; 0.46]

Inequality of Opportunity for Income in Africa

Table 2.

Notes: Bootstrap conﬁdence intervals between brackets. Coverage: Men 20–69 years old. Source: see Table 1. a Per capita consumption in international $ (Source: Penn World Tables 6.1, PPP level of consumption for the reference year, except for Ghana 1998 for which the PPP 1988 deﬂator is used). b Per capita consumption in international $ (Source: Maddison, 2003, for the reference year, except for Ghana 1998 for which the PPP deﬂator is 1988 one).

111

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broadly consistent with those which are compiled for the same countries and surveys by the United Nations WIDER database8 and the World Bank report on Equity and Development (World Bank, 2005). Of the ﬁve countries studied, Ghana in 1987–1988 has by far the lowest income inequality, while Ghana in 1998 and the four other countries cannot be statistically distinguished.9 Lorenz curves dominance conﬁrms this partial ranking.

4. RESULTS: DIFFERENCES IN INEQUALITY OF OPPORTUNITY FOR INCOME Table 3 shows the maximin index for which both Roemer’s and Van de Gaer’s give the same results, given the monotonicity of the transition matrix linking types of social origins and income. Having a farmer father is always the most disadvantaged social origin, whatever the country that is considered, and whatever the within-type income decile. The index is presented in both its social welfare version (PPP levels with Maddison’s exchange rates) and its inequality (normalized by the mean) version. The social welfare version produces the same ranking of countries as average consumption per capita in PPP (see Table 2): this means that between-countries differences in inequality (of opportunity) are not high enough to modify the differences in income level or poverty. Generalized Lorenz dominance unambiguously conﬁrms this diagnosis (see Fig. 1). The normalized index gives another ranking of countries: 1987–1988 Ghana comes ﬁrst, followed by Ivory Coast, Uganda, 1998 Ghana, Guinea, and lastly Madagascar. For instance, it expresses that the mean income of sons with a father farmer reaches 91% of the country mean income in Ghana, whereas the same ratio is only 80% in Madagascar. These indexes are composed of two basic elements: the ﬁrst is the earnings scale between social origins, shown in Table 4; the second is the vector of weights of social origins in the population, shown in Table 1. Table 4 reveals that it is in 1987–1988 Ghana that mean income differentials by social origins are the lowest, and in Madagascar the highest. In 1987–1988 Ghana, the earnings scale goes from 100 through 127 to 151, while in Madagascar it climbs from 100 through 185 to 317. The inequality of opportunity partial ordering provided by Lorenz curves dominance distinguishes only three groups of countries: the two extreme

Ivory Coast

Inequality of Opportunity for Income Indexes. Ghana 1988

Ghana 1998

Guinea

Madagascar

Uganda

946 0.91 0.07 [0.07; 0.07] 0.012 [0.011; 0.013]

876 0.82 0.13 [0.12; 0.13] 0.033 [0.032; 0.034]

416 0.83 0.14 [0.14; 0.15] 0.052 [0.051; 0.055]

258 0.80 0.17 [0.16; 0.18] 0.087 [0.083; 0.094]

467 0.86 0.11 [0.11; 0.12] 0.040 [0.038; 0.043]

Adding region of birth: Born in the capital town district (six groupsb) Maximin indexa 1,339 940 858 Gini index 0.13 [0.12; 0.14] 0.08 [0.07; 0.08] 0.15 [0.14; 0.15] Theil-T index 0.050 [0.047; 0.054] 0.015 [0.014; 0.017] 0.045 [0.043; 0.047]

412 0.15 [0.14; 0.15] 0.056 [0.054; 0.059]

251 0.18 [0.17; 0.19] 0.092 [0.086; 0.97]

n.a. n.a. n.a.

Adding region of birth: Born in the North or in the capital town (ﬁve groupsc) Maximin indexa 1,157 804 574 Gini index 0.15 [0.15; 0.16] 0.09 [0.09; 0.10] 0.17 [0.16; 0.17] Theil-T index 0.054 [0.051; 0.057] 0.016 [0.015; 0.016] 0.049 [0.047; 0.051]

n.a. n.a. n.a.

n.a. n.a. n.a.

n.a. n.a. n.a.

Fathers’ position in three groups Maximin indexa 1,388 Normalized by mean 0.87 Gini index 0.11 [0.10; 0.12] Theil-T index 0.045 [0.042; 0.049]

Inequality of Opportunity for Income in Africa

Table 3.

Notes: Bootstrap conﬁdence intervals between brackets. Coverage: Men 20–69 years old. Source: see Table 1. a Per capita consumption in international $ (Source: Maddison, 2003, for the reference year, except for Ghana 1998 for which the PPP 1988 deﬂator is used). b Social origins (three groups) X being or not being born in the capital town district. c Father farmer and being born in the North; father farmer and being born in the capital town district; father farmer and being born elsewhere; uneducated nonfarmer father, educated nonfarmer father.

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DENIS COGNEAU AND SANDRINE MESPLE´-SOMPS

114

Generalized Lorenz Curves

1000 500 0

Per capita consumption International $

1500

Ivory Coast Ghana 1988 Ghana 1998 Guinea Madagascar Uganda

0

Fig. 1.

.1 .2 .3 .4 .5 .6 .7 .8 .9 Men 20-69 ranked by conditional (to social origin) mean income

1

Generalized Lorenz Curves between three Groups of Fathers’ Position.

Table 4.

(1) Father farmer (2) Father Nonfarmer, low education (3) Father nonfarmer, high education F test (2)=(3) (p value)

Conditional Means Differences.

Ivory Coast

Ghana 1988

Ghana 1998

Guinea

Madagascar Uganda

100 170

100 127

100 130

100 184

100 185

100 146

270

151

181

262

317

228

0.00

0.01

0.00

0.00

0.00

0.00

Note: Coverage: Men 20–69 years old. Source: see Table 1. Difference with (1) signiﬁcant at 1%.

cases, 1987–1988 Ghana and 1993 Madagascar, and the group formed by the four other country cases whose Lorenz curves cross each other (Fig. 2). This dominance result is reﬂected in the top panel of Table 3 by the different orderings obtained within the middle-group of countries, according

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115

-.02 -.04 -.06 -.08

Ivory Coast-Ghana 88 Ghana 98-Ghana 88 Guinea-Ghana 88 Madagascar-Ghana 88 Uganda-Ghana 88

-.1

Difference in cumulated consumption

0

Lorenz Curve Differences

0

Fig. 2.

.1 .2 .3 .4 .5 .6 .7 .8 .9 Men 20-69 ranked by conditional (to social origin) mean income

1

Lorenz Curves between three Groups of Fathers’ Position.

to which inequality index I is chosen to implement the Van de Gaer formula (2). According to the Gini index, Ivory Coast is the second least unequal country followed by Uganda, 1998 Ghana, and Guinea, whereas according to the Theil-T index it is 1998 Ghana that comes second, followed by Uganda, Ivory Coast, and Guinea. Furthermore, when looking at conﬁdence intervals, only the relative position of Guinea seems statistically reliable for these two indexes. The second and third panels of Table 3 provide another set of results about the inﬂuence of the individuals’ region of birth. The middle panel shows that being born in the capital town district (interacted with father’s occupation and education level) does not give rise to a very signiﬁcant advantage and thus does not modify very much the diagnosis about inequality of opportunity, whatever the country that is considered (in Uganda, the region of birth is not available). In the bottom panel, we additionally distinguish the Northern region of birth category that is most relevant in the cases of Ghana and Ivory Coast (including born in Burkina-Faso or Mali for the latter). It comes out that being born in the Northern disadvantaged regions signiﬁcantly restricts income opportunities in Ivory Coast and 1998 Ghana. The analysis of the

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DENIS COGNEAU AND SANDRINE MESPLE´-SOMPS

1988–1998 period of economic recovery in this latter country has indeed revealed that growth resumption has not been evenly distributed between the North and the South (Shepherd & Gyimah-Boadi, 2005).10 In the case of Ivory Coast, this lack of opportunity of Northerners is one element of explanation for the political crisis that has lead to the partition of the country since 2002 and is not yet entirely solved. It is for 1998 Ghana that this dimension comes out as most meaningful. In the end, the ordering of countries in three classes seems robust. This ordering suggests that inequality of opportunity for income is only imperfectly correlated with average wealth: although Madagascar is the poorest of our set, Ghana is not the most afﬂuent country when compared to Ivory Coast. Countries’ ranks are more consistent with overall income inequality: Table 2 indeed shows that 1987–1988 Ghana is the country with the most equal income distribution, with a Gini index of 0.40, while Madagascar stands out as the most unequal. Furthermore, a look at Ghana across time suggests that economic recovery was accompanied with an increase in both overall inequality and inequality of opportunity. The last part of our results introduces the social position as an intermediary variable for analyzing the link between social origin and income. As already mentioned in Section 2, we coded the sons’ social position as the fathers,’ except for an inactive class that gathers students in younger cohorts, unemployed, and retired people in older cohorts. The distribution of population among the four social positions for each country was already given in Table 1. Table 5 shows the outﬂow tables of intergenerational mobility between social origins and social positions. Conditional probabilities are somewhat difﬁcult to compare at face value, as the weights of each social destination vary from one country to another. However, the share of farmers among sons is roughly comparable between Ivory Coast, Ghana, and Guinea. Then, the comparison of the ﬁrst columns of the corresponding outﬂow tables reveals that nonfarmer sons have a higher probability of working in agriculture in Ghana than in the two other countries. Once the higher weight of farmers is taken into account, the other former British colony, Uganda, shares this characteristic with Ghana. The outﬂow table for Madagascar reveals two other speciﬁc features: a very low rate of exit from agriculture for farmers’ sons and also the highest rate of reproduction in the nonfarmer educated social class. Table 6 presents another element of our decomposition: the earnings scales according to social position. Like for earnings scales according to father’s position, Ghana again stands out with the narrowest earnings scales

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Table 5.

Intergenerational Social Mobility: Outﬂow Tables. 1

2

3

In.

Total

Ivory Coast 1 2 3 Total

63 17 5 56

18 37 13 20

9 20 38 12

10 26 44 13

100 100 100 100

Ghana 1988 1 2 3 Total

67 30 27 57

10 21 5 11

19 39 52 26

4 10 16 7

100 100 100 100

Ghana 1998 1 2 3 Total

66 33 16 50

8 18 4 9

19 36 54 29

7 13 25 12

100 100 100 100

Guinea 1 2 3 Total

68 19 3 57

19 38 17 23

6 23 42 10

7 20 38 10

100 100 100 100

Madagascar 1 2 3 Total

86 31 20 75

7 27 6 9

5 33 55 11

3 9 19 5

100 100 100 100

Uganda 1 2 3 Total

71 45 26 64

13 27 16 15

9 19 36 12

7 9 22 9

100 100 100 100

Notes: Father’s position in rows, son’s positions in columns.1, Farmer; 2, nonfarmer with no more than primary education; 3, nonfarmer with post-primary education; In., inactive (student, unemployed, retiredy). Coverage: Men 20–69 years old. Source: see Table 1.

whatever the year (1988 or 1998) considered. As already noticed in previous works using the same surveys, returns to education are fairly limited in that country (Glewwe & Twum-Baah, 1991; Schultz, 1999; Cogneau et al., 2006). Among other countries, Guinea exhibits the widest earnings scale, while the

118

Table 6.

DENIS COGNEAU AND SANDRINE MESPLE´-SOMPS

Mean Income Differences between Sons According to Social Positions.

(1) Farmer (2) Nonfarmer, low education (3) Nonfarmer, high education (4) Inactive Between-group component of Theil-T (%)

Ivory Coast

Ghana 1988

Ghana 1998

Guinea

Madagascar Uganda

100 158

100 136

100 138

100 236

100 173

100 200

321

157

199

368

314

285

192 27

106 7

149 12

208 31

214 25

198 21

Note: Coverage: Men 20–69 years old. Source: see Table 1. Difference with (1) signiﬁcant at 1%.

three remaining countries are rather close to each other from that point of view. The between-groups component of the Theil-T index decomposition tells a rather similar story: while in Ghana income differentials between social positions hardly explain more than 10% of income inequality between 20–69 years old men, in all other countries this share exceeds 20% and reaches 31% in the case of Guinea. We lastly turn to the decomposition exercises. Table 7 (respectively, Table 8) shows in columns the three terms of decomposition of Eq. (6) when passing from a given country to 1988 Ghana (respectively, 1993 Madagascar), the country where inequality of opportunity for income has been estimated as the lowest (respectively, the highest). For each pair-wise comparison, the two rows correspond to the two possible paths for implementing the same decomposition in the same order: the ﬁrst row simply starts from the country indicated in the ﬁrst column, whereas the second starts from the chosen benchmark country (Ghana 1988 or Madagascar). The ﬁrst term corresponds to the inﬂuence of the mobility matrices inner structure, i.e., ORs. As revealed by the signs and magnitudes of the ﬁrst term, Ghana (especially in 1988) and Uganda are the countries where intergenerational mobility is the most ﬂuid, and Madagascar where it the most rigid, at least from the standpoint of income opportunities; Ivory Coast and Guinea stand in-between. The second term assesses the impact of moves in the margins of the mobility tables; it is linked to the differences in economic structures (weight

Decomposition of Inequality of Opportunity: The Impact of Mobility Matrices (Benchmark Country: Ghana 1988).

Observed Difference due to With Ghana 1988 Difference With Ghana Difference (2)+(3) Ghana 1988 ORs (1) ORsa due to Margins (2) 1988 Mob. Matrixb due to Earnings Structures (3) Ghana 1988 Ghana 1998

0.07 0.13

Ivory Coast

0.11

Guinea

0.14

Madagascar

0.17

Uganda

0.11

0.01 0.01 0.02 0.02 0.03 0.02 0.05 0.02 0.00 0.01

0.12 0.09 0.11 0.12 0.11

0.03 0.01 +0.09 +0.05 +0.05 +0.03 +0.05 +0.03 +0.04 +0.02

0.09 0.18 0.16 0.17 0.15

0.02 0.04 0.11 0.07 0.09 0.08 0.10 0.11 0.08 0.05

0.05 0.05 0.02 0.02 0.04 0.05 0.05 0.08 0.04 0.03

0.07

Inequality of Opportunity for Income in Africa

Table 7.

Notes: For each country, the second row corresponds to the decomposition starting from the reference country sample (Ghana 1988) with the opposite sign. Coverage: Men 20–69 years old. Source: see Table 1. a Inequality of opportunity index obtained through a reweighing of observations according to a counterfactual social mobility matrix with the same margins (distribution of father’s and son’s positions) as the country under review but Ghana 1988 ORs. b Inequality of opportunity index obtained through reweighing of observations according to the Ghana 1988 social mobility matrix (both ORs and margins).

119

120

Table 8.

Decomposition of Inequality of Opportunity: The Impact of Mobility Matrices (Benchmark Country: Madagascar 1993). Difference due to ORs (1)

With Madagascar ORsa

Difference due to Margins (2)

With Madagascar Mob. Matrixb

Difference due to Earnings Structures (3)

(2)+(3)

Ghana 1988

0.07

0.09

0.13

Ivory Coast

0.11

Guinea

0.14

0.03 0.05 0.06 0.09 +0.05 +0.06 +0.04 +0.04

0.06

Ghana 1998

+0.02 +0.05 +0.02 +0.04 +0.02 +0.01 +0.00 +0.00

+0.11 +0.10 +0.08 +0.09 0.01 0.01 0.01 0.01

+0.08 +0.05 +0.02 +0.00 +0.04 +0.05 +0.03 +0.03

Madagascar Uganda

0.17 0.11

+0.02 +0.03

+0.01 +0.02

+0.05 +0.04

0.15 0.13 0.14

0.16

0.01 0.01

0.09 0.18 0.18

0.15

Madagascar 1993

0.17

Notes: For each country, the second row corresponds to the decomposition starting from the reference country sample (Madagascar 1993) with the opposite sign. Coverage: Men 20–69 years old. Source: see Table 1. a Inequality of opportunity index obtained through a reweighing of observations according to a counterfactual social mobility matrix with the same margins (distribution of father’s and son’s positions) as the country under review but Madagascar 1993 ORs. b Inequality of opportunity index obtained through reweighing of observations according to the Madagascar 1993 social mobility matrix (both ORs and margins).

DENIS COGNEAU AND SANDRINE MESPLE´-SOMPS

Observed

Inequality of Opportunity for Income in Africa

121

of agriculture) and in educational development. The third term of the decomposition, that we call the earnings structure effect, corresponds to the moves in average earnings attached to each cells of the mobility matrices. The two terms have in most cases large magnitudes. They also have opposite signs, the only exception arising in the historical evolution of Ghana over the 1990s. This latter feature suggests the presence of compositional and/or general equilibrium effects linking labor supply composition to earnings structures. Because of this general equilibrium issue, we also look at the sum of the second and third terms, and compare column (1) with column (2)+(3). In most cases, the earnings structure effect dominates the margins effect in their sum; or in the Bourguignon et al. (2002) terminology, price effects dominate population effects, one exception arising when Ivory Coast and Guinea are compared to Madagascar. When comparing Uganda or 1998 Ghana with Madagascar, the mobility effect dominates the sum of the tables’ margins (population) and earnings structure (price) effects. It also contributes to half of the difference between 1988 Ghana and 1985–1988 Ivory Coast (and between 1987–1988 and 1998 Ghana for one path). In the other cases, it is dominated by the joint inﬂuence of population distribution and earnings structure. However, mobility effects always have the same sign as the sum of population and earnings effects. This latter feature once again suggests that compositional effects and/or general equilibrium forces may be at play that would determine at the same time the earnings differentials, the occupational and educational structure, and the intergenerational opportunities.

5. CONCLUSION A new analysis of large-sample surveys in ﬁve comparable Sub-Saharan African countries, all providing information on individuals’ parental background and region of birth, allows measuring for the ﬁrst time inequality of opportunity for income in ﬁve comparable countries of Africa: Ivory Coast, Ghana, Guinea, Madagascar, and Uganda. We compute inequality of opportunity indexes in keeping with the main proposals in the literature, and propose a decomposition of between-country differences that distinguishes the respective impacts of intergenerational mobility between social origins and positions, of the distribution of education and occupations in the sons’ generation, and of the earnings structure. Among our ﬁve countries, Ghana in 1988 has by far the lowest income inequality

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between individuals of different social origins, while Madagascar in 1993 displays the highest inequality of opportunity from the same point of view. Ghana in 1998, Ivory Coast in 1985–1988, Guinea in 1994, and Uganda in 1992 stand in-between and cannot be ranked without ambiguity. Inequality of opportunity for income seems to correlate with overall income inequality more than with national average income, like in the famous comparison between Sweden and United States undertaken by Bjo¨rklund and Ja¨ntti (2001). Decompositions reveal that the two former British colonies (Ghana and Uganda) share a much higher intergenerational educational and occupational mobility than the three former French colonies. Further, Ghana distinguishes itself from the four other countries, because of the combination of widespread secondary schooling, low returns to education, and low income dualism against agriculture. Nevertheless, it displays marked regional inequality insofar as being born in the Northern part of this country produces a signiﬁcant restriction of income opportunities. Intergenerational social mobility, school extension, and income dualism are not necessarily independent phenomena, if only through general equilibrium effects. More research is warranted about the long-term dynamics that have brought about this differentiation between countries, and in particular between such close neighbors as Ivory Coast and Ghana. Those long-term dynamics would involve the combined impacts of geography, pre-colonial conditions and colonial powers’ policies, and consecutive or disruptive post-colonial State trajectories.

NOTES 1. In terms of inequality of opportunity, the case of Brazil has been particularly investigated (Dunn, 2007; Bourguignon, Ferreira, & Menendez, 2007; Cogneau & Gignoux, 2008). 2. Moreno-Ternero (2007) ﬁrst proposed the application of an inequality index other than the minimum function, and Rodrı´ guez (2008) proposed an extension to more general partial orderings. 3. In implementing these decompositions, the earnings structures Ec(Y|o,s) and E c0 ðYjo; sÞ do not have to be estimated, as all terms are obtained through a reweighing of the relevant samples (c or cu) by the appropriate systems of weights: pc!c0 ðs; oÞ=pc ðs; oÞ then pc0 ðs; oÞ=pc ðs; oÞ, like in Di Nardo et al. (1996). 4. Documentation and more details can be found at the Website of the World Bank’s Africa Household Survey Databank: http://www4.worldbank.org/afr/ poverty/databank/default.cfm. 5. Percentages of missing data are in Table A2 in appendix. These rates are quite low (maximum 4% for Guinean sample), as are also refusal rates for each survey. We

Inequality of Opportunity for Income in Africa

123

checked there were no signiﬁcant differences in terms of outcome level and sons’ positions between the whole sample and the sample with information on fathers. 6. Details are available from the authors. The consumption variable includes all food and nonfood current expenditures, home-produced food consumption, and imputed rent for house owners. It excludes too infrequent expenditures such as durable goods and health, as well as net transfers. It is adjusted for infra-annual inﬂation. Ghana 1998 consumption is measured at 1988 prices and translated in international dollars using the same PPP exchange rate as Ghana 1987–1988. 7. It should be pointed out that international statistics base most of the estimates for Africa on per capita consumption, while most of the estimates for Latin America are based on per capita income. As income inequality is most often higher than consumption inequality, due to transient components and measurement errors, international comparisons tend to understate the inequality level of Africa in comparison to Latin America. 8. UNU/WIDER–UNDP, 2000. World Income Inequality Database Version 1.0: http://wider.unu.edu/wiid/wwwwiid.htm. 9. An analysis of the sensitivity of income inequality indexes to the magnitude of measurement errors, following the lines proposed by Chesher and Schluter (2002), does not lead to question the more equal income distribution of Ghana in 1987–1988 (see Cogneau et al., 2006). 10. In fact, our data shows no growth of average consumption per capita over the period. This feature is in line with national accounts data collected by World Bank (World Bank, 2006). This data combines a positive GDP per capita growth (coming from investment and exports) with a stabilization of consumption per capita.

ACKNOWLEDGMENTS This research received funding from the Agence Franc- aise de De´veloppement (AFD) Research Department. The authors thank Jean-David Naudet for his support and clever remarks. They also thank Thomas Bossuroy, Philippe De Vreyer, Charlotte Gue´nard, Victor Hiller, Phillippe Leite, Laure Pasquier-Doumer, and Constance Torelli for their contributions in the construction of datasets and their participation in the ﬁrst stage of this study. The usual disclaimer applies.

REFERENCES Benavot, A., & Riddle, P. (1988). The expansion of primary education, 1870–1940: Trends and issues. Sociology of Education, 61(3), 191–210. Bishop, Y., Fienberg, S., & Holland, P. (1975). Discrete multivariate analysis: Theory and practice. Cambridge, MA: MIT Press. Bjo¨rklund, A., & Ja¨ntti, M. (2001). Intergenerational income mobility in Sweden compared to the United States. American Economic Review, 87(5), 1009–1018.

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Blinder, A. S. (1973). Wage discrimination: Reduced form and structural estimates. Journal of Human Resources, 8(4), 436–455. Bourguignon, F., Ferreira, F. H. G., & Leite, P. (2002). Beyond Oaxaca-Blinder: Accounting for differences in household income distributions across countries. DELTA Working Paper no. 2002–2004; William Davidson Institute Working Paper no. 478; World Bank Working Paper no. 2828. Bourguignon, F., Ferreira, F. H. G., & Menendez, M. (2007). Inequality of opportunity in Brazil. Review of Income and Wealth, 53(4), 585–618. Chesher, A., & Schluter, C. (2002). Welfare measurement and measurement error. Review of Economic Studies, 69, 357–378. Cogneau, D. (2006). Equality of opportunity and other equity principles in the context of developing countries. In: G. Kochendorfer-Lucius & B. Pleskovic (Eds), Equity and development, InWent/World Bank Berlin workshop series (pp. 53–68). Washington, DC: World Bank Publications. Cogneau, D., Bossuroy, T., De Vreyer, P., Gue´nard, C., Hiller, V., Leite, P., Mesple´-Somps, S., Pasquier-Doumer, L., & Torelli, C. (2006). Inequalities and equity in Africa. Working paper DIAL DT 2006/11 and AFD Notes et Documents 31. Cogneau, D., & Gignoux, J. (2008). Earnings inequalities and educational mobility in Brazil over two decades. In: S. Klasen & F. Nowak-Lehrman (Eds), Poverty, inequality and policy in Latin America, CESifo Series (forthcoming). Harvard, MA: MIT Press. Deaton, A. (1997). The analysis of household surveys: A microeconometric approach to development policy. Baltimore, MD: Johns Hopkins University Press, World Bank, (forthcoming). Deininger, K., & Squire, L. (1996). A new data set measuring income inequality. The World Bank Economic Review, 10(3), 565–591. Di Nardo, J., Fortin, N., & Lemieux, T. (1996). Labor market institutions and the distribution of wages, 1973–92: A semiparametric approach. Econometrica, 64(5), 1001–1044. Dunn, C. E. (2007). The intergenerational transmission of lifetime earnings: Evidence from Brazil. The B.E. Journal of Economic Analysis & Policy, 7(2), (Contributions), Article 2. Gajdos, T., & Maurin, E. (2004). Unequal uncertainties and uncertain inequalities: An axiomatic approach. Journal of Economic Theory, 116(1), 93–118. Glewwe, P., & Twum-Baah, K. A. (1991). The distribution of welfare in Ghana 1987–99. LSMS Working Paper no. 75, 94 pp. Heston, A., Summers, R., & Aten, B. (2002). Penn World Table Version 6.1, Center for International Comparisons at the University of Pennsylvania (CICUP). Juhn, C., Murphy, K. M., & Pierce, B. (1993). Wage inequality and the rise in returns to skill. Journal of Political Economy, 101(3), 410–442. Lam, D. (1999). Generating extreme inequality: Schooling, earnings and intergenerational transmission of human capital in South Africa and Brazil. Research Report, Population Studies Center, University of Michigan. Leite, P., Mc Kinley, T., & Osorio, R. (2006). The post-apartheid evolution of earnings inequality in South-Africa, 1995–2004. UNDP International Poverty Centre, Working Paper no. 32. Louw, M., Van Der Berg, S., & Yu, D. (2006). Educational attainment and intergenerational social mobility in South Africa. Stellenbosch Economic Working Paper no. 09/06.

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Maddison, A. (2003). The world economy: Historical statistics. Development Center Studies. Paris: OECD. Moreno-Ternero, J. D. (2007). On the design of equal-opportunity policies. Investigaciones Econo´micas, 31, 351–374. Rodrı´ guez, J. G. (2008). Partial equality-of-opportunity orderings. Social Choice and Welfare, forthcoming. Oaxaca, R. (1973). Male-female wage differentials in urban labor markets. International Economic Review, 14(3), 673–709. Roemer, J. (1996). Theories of distributive justice. Cambridge, MA: Harvard University Press. Roemer, J. (1998). Equality of opportunity. Cambridge, MA: Harvard University Press. Schultz, T. P. (1999). Health and schooling investments in Africa. Journal of Economic Perspectives, 13(3), 67–88. Shepherd, A., & Gyimah-Boadi, E. (2005). Bridging the north-south divide in Ghana? Background Paper for the 2005 World Development Report, mimeo, World Bank, Washington, DC. Shorrocks, A. (1978). The measurement of mobility. Econometrica, 46(5), 1013–1024. Van de Gaer, D. (1993). Equality of opportunity and investment in human capital. Catholic University of Leuven, Faculty of Economics, no. 92. Van de Gaer, D., Schokkaert, E., & Martinez, M. (2001). Three meanings of intergenerational mobility. Economica, 68(272), 519–538. World Bank. (2005). World development report 2006: Equity and development. New York: Oxford University Press. World Bank. (2006). World development indicators CD-Rom. Washington, DC: World Bank.

APPENDIX. DECOMPOSITION OF INEQUALITY OF OPPORTUNITY INDEXES USING THE LOG-LINEAR MODEL The log-linear model provides a useful parameterization of a contingency table such as a country c social mobility table (Bishop, Fienberg, & Holland, 1975). ln½ pc ðs; oÞ ¼ mc þ ac ðoÞ þ bc ðsÞ þ gc ðs; oÞ

(A.1)

This parameterization is unique under the following constraints: X X X X ac ðoÞ ¼ 0; bc ðsÞ ¼ 0; gc ðs; oÞ ¼ 0; 8o; gc ðs; oÞ ¼ 0; 8s o

s

s

o

In this so-called saturated, i.e., unconstrained, form, the log-linear is purely descriptive. Its coefﬁcients merely provide an arithmetic decomposition of the table frequencies: mc is equal to the mean of log joint

DENIS COGNEAU AND SANDRINE MESPLE´-SOMPS

126

probabilities, ac(o) and bc(s) characterize, respectively, row and column marginal probabilities, and the gc(s, o) characterize the ORs. For instance, in the case where social origin o and social position s are dichotomic (o=0, 1 and s=0, 1), i.e., if the social mobility table is a 2 rows and 2 columns matrix, one obtains m¼

1 ½ln pc ð0; 0Þ þ ln pc ð0; 1Þ þ ln pc ð1; 0Þ þ ln pc ð1; 1Þ 4

ac ð1Þ ¼

1 ½ln pc ð1; 0Þ þ ln pc ð1; 1Þ m 2

bc ð1Þ ¼

1 ½ln pc ð0; 1Þ þ ln pc ð1; 1Þ m 2

gc ð1; 1Þ ¼

1 pc ð0; 0Þpc ð1; 1Þ ln 4 pc ð0; 1Þpc ð1; 0Þ

In the general case where social origin and social position are polytomic categorical variables, ORs read ln½ORc ðs; o; s0 ; o0 Þ ¼ gc ðs; oÞ þ gc ðs0 ; o0 Þ gc ðs0 ; oÞ þ gc ðs; o0 Þ (A.2) A ‘‘nonsaturated,’’ i.e., constrained, version of the log-linear model allows us to construct a ﬁctional mobility table where row and column marginal probabilities are those of country c and ORs are those of country cu ln½ pc!c0 ðs; oÞ ¼ mc!c0 þ ac!c0 ðoÞ þ bc!c0 ðsÞ þ gc0 ðs; oÞ

(A.3)

Under the assumption that frequency counts nc(s, o) follow a multinomial distribution, this model can be estimated by maximum likelihood. The saturated model for country cu provides the gcu(s, o) which characterize the country cu mobility matrix odds-ratios ORcu. The constrained log-linear model of Eq. (A.3) provides predicted probabilities of a ﬁctional mobility table with country c marginal distributions and country cu ORs, and conditional probabilities: pc!c0 ðsjoÞ ¼ pc!c0 ðs; oÞ=pc ðoÞ. This allows us to deﬁne another counterfactual conditional expectation of income for each social origin o: X E c ðYjo; sÞpc!c0 ðsjoÞ (A.4) M c!c0 ðYjoÞ ¼ s

We may last deﬁne a second counterfactual conditional expectation of income where conditional probabilities of country cu are used to weight the

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earnings structure of country c: M c!c0 ðYjoÞ ¼

X

E c ðYjo; sÞpc0 ðsjoÞ

(A.5)

s

Then the decomposition of Eq. (6) is completely deﬁned.

Table A1. Country

Ivory Coast

Ghana

Guinea

Madagascar Uganda

a

Surveys.

Name of the Survey

Enqueˆte permanente aupre`s des me´nages (EPAM)

Period

Household Sample Size for Analysis

February 1985–April 1989a

4,090

Coˆte d’Ivoire Living Standards Surveys (CILSS) Ghana Living Standards September Survey, rounds 1 and 4 1987–July (GLSS1 and GLSS4) 1988 April 1998– March 1999 Enqueˆte inte´grale sur les January conditions de vie des 1994– me´nages (avec modules February budget et consommation) 1995 (EIBC) Enqueˆte permanente aupre`s April 1993– des me´nages (EPM) April 1994 National Integrated March 1992– Household Survey (NHIS) March 1993

3,148

5,923

3,971

3,700 8,205

The four surveys approximately cover the whole period. In the ﬁrst three years, half of the sample has been interviewed again the following year (panel data). For panelized households, only the most recent information was kept, so that the ﬁnal stacked sample contains around 800 households for each year of the 1985–1987 period and 1,600 for 1988–1989.

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Table A2.

Sample Size and Missing Social Origins and Social Positions. Ivory Coast

Sample size Father’s position missing (%) Son’s position missing (%)

Ghana Ghana Guinea Madagascar Uganda 1988 1998

4,843 3.7

2,702 2.1

4,625 4.0

5,513 4.4

4,427 1.6

8,530 1.6

2.3

0.1

2.0

0.3

0.6

0.6

Coverage: Men 20–69 years old. Source: see Table 1.

EQUALIZING INCOME VERSUS EQUALIZING OPPORTUNITY: A COMPARISON OF THE UNITED STATES AND GERMANY Ingvild Alma˚s ABSTRACT Germany has lower posttax income inequality than the United States and hence is doing better according to a strict egalitarian fairness ideal. On the other hand, the United States is doing better than Germany according to a libertarian fairness ideal, which states that people should be held fully responsible for their income. However, most people hold intermediate (responsibility-sensitive) positions, and this paper studies fairness of the income distributions in Germany and the United States according to these positions. We ﬁnd that only if peoples’ preferences are characterized by substantial degree of individual responsibility, the United States is considered less unfair than Germany. If we hold people responsible for the unexplained variation, the United States is considered fairer than Germany for all levels of responsibility sensitiveness. If we, on the other hand, demand compensation for the unexplained variation, Germany is fairer than the United States for all levels of responsibility. The latter may be seen as the preferred approach as it follows a ‘‘beneﬁt of the doubt’’ Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 129–156 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16006-9

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130

strategy. To the best of our knowledge, this paper presents the ﬁrst crosscountry fairness comparison based on responsibility-sensitive ideals.

1. INTRODUCTION Posttax income inequality is greater in the United States than in most continental European countries. Based on this, the United States has been considered to have a more unfair income distribution than continental Europe. Drawing conclusions about fairness from inequality alone can only be justiﬁed by a strict egalitarian fairness ideal. From experiments and surveys, we learn that a majority of people consider fairness ideals other than the strict egalitarian, and it is therefore important to study other fairness ideals when evaluating income distributions (Cappelen, Hole, Sørensen, & Tungodden, 2007; Konow, 2003; World Value Survey, 2005). The majority of citizens in both Germany and the United States hold fairness ideals other than the strict egalitarian. However, we have reasons to believe that Germans are more favorable towards redistribution than the citizens of the United States (Alesina & Glaeser, 2001; World Value Survey, 2005; Alesina & Angeletos, 2005; Luttens & Valfort, mimeo). There are two potential reasons for this. First, citizens of the United States and Germany may have different beliefs about the income-generating process. Second, the citizens of the two countries may have different perceptions of fairness. The ﬁrst can be referred to as a difference in positive perceptions whereas the latter can be referred to as a difference in normative perceptions. The main contributions of this paper are twofold. First, the pretax income-generating process is estimated, and subsequently the question of whether the income-generating processes of the two countries differ, is analyzed. That is, we analyze whether the citizens of the two countries have reasons to hold different beliefs about the income-generating process. Second, on the basis of the estimated income-generating processes, this paper evaluates, using different redistributive fairness ideals, the posttax income distribution in the two countries. This evaluation could be done by using two different approaches, both of which we have seen examples of in the earlier literature. First, we may group people according to their responsibility variables, and consider within-group inequality as unfairness. This would be similar to the procedure followed in Devooght (2008) and Alma˚s et al. (2007). Second, we may deﬁne groups according to their nonresponsible characteristics, and consider unfairness to be the variation across these groups. This would be in line with the Roemer tradition (Roemer, 1998).

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The difference between the two approaches stems from the treatment of the unexplained variation. The former approach treats the unexplained variation as a nonresponsibility variable and hence gives us the upper bound of unfairness, whereas the latter approach treats the unexplained variation as a responsibility variable and hence gives us the lower bound of unfairness. In this paper, we follow both approaches, and we ﬁnd that the gap between the two is substantial and, furthermore, that the fairness ranking of the two countries depends on the treatment of the unexplained variation. For the lower bound of unfairness, the United States is considered fairer than Germany for all levels of responsibility, whereas for the upper bound of unfairness, Germany is considered fairer than the United States for all levels of responsibility. The upper bound of unfairness may be seen as the preferred approach if researchers and political decision makers refuse to hold people responsible for processes that we are unable to measure correctly. This approach can be referred to as the ‘‘beneﬁt of the doubt’’ approach. Given that the fairness preferences differ between the two countries, there is the important question of which fairness preferences should form the basis for a comparison between Germany and the United States. We may want to allow for different country-speciﬁc degrees of responsibility to form this basis or we may want to use the same degree of responsibility in both countries. The main approach in this paper is to use one and the same norm in the evaluation of both countries. It is shown, however, that the results are robust to the change of responsibility sensitivity in one country, as long as the unexplained variation is treated in the same way in both countries. In order to evaluate the degree of unfairness of an income distribution, we need to identify the fair income distribution. Our main focus is on fair incomes calculated using the generalized proportionality principle (Bossert, 1995; Konow, 1996; Cappelen & Tungodden, 2007). According to this principle, each individual should be given the proportion of the total income in the society that is dependent on his or her responsibility variables, but independent of his or her nonresponsibility variables.1 In the ranking of the different distributions, a generalized version of the Gini index and the Lorenz framework is used (Alma˚s et al., 2007).2 The outline of the paper is as follows. Section 2 gives some background information on preferences for redistribution. Section 3 gives a description of the framework. Section 4 presents the estimation results. Section 5 evaluates the posttax income distribution in the two countries. Section 6 shows the results of applying principles other than the proportional to identify fair incomes. Section 7 concludes.

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2. PREFERENCES FOR REDISTRIBUTION

80 60 40 20 0

Cumulative population percent

100

The political climate is different in the United States and Europe, and we have some evidence that Europeans tend to be less responsibility sensitive than Americans (World Value Survey, 2005). Figs. 1–3 are drawn on the basis of the World Value Survey and illustrate that preferences or beliefs differ between Germany and the United States. Fig. 1 shows the cumulative percentage of the population in Germany and the United States, respectively, on a subjective scale from 1 to 10. Placing oneself close to the value 1 indicates that one believes that people should be more responsible for themselves, whereas the value 10 indicates that one believes that the government should ensure that everyone is provided for. As the cumulative distribution of the United States is steeper, the citizens of the

0

2

4 6 Degree of responsibility Germany

8

10

USA

Fig. 1. Public Versus Private Responsibility. The Figure Gives the Cumulative Population Proportion on a Scale from 1 to 10. The Respondents are Asked the Following Question. ‘‘How Would you Place Your Views on This Scale? 1 Means you Agree Completely with the Statement on the Left [First Statement]; 10 Means you Agree Completely with the Statement on the Right [Second Statement]; and if Your Views Fall Somewhere in Between, you can Choose any Number in Between.’’ Statements: ‘‘People Should Take More Responsibility to Provide for Themselves vs. The Government Should Take More Responsibility to Ensure that Everyone is Provided for’’. The data are Taken from the World Value Survey (2005). Survey Year 1999.

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United States tend to be more in favor of individual responsibility than the Germans. Hence we would expect them, on average, to hold fairness ideals closer to the libertarian ideal of no redistribution. However, we see that it is not the case that all citizens in the United States are purely libertarian – the median respondent chooses a value of 4. The value 4 indicates that the government should be partly responsible for individuals’ outcome, and hence it departs from the libertarian ideal, which states that the less redistribution or the less governmental interference, the better. Fig. 1 indicates that the median voter in the United States would advocate using a slightly more responsibility-sensitive measure than the median voter in Germany would. Fig. 2 shows that very few citizens, in both Germany and the United States, would ﬁnd it unfair that a secretary who was more reliable, more efﬁcient, and quicker than another also earned more, despite the fact that the two secretaries were doing practically the same job. Hence, we can Germany

USA

Not fair

Fair

Graphs by Country

Fig. 2. Fairness. The Respondents are Asked the Following Question. ‘‘Imagine Two Secretaries, of the Same Age, Doing Practically the Same Job. One Finds out that the Other Earns Considerably More than she Does. The Better-Paid Secretary, However, is Quicker, More Efﬁcient, and More Reliable at her Job. In Your Opinion, is it Fair or not Fair that one Secretary is Paid More Than the Other?’’ 12.3 percent in Germany ﬁnd it Unfair, whereas 8.8 percent in the United States ﬁnd it Unfair. The Data are Taken from the World Value Survey (2005). Survey Year 1999.

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conclude that almost all citizens, in both Germany and the United States, would like to hold people responsible for at least some of the differences described; reliability, efﬁciency, and how fast people work. In other words, Fig. 2 indicates that very few citizens, in both Germany and the United States, are strictly egalitarian. There are more German than American citizens who ﬁnd the unequal earnings of the two secretaries unfair, however, and this indicates that there are more strictly egalitarian citizens in Germany than in the United States. Fig. 3 illustrates that perceptions of why people are in need are different in the two countries. We see that in the United States, more people tend to think that people are in need because of laziness and lack of willpower, whereas in Germany, more people tend to think that people are in need because of an unfair society. Hence, the citizens of the United States believe, to a larger extent than German citizens, that people in need are responsible Germany

Laziness and lack of willpower

USA

Unfair society

Graphs by Country

Fig. 3. Laziness and Lack of Willpower Versus Unfair Society. The Respondents are Asked the Following Question. ‘‘Why, in Your Opinion, are there People in this Country who live in Need? Here are Two Opinions: Which Comes Closest to Your View? 1. Poor Because of Laziness and Lack of Willpower. 2. Poor Because of an Unfair Society.’’ 61.2 percent in the United States Says that it is Because of Laziness or Lack of Willpower, whereas 38.8 percent in Germany State this Opinion. The Data are Taken from the World Value Survey (2005). Survey Year 1995.

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for their own situation. This revealed difference between the two countries is positive rather than normative, however, as the question relates to the reasons why people are in need and not to the judgment of what they should and should not be responsible for. Subsequently, we have reasons to believe that the citizens in the United States would tend to hold people more responsible because they have different beliefs about the income-generating process than the German citizens do. To sum up, Figs. 1–3 illustrate that the citizens of Germany tend to be less responsibility sensitive than those of the United States. Fig. 1 does not indicate whether the difference is because of different beliefs or different preferences for redistribution. Fig. 2 indicates that the difference in responsibility sensitivity is because of different preferences for redistribution, whereas Fig. 3 indicates that the difference is because of different beliefs about the income-generating process. Hence, we have reasons to believe that the difference in responsibility sensitivity between the United States and Germany is a result of both different beliefs and different preferences for redistribution. In this paper, we seek to identify the pretax income-generating process, which would allow us to discuss whether the citizens of the two countries have reasons to hold different beliefs about the income-generating process. Based on the estimated income-generating processes, we provide a normative evaluation of the posttax income in the two societies.

3. FRAMEWORK The framework applied follows a three step procedure. First, the pretax income-generating process is estimated based on individual characteristics such as hours worked per week, education, age, gender, and immigration status. Second, based on the estimated pretax income-generating process, fair incomes are calculated by using the generalized proportionality principle. Third, the difference between the fair income and the posttax income is evaluated using a difference-based approach. The three steps are explained below in descending order.

3.1. Step 3: Evaluating the Posttax Income Distribution In the standard approach to inequality measurement, each individual has a fair income equal to the mean income in the society, and the fair income is

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therefore the same for all individuals. Hence, the fair income is not individual speciﬁc and we can restrict ourselves to focusing on the posttax income distributions. When considering responsibility-sensitive fairness ideals, however, the fair income is individual speciﬁc. Hence, the overall fairness in the societies can only be identiﬁed by considering the joint distribution of individual-speciﬁc fair and posttax income. The joint distribution of fair and posttax income can be expressed one-dimensionally, by focusing on the distribution of individual-speciﬁc differences between posttax and fair income. Alma˚s et al. (2007) propose a generalized Gini index and difference-based Lorenz curves, which is anonymous in the difference between posttax income and fair income. We brieﬂy present these concepts below. 3.1.1. The Difference-Based Lorenz Curve In the classical inequality framework, the Lorenz curve relies on a ranking of the population from the person with the lowest income to the person with the highest income. Acknowledging that the fair income in the standard framework is the mean income, the ordering underlying the classical Lorenz curve is identical to the ordering of differences, i.e., ordering of deviations from the fair income: y1 m y 2 m y N m

(1)

where yk is the posttax income of individual k and N the total number of individuals in the society. m is the mean income in the society, which is also the fair income in the classical approach. The ordering of the deviation from the mean has a direct analogy in a fairness framework. When allowing for the possibility that the fair income for individual i (zi), is different from the mean income (m), (1) has to be reformulated in the following way: y 1 z 1 y 2 z 2 yN z N

(2)

Denoting the difference d, the ordering can be expressed as follows: d1 d2 dN

(3)

As in the classical approach, the differences are negative for low ranks and positive for high ranks. Considering a given society, the

Equalizing Income versus Equalizing Opportunity

difference-based Lorenz curve is given by: Pk di lðd; k; NÞ ¼ i¼1 ; k ¼ 1; 2; . . . ; N mN

137

(4)

where d is the vector of differences in this given society. For an empty responsibility set, i.e. a strict egalitarian norm with no responsibility sensitivity, the difference-based Lorenz curve is identical to the one given by the differences in (1). On the other hand, for the libertarian view of no responsibility and if there is no redistribution in the society, the difference-based Lorenz curve takes the value zero for any k, and hence the difference-based Lorenz curve intersects the horizontal axis. Imposing scale invariance and anonymity on the differences, a generalized Pigou–Dalton criterion, and a weak condition of unfairism, the differencebased Lorenz curves have properties exactly analogous to those of the classical Lorenz curve (see Alma˚s et al., 2007) for a detailed discussion).3 Hence, if the difference-based Lorenz dominance criterion is satisﬁed, we know that any unfairness measure that satisﬁes the generalized Pigou– Dalton criterion gives the same unfairness ranking for the two distributions. On the basis of the difference-based Lorenz curve, it is straightforward to generalize the classical and extensively used Gini:4 XX 1 Gu ¼ jd i d j j (5) 2NðN 1Þm i j If considering an empty responsibility set, i.e., no responsibility sensitivity, the unfairness measure will collapse to the classical Gini index. If, on the other hand, considering the libertarian ideal that holds people fully responsible for their income, the generalized Gini index will take the value zero if there is no redistribution in the society. In the empirical analysis, we study the difference-based Lorenz curve and the generalized Gini index according to different fairness ideals.

3.2. Step 2: Identifying Fair Posttax Income Individuals may differ both with respect to variables for which we want to hold them responsible and with respect to variables for which we do not want to hold them responsible. However, individual fair income should only depend on the individual nonresponsibility variables. A fair redistribution should eliminate inequalities because of nonresponsibility variables, unfair

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inequality, but preserve inequalities because of responsibility variables, fair inequality. Therefore, a fair posttax income distribution must satisfy at least two requirements. First, any two individuals with the same responsibility characteristics should have the same fair income. Second, there should be no redistribution for equal nonresponsibility characteristics. These two requirements may be seen as the core elements of a responsibility-sensitive framework (Arneson, 1989; Cohen, 1989; Roemer, 1996, 1998).5 There exist different formulations of the two above-mentioned criteria in the literature. Strong versions of these criteria can be referred to as ‘‘equal income for equal responsibility variables’’ (EIER) and ‘‘equal transfer for equal nonresponsibility variables’’ (ETEN) formulations. EIER states that two people with the same responsibility characteristics should get the same income. ETEN states that two people with the same nonresponsibility characteristics should get the same transfers.6 EIER thus describes how we should compensate for nonresponsibility factors, whereas ETEN suggests that we should treat individuals with equal nonresponsibility factors identically. These two criteria can generally not be met jointly. However, if deﬁning an alternative formulation of the latter, the two can be met jointly. Cappelen and Tungodden (2007) suggest the following formulation of the latter. If we deﬁne groups as people with the same responsibility characteristics and all these groups have the same proﬁle of nonresponsibility characteristics, then we should have no redistribution between groups. This criterion and EIER are met jointly if following an old idea of proportionality and applying the generalized proportionality principle (Bossert, 1995; Konow, 2003; Cappelen & Tungodden, 2007).7 The main idea captured in this principle is that the proportion of total posttax income that the individual should get is dependent on individual responsibility characteristics but independent of individual nonresponsibility characteristics. The generalized proportionality principle is the focus of the empirical analysis of this paper. Consider again a given society with individual i’s responsibility characteristics given by vector ei, and the nonresponsibility characteristics given by the vector ti, respectively. According to the generalized proportionality principle, individual i’s fair income, zi, is given by: gðei Þ z i ¼ PN Y j gðej Þ

(6)

where Y is the sum of posttax income in society, subscript i and j indicate that the variable belongs to individual i and j, respectively.

Equalizing Income versus Equalizing Opportunity

g(ei) is given by: gðei Þ ¼

1X f ðei ; tj Þ N j

139

(7)

where f captures the income-generating process. f(ei, ti) is the pretax income of an individual with responsibility characteristics ei and nonresponsibility characteristics ti. f(ei, tj) is the virtual pretax income of a person with the responsibility characteristics of person i and the nonresponsibility characteristics of person j. There are well-known alternatives to the proportionality principle in the literature, which also have desired properties. Two of these are applied in the robustness analysis of this paper, namely the egalitarian equivalent principle and the conditional egalitarian principle (Bossert, 1995; Kolm, 1996; Bossert & Fleurbaey, 1996). The egalitarian equivalent principle ensures that EIER is met jointly with a different formulation of the latter requirement. The criterion for the egalitarian equivalent mechanism to be jointly met with EIER is that if all individuals have the same nonresponsibility characteristics, then there should be no redistribution between groups. This is clearly a weaker criterion than the one formulated in Cappelen and Tungodden (2007). This formulation as well as the formulation by Cappelen and Tungodden (2007) ensures that we do not interfere with differences because of responsibility characteristics. The fair income is, according to the egalitarian equivalent principle, independent of individual nonresponsibility characteristics but dependent on individual responsibility characteristics. The conditional egalitarian principle, on the other hand, ensures that ETEN is satisﬁed. ETEN is the strongest requirement and implies both the alternative criterion underlying the generalized proportionality principle and the alternative criterion underlying the egalitarian equivalent principle. However, the conditional egalitarian principle does not ensure that EIER is met. It ensures, however, that a weaker formulation related to EIER is satisﬁed: It ensures that for all people with responsibility characteristics equal to a reference responsibility characteristics, get the same income. We describe the egalitarian equivalent and the conditional egalitarian principles in more detail in Section 6. (See also Fleurbaey, 2008, for a detailed discussion of how these two core requirements can be formulated and met.) 3.3. Step 1: Estimating the Pretax Income-Generating Process In order to calculate the fair income deﬁned in Eq. (6), we need to estimate the pretax income-generating process. The income equation is assumed to

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have the standard log-linear form: ln qi ¼ a þ bei þ gti þ i

(8)

where qi is the pretax income of individual i. The dependent variable in Eq. (8) is pretax income, whereas the explanatory variables included are: age in years, the square of age, hours worked per week, gender, immigration status, and educational dummies. The choice of explanatory variables is made on the grounds of availability, and all variables that inﬂuence income, for which we have data, are included. The educational categories differ between the two countries, and in order to estimate the effect of education as precisely as possible, the countries’ own educational categories are used.8 The educational system of the United States is represented by 14 educational categories, whereas the German educational system is represented by 11 categories. The pretax income is only applied in order to identify the fair income for each individual. In order to get consistent estimates, we focus on single households, and hence individual characteristics and household characteristics are identical. Both pretax and posttax data are taken from the Luxembourg Income Study database wave ﬁve (Luxembourg Income Study, 2000). In order to make the data nationally representative, household weights are used throughout the analysis. The documentation for all country surveys can be found on the web page of the Luxembourg Income Study (Luxembourg Income Study, 2007). Table 1 gives the descriptive statistics of the variables used in the empirical analysis of the paper. Using Eqs. (6) and (8), the fair income for individual k when considering the upper bound of unfairness, is expressed as follows: PN expðbek þ gti þ i Þ expðbek Þ U Y ¼ PN Y (9) zk ¼ P iP N N j expðbej Þ j i expðbej þ gti þ i whereas the fair income for the lower bound of unfairness is expressed as follows: PN expðbek þ gti þ k Þ expðbek þ k Þ L Y ¼ PN Y (10) zk ¼ P iP N N j expðbej þ j Þ j i expðbej þ gt i þ j Five different degrees of responsibility, i.e. ﬁve different responsibility cuts, are considered. A crucial point in the philosophical as well as the political debate is where to draw the responsibility cut. In the philosophical

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Table 1.

Germany Pretax income Posttax income Age Hours Education (%) Less high school Grad high school Postsecondary University degree Female (%) Foreign (%) Observations USA Pretax income Posttax income Age Hours Education (%) Less high school Grad high school Postsecondary University Degree Female (%) Foreign (%) Observations

Descriptive Statistics.

Mean Value

Std. Dev

Minimal Value

Maximum Value

54987.01 44720.3 40.94 39.00

38749.48 25,274 12.16 13.31

207.01 2723.532 20 .5

435,898 244642.3 80 90

43712.87 31825.88 12.86 11.06

24 241 20 1

391,994 347,304 90 111

1.61 61.98 13.01 23.41 37.18 5.62 2367 38932.88 34922.51 42.53 41.49 4.03 6.87 58.13 30.98 47.24 13.72 10,427

Note: The table shows the descriptive statistics of the variables used in the empirical analysis.

literature, a prominent answer has been that individuals should be held responsible for factors under their control, but not for factors beyond their control (Cohen, 1989). In the political debate however, we need to be more pragmatic, and we will not always be able to separate completely the factors under control and beyond control. In this paper, we take a pragmatic view and leave aside the question of how a particular responsibility cut should be justiﬁed. We rather analyze the implications of various responsibility cuts and leave it to the reader, the voters, and the political decision makers to determine which is more appealing. Table 2 gives an overview of the responsibility variables for each of the ﬁve responsibility cuts. First, we study the empty responsibility set characterizing the classical inequality measures. Second, we consider the

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Table 2. Responsibility Factors Strict egalitarian RS1 upper RS2 upper RS3 upper Libertarian RS1 lower RS2 lower RS3 lower

Hours

X X X X X X X

Different Responsibility Cuts.

Education

Age

Female

Immigrant

Error Term

X X X

X X

X

X

X X

X

X X X X

Note: The table shows the different responsibility cuts underlying the different fairness measures.

case where people are only held responsible for hours worked, and then we extend this to hours worked and education, education being represented by the countries’ own educational categories. Furthermore, the two age components, hours worked, and the educational dummies are contained in the responsibility set. The last responsibility cut is the libertarian cut, where all income-generating variables are considered to be responsibility variables and the fair distribution is simply the pretax income distribution.

4. ESTIMATING THE INCOME-GENERATING PROCESS The estimation results are given in Table 3. We see that the citizens of the two countries have reasons to have different beliefs. First, age has a larger positive effect in Germany than in the United States. Furthermore, the income age relationship is more curved in Germany as the coefﬁcient of the square of age is larger in Germany than in the United States. Second, perhaps surprisingly, it pays more to work harder in Germany than in the United States. This contradicts the common perception that it pays more to work harder in the United States than in continental European countries, i.e. the story in Alesina and Angeletos (2005). Third, education has a more substantial effect in the United States where we see a clear positive effect of education although not signiﬁcant for all levels. Fourth, the negative effect of being a foreigner is larger in Germany

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Table 3. Dep var: Gross Income Age Age sq Hours Secondary, 1st stage Secondary, 2nd stage Academy/fachoberschule Techn. col. (fachhochschule) University Foreign university Technical school gdr University gdr Other diploma No diploma 7th or 8th grade 9th grade 10th grade 11th grade 12th grade, no diploma High school graduate Some college, no degree Vocational program Academic program Bachelor’s degree Master’s degree Professional school degree Doctorate degree Foreign Female Constant No. of obs. Adj. R-sq

Regression Results.

Germany

St. Dev.

USA

St. Dev.

0.143 0.0015 0.0349 0.1183 0.1498 0.1729 0.4276 0.2858 0.0031 0.0282 0.2729 0.2021 0.3308

(0.0151) (0.0002) (0.0027) (0.0492) (0.0915) (0.1025) (0.0814) (0.1058) (0.1515) (0.1541) (0.1604) (0.4385) (0.2375)

0.0796 0.0008 0.0234

(0.0047) (0.00005) (0.0012)

0.0340 0.2038 0.0333 0.0997 0.1979 0.4089 0.5806 0.6844 0.6720 0.9561 1.0980 1.0886 1.1619 0.0306 0.3033 7.020 10,427 0.33

(0.1039) (0.0960) (0.0831) (0.0781) (0.1294) (0.0617) (0.0629) (0.0670) (0.0709) (0.0625) (0.0670) (0.1095) (0.0933) (0.0268) (0.0174) (0.1195)

0.1291 0.1741 6.132 2,367 0.43

(0.2890) (0.0451) (0.2874)

Note: The table shows the regression results for the two countries in the study (Robust least squares estimation). The logarithm of pretax household income is the dependent variable. Standard errors are given in parentheses. Indicates that the coefﬁcient is signiﬁcant at 10% level. Indicates that the coefﬁcient is signiﬁcant at 5% level. Indicates that the coefﬁcient is signiﬁcant at a 1% level.

than in the United States, whereas the negative effect of being a female is larger in the United States than in Germany. Interestingly, the unexplained variation is larger in the United States than in Germany. Again, we have reasons to believe that this contradicts the

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Alesina and Angeletos story that luck determines income to a larger extent in continental Europe whereas working hard determines income to a larger extent in the United States. The adjusted R-squared is 0.43 for Germany and 0.33 for the United States.

5. MEASURING UNFAIRNESS

0

.05

.1

.15

.2

.25

.3

.35

.4

Fig. 4 illustrates how measured unfairness changes according to the generalized Gini index for the two countries when changing the responsibility cut. Because of the larger fraction of unexplained variation, the distance between the upper and the lower bound of the United States is larger than

Inequality

RS1

RS2

RS3

Libertarian

Responsibility cut Germany upper bound Germany lower bound

USA upper bound USA lower bound

Fig. 4. Unfairness and Inequality. The Figure Shows Measured Unfairness for Four Different Responsibility Cuts as Well as the Classical Inequality Measure. Inequality Refers to the Classical Gini Index, RS1, Refers to the Generalized Gini Index Considering Hours Worked to be a Responsibility Variable, RS2 Refers to the Generalized Gini Index Considering Hours Worked and Education to be Responsibility Variables, and RS3 Refers to the Generalized Gini Index Considering Hours Worked, Education, and Age to be Responsibility Variables. Libertarian Corresponds to the Generalized Gini Index Considering Individuals Responsible for all Income Differences.

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that of Germany. According to the upper bound of unfairness, Germany is considered to be less unfair than the United States for all the intermediate responsibility-sensitive measures as well as for the classical inequality measure. The ranking of the two countries only changes when going from the most responsibility-sensitive measure to the libertarian measure of full responsibility, however. Hence, when regarding the unexplained variation as a nonresponsibility variable, the United States is a more unfair society unless one endorses the libertarian fairness ideal. According to the lower bound of unfairness, however, the United States is considered to be less unfair than Germany for all the intermediate responsibility-sensitive measures, i.e., when regarding the unexplained variation as a responsibility variable, the United States is a less unfair society unless one endorses the strict egalitarian fairness ideal. Hence, the fairness ranking of the two countries is very sensitive to the treatment of the unexplained variation. The results for each of the ﬁve responsibility cuts and its robustness is discussed below.

5.1. Classical Inequality Not surprisingly, Germany does better than the United States according to the classical inequality measure of no responsibility, i.e., the classical Gini index. It is relevant to discuss whether this ﬁnding is robust; would other inequality measures reveal the same ﬁnding? Fig. 5 shows that the distribution of Germany Lorenz dominates that of the United States, and therefore we have the robust conclusion that Germany is less unfair than the United States according to the strict egalitarian fairness ideal.9

5.2. Hours Worked as the Only Responsibility Variable Many people would not ﬁnd it unfair that a person who worked more than another also earned more. Hence, it is reasonable to consider hours worked per week as a responsibility variable. When measuring unfairness according to a responsibility-sensitive fairness ideal where hours worked is the only responsibility variable, it is not straightforward to rank Germany and the United States for either the upper or the lower bound of unfairness. For the upper bound, the generalized Gini index reveals that Germany is less unfair than the United States. However, as the Lorenz curves cross (see Fig. 6), there exists another unfairness measure satisfying the generalized

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−.15

−.1

−.05

0

Difference based Lorenz curves Classical

−.25

−.2

Germany

USA 0

20

40

60

80

100

Percentile of population

Fig. 5.

Difference-Based Lorenz Curves when the Responsibility Set is Empty.

Pigou–Dalton criterion that would evaluate the United States as less unfair than Germany. More speciﬁcally, this unfairness measure gives more weight to the most unfairly treated than the generalized Gini index does, as the Lorenz curve of the United States is closer to the horizontal axis for the lower tail of the difference distribution. For the lower bound of unfairness, the generalized Gini index for the United States (0.269) is very similar to that of Germany (0.266). Neither in this case do we have a robust ranking of the two countries.

5.3. Education and Hours Worked as Responsibility Variables Whether individuals should be accountable for their education is likely to be a debated issue in many countries. Successful education usually demands both effort and talent.10 Adding education to the responsibility set gives slightly lower measured unfairness for the upper bound, in both Germany and the United States. As can be seen in Fig. 7, for all practical purposes, we have the robust conclusion that the income distribution of Germany is less

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−.15

−.1

−.05

0

Difference based Lorenz curves Hours worked resp var

Germany lower bound

−.2

USA lower bound

−.25

Germany upper bound USA upper bound 0

20

40

60

80

100

Percentile of population

Fig. 6.

Difference-Based Lorenz Curves when Hours Worked is Considered to be a Responsibility Variable.

unfair than that of the United States, for the upper bound. However, the Lorenz curves cross slightly at the lower tail, and if we constructed an unfairness measure that gives absolute priority to the most unfairly treated, we would reach the opposite conclusion. The decrease in unfairness is more substantial for the lower bound of unfairness. Furthermore, for the lower bound, we have a clear crossing of the two Lorenz curves, and thus we have no robust ranking of the two countries according to the lower bound of unfairness.

5.4. Age as a Responsibility Variable Age is one of the few variables in life that individuals cannot affect. It might therefore be surprising to suggest that people should be held responsible for their age. However, if people are asked whether they would like society to accept differences in income because of age, many would answer that they would. If life expectancy is the same for all citizens, income differences

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−.1

−.05

0

Difference based Lorenz curves Hours and education resp var

−.15

USA lower bound Germany lower bound

−.2

Germany upper bound

−.25

USA upper bound

0

20

40

60

80

100

Percentile of population

Fig. 7.

Difference-Based Lorenz Curves when Hours Worked and Education are Considered to be Responsibility Variables.

because of age do not affect permanent income, and this may be one reason why people ﬁnd inequality because of differences in age justiﬁable. The earliest literature proposing to eliminate inequality because of certain factors before evaluating income distributions, focused on eliminating differences because of age.11 We see that although the country ranking stays the same, the measured unfairness decreases slightly for the upper bound of unfairness in both countries when age is included as a responsibility variable. Again, we have a slight crossing of the two Lorenz curves. However, the area between the curves where the United States is closer to the horizontal axis is small, and only if we gave full priority to the most unfairly treated we would get the conclusion that the United States is fairer than Germany. However, we cannot conclude as robustly as for the classical inequality measurement (see Fig. 8). For the lower bound, the decrease in measured unfairness of including age as a responsibility variable is more substantial. For this bound, we have a clear crossing of the Lorenz curves, and hence, we have no robust ranking of the two countries.

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−.1

−.05

0

Difference based Lorenz curves Hours age and education resp var

USA lower bound

−.2

−.15

Germany lower bound

Germany upper bound

−.25

USA lower bound

0

20

40

60

80

100

Percentile of population

Fig. 8.

Difference-Based Lorenz Curves when Hours Worked, Education, and Age are Considered to be Responsibility Variables.

5.5. The Libertarian Responsibility Cut For the upper bound, the ranking of the two countries is reversed when considering the libertarian fairness ideal: the United States is less unfair than Germany. However, as Fig. 9 shows, the Lorenz curves cross when applying the libertarian ideal of no redistribution, and hence we have a mixed picture: There exist other fairness measures that would conclude that Germany is less unfair than the United States. As such, the conclusion that the United States is fairer than Europe for the libertarian fairness ideal is not entirely clear. It is interesting to note that the shape of the Lorenz curve for the United States is very different from the Lorenz curves for the European country for this responsibility cut. We see that the unfairness in the United States relates to the lower tail of the difference distribution. That is, the most unfairly treated are treated worse in the United States than in Germany. However, the major part of the population, in the middle of the fairness ranking given in Eq. (3), seems not to be unfairly treated in the United States as the Lorenz curve is ﬂat in the major part characterizing the middle of the distribution. In Germany, however,

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−.1

−.05

0

Difference based Lorenz curves Libertarian norm

USA

−.15

Germany 0

Fig. 9.

20

40 60 Percentile of population

80

100

Difference-Based Lorenz Curves when all Variables in Addition to the Error Term are Considered to be Responsibility Variables.

the citizens in the center of the fairness ranking in Eq. (3) are treated unfairly, and we see that the Lorenz curve for Germany has no ﬂat part. The measured unfairness is lower in both countries if applying the libertarian fairness ideal than for all other fairness ideals. The measured unfairness is substantially lower for the United States than for Germany, however. That is, the posttax income distribution is closer to the pretax income distribution in the United States than in Germany according to the generalized Gini index.

6. THE EGALITARIAN EQUIVALENT AND CONDITIONAL EGALITARIAN FAIR INCOME: A ROBUSTNESS ANALYSIS We now turn to two fairness principles other than the generalized proportionality principle, namely the egalitarian equivalent and the conditional egalitarian principle. The fair income calculated from the egalitarian equivalent (Bossert, 1995) principle is given by: 1X ¼ f ðei ; t~Þ ð f ðej ; t~Þ f ðej ; tj ÞÞ (11) zEE i N j

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where t~ is the reference nonresponsibility vector. The reference vector we consider is the vector of average nonresponsibility variables in society, t:12 The empirical counterpart of this is the fair income: 1X ¼ expðbei þ gtÞ ðexpðbej þ gtÞ expðbej þ gtj þ j ÞÞ zEE i N j where t is the vector of mean nonresponsibility variables. The conditional egalitarian (Kolm, 1996) fair income of individual i is given by: 1X zCE ¼ f ðe ; t Þ f ð~ e ; t Þ þ f ð~e; tj Þ (12) i i i i N j where e~ is the reference responsibility vector. We consider the reference to be the vector of average responsibility variables in society, e .13 The empirical counterpart of this is the fair income given by: expðbeÞ X zCE ¼ expðgtj þ i Þ þ expðgti þ i Þ½expðbei Þ expðbeÞ i N j Table 4 presents generalized Gini indexes according to the different ideals applying the generalized proportional, the egalitarian equivalent, and the conditional egalitarian approach, respectively. When considering the upper bound of unfairness, Germany is less unfair than the United States for all levels of responsibility except for the libertarian case of full responsibility. When considering the lower bound of unfairness, the United States is less unfair than Germany for all levels of responsibility, except the egalitarian case of zero responsibility. Hence, the results are robust to the choice of fairness principle.

7. CONCLUDING REMARKS The paper compares the income distributions of Germany and the United States using a ﬂexible group of fairness measures that allows us to calculate unfairness for different fairness ideals. The fairness ideals that are discussed are the libertarian ideal, the strict egalitarian ideal, and different responsibility-sensitive intermediate positions. The main ﬁndings of this paper are twofold. First, for the extreme case of no responsibility, Germany has lower unfairness than the United States, whereas in the extreme case of

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Table 4. Resp. set Empty RS1 gp upper RS2 gp upper RS3 gp upper All RS1 gp lower RS2 gp lower RS3 gp lower RS1 ee upper RS2 ee upper RS3 ee upper RS1 ee lower RS2 ee lower RS3 ee lower RS1 ce upper RS2 ce upper RS3 ce upper RS1 ce lower RS2 ce lower RS3 ce lower

Measured Unfairness for Different Responsibility Cut. Germany

USA

0.285 0.316 0.310 0.306 0.215 0.269 0.248 0.221 0.296 0.289 0.284 0.259 0.240 0.218 0.283 0.279 0.287 0.228 0.224 0.216

0.368 0.375 0.367 0.366 0.185 0.266 0.211 0.195 0.366 0.350 0.347 0.258 0.205 0.191 0.341 0.304 0.301 0.221 0.195 0.189

Note: gp indicates that the generalized proportionality principle has been used in the calculation. ee and ce indicate that the egalitarian equivalent and the conditional egalitarian principles have been used. All unfairness measures are calculated using the generalized Gini index given in Eq. (5).

full responsibility, the United States has lower fairness than Germany. However, in the latter case, the conclusion that Germany is less unfair than the United States when considering a libertarian fairness ideal, depends on the choice of the Gini index. Another unfairness measure giving a larger weight to the most unfairly treated, would conclude that Germany is fairer than the United States also in this case. Second, for the intermediate positions of responsibility sensitivity, the picture is mixed. When applying the upper bound of unfairness, Germany is measured to be fairer than the United States for all levels of responsibility sensitivity. However, when applying the lower bound of unfairness, the United States is measured to be fairer than Germany for all levels of responsibility sensitivity. Hence, the fairness ranking depends on the treatment of the unexplained variation. As responsibility sensitivity seems to get more and more attention from researchers, however, the hope is that more descriptives on cross-country comparable income-generating processes

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will be gathered in the near future as ‘‘description can be motivated by predictive interest or by prescriptive interest’’ (Sen, 1980). When more descriptives are in place, we will be able to narrow the bounds and hence make more accurate measurements of the unfairness in different societies. As has been advocated by Devooght (2008) and Alma˚s et al. (2007), it is plausible that researchers as well as political decision makers do not want to hold people responsible for variation they cannot explain. Hence, the preferred strategy might be to apply the upper bound of unfairness – a strategy that can be referred to as the ‘‘beneﬁt of the doubt’’ strategy. However, it is possible to argue against the ‘beneﬁt of the doubt’ strategy and consider the lower bound of unfairness to be the preferred strategy. Because we have no measure of luck and only an imperfect measure of abilities (education), we have reasons to believe that both luck and abilities are captured in the unexplained variation. For Germany, it may be reasonable to include both these variables in the nonresponsibility set. However, for the United States, this may be less obvious. When the unexplained variation in the responsibility is included when evaluating fairness in the United States, but not in Germany, the fairness ranking of the two changes compared with the above-mentioned ‘‘beneﬁt of the doubt’’ strategy. The United States is then fairer than Germany for all levels of responsibility. More generally, we might want to depart from the common norm for both countries and include more factors in the responsibility set for the United States than for Germany, as the citizens of the United States seem to be more responsibility sensitive (cf. the evidence from the World Value Survey, 2005). However, unless we include the unexplained variation in the responsibility set, applying a larger responsibility set in the United States than in Germany does not change the results. The robustness of the ﬁndings is discussed through Lorenz curves and through applying different fairness principles, i.e. the egalitarian equivalent and the conditional egalitarian principles. As we have seen the Lorenz curves cross for some of the responsibility cuts, and hence we do not have robust ranking of the two countries for these cuts. However, all results are robust to the choice of fairness principle. For further checks of robustness one way to go could be to introduce formal inference tests (see for example Cowell, 1999; Davies & Paarsch, 1998; Bishop, Formby, & Thistle, 1991; Bishop, Chow, & Zheng, 1995 and Bishop, Formby, & Zheng, 1998, for discussions of inference tests and Donald & Barrett, 2004 for a proposal of nonparametric tests). It is open for future research to apply these formal tests of inference to the generalization of the Lorenz curves and the Gini index applied in this paper.

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NOTES 1. In the robustness analysis, two other extensively discussed principles are applied: the egalitarian equivalent and the conditional egalitarian principle (Bossert, 1995; Kolm, 1996; Bossert & Fleurbaey, 1996). 2. The framework has some parallels to earlier work on formalizing criteria to rank distributions according to norm (fair) incomes (Garvy, 1952; Paglin, 1975; Jenkins & O’Higgins, 1989; Devooght, 2008; Bishop, Formby, & Smith, 1997). 3. The generalized Pigou–Dalton criterion states that if you have two distributions, A and B with identical fair incomes, and all differences in A and B are identical except for the differences for individual i and j, which can be expressed by the B B A following: d A i od i d j od j , then A is more unfair than B. The condition of unfairism states that if A and B have the same mean and all individual differences are identical, then A and B are equally fair. 4. Index it follows that the generalized Gini has a maximum value of 2. A related measure was discussed in Wertz (1979). 5. Note that if there are no responsibility variables, this approach is consistent with the classical framework based on a strict egalitarian fair distribution. On the other hand, if there are no nonresponsibility variables, the framework will be consistent with the libertarian fairness ideal giving no weight to inequality concerns. 6. See Bossert (1995) and Fleurbaey (1994, 1995a, 1995b, 1995c). 7. The concept of proportionality has a long tradition and can be traced back to Aristotle (Barnes, 1984), who proposed to distribute in proportion to individual effort. 8. We have also estimated the relationship by using four comparable educational categories in the two countries. This speciﬁcation gives less explanatory power, but the results of the fairness comparison do not change. 9. The conclusion is robust in the sense that all classical inequality measures that satisfy the uncontroversial generalized Pigou–Dalton criterion will reveal a higher inequality in the United States than in Germany (see Alma˚s et al., 2007 for a discussion of this). 10. The question on equalizing the opportunities for education in the United States is discussed extensively in Betts and Roemer (2007). 11. See Paglin (1975) and the many subsequent comments in the American Economic Review in the late 1970s and 1980s. 12. If all variables are considered to be nonresponsibility variables, tincludes average hours worked per week, average level of education (operationalized through including average dummies, i.e. between zero and one for all educational levels), average age, average sex, i.e. a dummy equal to 0.5 for females, and average immigration status, with a dummy variable equal to the average of the immigration dummy. 13. The average responsibility variables are calculated in the same way as t.

ACKNOWLEDGMENT The author thanks Alexander Cappelen, Marc Fleurbaey, Jo Thori Lind, Kalle Moene, John Roemer, Bertil Tungodden, and participants at the ECINEQ conference in Berlin 2007. The usual disclaimer applies.

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REFERENCES Alesina, A., & Angeletos, G.-M. (2005). Fairness and redistribution. American Economic Review, 95(4), 960–980. Alesina, A., & Glaeser, E.-l. (2001). Why doesn’t the United States have a European-style welfare state? Brookings Papers on Economic Activity, 2, 187–254. Alma˚s, I., Cappelen, A. W., Lind, J. T., Sø rensen, E. Ø., & Tungodden, B. (2007). Measuring unfair inequality. Evidence from Norwegian data. NHH, Department of Economics working paper, no. 32. Arneson, R. J. (1989). Equality and equal opportunity for welfare. Philosophical Studies, 56, 159–194. Barnes, J. (1984). Nicomachean ethics. In: The complete works of Aristotle. Princeton, NJ: Princeton University Press. Betts, J., & Roemer, J. (2007). Equalizing opportunity for racial and socioeconomic groups in the United States through educational ﬁnance reform. In: P. Peterson (Ed.), Schools and the equal opportunity problem. Cambridge: The MIT Press. Bishop, A., Chow, K. V., & Zheng, B. (1995). Statistical inference and decomposable poverty indices. Bulletin of Economic Research, 47, 329–340. Bishop, A., Formby, J. P., & Zheng, B. (1998). Inference tests for Gini-based tax progressivity indices. Journal of Business and Economic Statistics, 16, 322–330. Bishop, J. A., Formby, J. P., & Smith, W. J. (1997). Demographic change and income inequality in the United States, 1976–1989. Southern Economic Journal, 64, 34–44. Bishop, J. A., Formby, J. P., & Thistle, P. D. (1991). Rank dominance and international comparisons of income distributions. European Economic Review, 35(7), 1399–1409. Bossert, W. (1995). Redistribution mechanisms based on individual characteristics. Mathematical Social Sciences, 29, 1–17. Bossert, W., & Fleurbaey, M. (1996). Redistribution and compensation. Social Choice and Welfare, 13, 343–355. Cappelen, A. W., Hole, A. D., Sørensen, E.Ø., & Tungodden, B. (2007). The pluralism of fairness ideals: An experimental approach. American Economic Review, 97, 818–827. Cappelen, A. W., & Tungodden, B. (2007). Fairness and the proportionality principle. NHH, Department of Economics working paper, no. 31. Cohen, G. A. (1989). On the currency of egalitarian justice. Ethics, 99, 906–944. Cowell, F. A. (1999). Estimation of inequality indices. In: J. Silber (Ed.), Handbook of income inequality measurement. With a foreword by Amartya Sen, Recent Economic Thought (pp. 269–286). New York: Springer. Davies, D. G., & Paarsch, H. (1998). Economic statistics and social welfare comparisons: A review. In: A. Ullah & D. E. A. Giles (Eds), Handbook of applied economic statistics. Statistics: Textbooks and Monographs, Vol. 155, pp. 1–38. Devooght, K. (2008). To each the same and to each his own: A proposal to measure responsibility-sensitive income inequality. Economica, 75(298), 280–295. Donald, S. G., & Barrett, G. F. (2004). Consistent nonparametric tests for Lorenz Dominance. Econometric Society 2004 Australasian Meetings 321, Econometric Society, revised. Fleurbaey, M. (1994). On fair compensation. Theory and Decision, 36, 277–307. Fleurbaey, M. (1995a). The requisites of equal opportunity. In: W. A. Barnett (Ed.), Social choice, welfare, ethics (pp. 37–53). Cambridge: Cambridge University Press.

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Fleurbaey, M. (1995b). Three solutions for the compensation problem. Journal of Economic Theory, 65, 505–521. Fleurbaey, M. (1995c). Equality and responsibility. European Economic Review, 39, 683–689. Fleurbaey, M. (2008). Fairness, responsibility and welfare. Oxford: Oxford University Press (forthcoming). Garvy, G. (1952). Inequality of income: Causes and measurement. In Eight papers on size distribution of income, Studies in Income and Wealth, Volume 15. Jenkins, S. P., & O’Higgins, M. (1989). Inequality measurement using ‘norm incomes’: Were Garvy and Paglin onto something after all? Review of Income and Wealth, 35, 283–296. Kolm, S. (1996). Modern theories of justice. Cambridge: MIT Press. Konow, J. (1996). A positive theory of economic fairness. Journal of Economic Behavior and Organization, 46, 137–164. Konow, J. (2003). Which is the fairest one of all? A positive analysis of justice theories. Journal of Economic Literature, 41, 1188–1239. Luttens, R., & Valfort, M. A. (mimeo). Voting for redistribution under desert-sensitive altruism. Luxembourg Income Study. (2007). http://www.lisproject.org/. Date: 10/25/07. Luxembourg Income Study (LIS) Micro database. (2000). Harmonization of original surveys conducted by the Luxembourg income study, Asbl. Luxembourg, periodic updating, Luxembourg Income Study. Paglin, M. (1975). The measurement and trend of inequality: A basic revision. American Economic Review, 65, 598–609. Roemer, J. (1996). Theories of distributive justice. Cambridge: Harvard University Press. Roemer, J. (1998). Equality of opportunity. Cambridge: Harvard University Press. Sen, A. (1980). Description as choice. Oxford Economic Papers, New Series, 32(3), 353–369. Wertz, K. L. (1979). The measurement of inequality: Comment. American Economic Review, 69(4), 670–672. World value survey. (2005). http://www.worldvaluessurvey.org/. Date: 10/18/07.

INTERGENERATIONAL INCOME INEQUALITY AND DYNASTIC POVERTY PERSISTENCE: GERMANY AND THE UNITED STATES COMPARED Veronika V. Eberharter ABSTRACT Purpose: Using GSOEP-PSID the study analyzes the effects of redistribution policy on intergenerational income inequality, poverty intensity, intergenerational income mobility, and dynastic poverty persistence in Germany and the United States. Methodology: To evaluate the extent and the intensity of dynastic inequality and poverty the paper employs inequality measures and poverty indices. The contribution of a set of human capital and labor market variables on intergenerational income mobility and the risk of dynastic poverty persistence is analyzed with linear and nonlinear regression approaches and a binomial logit model. Findings: The empirical results partly corroborate that countries with a forced redistribution scheme succeed in reducing income inequality and poverty intensity, but at the expense of intergenerational income Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 157–175 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16007-0

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persistence and the relative risk of dynastic poverty persistence. In Germany, redistribution policy reduces income inequality and poverty intensity to a greater extent than in the United States, and the equalizing effect of public transfers increases with age. In the United States intergenerational income persistence and the relative risk of dynastic poverty persistence are more pronounced than in Germany. The contribution of gender, educational attainment, and labor market engagement to the intergenerational income mobility and the relative risk of dynastic poverty persistence is country speciﬁc and differ by age group. Research implications: The results call for further research of the interaction of family-life, labor market settings, and social policy in determining the degree of intergenerational income mobility and dynastic poverty persistence.

INTRODUCTION The reduction of income inequality and poverty is an important objective of social policy in most industrialized countries. Public transfers and taxes are the traditional tools used to reduce income inequality and poverty. How taxes and transfers are employed varies among countries in response to differing macroeconomic indicators, institutional settings, family role patterns, and the permeability of the social system. In particular, these factors lead to wellknown differences in tax and transfer policies between the United States and the European countries. Cross-section empirical evidence show lower poverty rates in countries with high-level or carefully targeted public transfers (Canada, Northern Europe) or heavily insured countries (Sweden, Belgium, and Germany). Compared to these countries the United States’ socialinsurance and redistributive tax system is weak, and contributes little to poverty reduction (Smeeding, Rainwater, & Burtless, 2001). From the point of the social and economic well-being the intergenerational effects of redistribution policy are of crucial interest. Previous analysis of the intergenerational effects of public transfers and welfare use on individual behavior and the economic outcome brought out ambiguous results: a number of studies showed that public transfers prevent persons from developing their resources or to take advantage of existing opportunities, and thus human capital such as education and work experience is not valued, and there is little motivation to pursue full-time employment (Gottschalk, 1992; An, Haveman, & Wolfe, 1993;

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Bane & Ellwood, 1994; Mayer, 1997; Pepper, 2000; Page, 2004). The analysis of Vartanian (1997, 1999) as well as Vartanian and MacNamara (2000, 2004) revealed that parental welfare use leads to an attenuation of labor market opportunities and encourages the welfare use of the children and thus perpetuates intergenerational poverty cycles. On the other hand, studies based on US data showed that public transfers and redistributive taxes succeed to narrow the intergenerational income gap, and to improve the living standard of the poor, so that the incomes of the children converge to the mean more quickly (Ellwood & Summers, 1986; Corak & Heisz, 1999; Corak, 2006). Eberharter (2008) found structural differences in the intergenerational occupational mobility between Germany and the United States, and only weak support for a higher social mobility in the United States. The purpose of this paper is to compare the impact of redistribution policy on intergenerational income inequality and poverty, as well as on intergenerational income mobility and the relative risk of dynastic poverty persistence in Germany and the United States. Given the different welfare policy regimes in both the countries the paper provides a test whether countries with a force redistribution scheme succeed or fail to break the intergenerational transmission of poverty. The paper addresses the following questions: To what extent does redistribution policy lower total income inequality and poverty in different phases of the life course? To what extent does redistribution policy determine intergenerational income mobility and the risk of intergenerational poverty persistence? To what extent do individual characteristics determine intergenerational income mobility and the risk of intergenerational poverty persistence? In Germany – a country with a forced redistribution policy – public transfers and redistributive taxes should reduce income inequality and poverty intensity to a greater extent than in the United States. In the United States – a country with a weaker welfare safety net and more liberal social policy concept – individual characteristics should determine intergenerational income mobility and dynastic poverty persistence to a greater extent than in Germany. Based on nationally representative data from the German SocioEconomic Panel (GSOEP) and the Panel Study of Income Dynamics (PSID), we analyze the effects of redistribution policy on the pre- and postgovernment income of parent–child pairs in different time windows to address to nonlinearities in the intergenerational income inequality and

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mobility setting (Hyson, 2003; Hertz, 2005). To evaluate the effects of redistribution policy on income inequality and poverty we use inequality measures sensitive to changes either at the bottom or at the top of the income distribution and poverty indexes presented in Foster, Greer, and Thorbecke (1984). We decompose the inequality measure to quantify the contribution of public transfers to total income inequality. To analyze intergenerational income mobility and its determinants we employ linear and nonlinear regression approaches on permanent income variables. Finally, we use a binomial logit model to quantify the determinants of the relative risk of dynastic poverty persistence. The paper proceeds as follows: Database and Sample Organization section reports the database. Methodology section characterizes the inequality and poverty measures, as well as the speciﬁcations of the econometric approaches. Empirical Results section presents the empirical results, and Conclusions concludes with a discussion of the implications of these ﬁndings and the directions for further research.

DATABASE AND SAMPLE ORGANIZATION The empirical analysis is based on data from the GSOEP and the PSID, which were made available by the Cross-National-Equivalent-File (CNEF) project at the College of Human Ecology at Cornell University, Ithaca, NY. The PSID began in 1980 and contains a nationally representative unbalanced panel of about 40,000 individuals in the United States. From 1997 onwards, the PSID data are available biyearly. The GSOEP began in 1984 and contains with a representative sample of about 29,000 German individuals and includes households in the former East Germany since 1990. Both surveys track socioeconomic variables of a household; each household member is asked detailed questions about age, gender, marital status, educational level, labor market participation, working hours, employment status, occupational position, and income situation.1 The income variables are measured on an annual basis and refer to the prior calendar year. For the underlying analysis we use the income variables referring to the wave of the interview, but questioned in the following wave. We analyze the economic and social situation of children living in the parental household and as adults in their own households. The data do not provide a sufﬁcient long time horizon to observe the parents and the children at identical life cycle situations, but cover a sufﬁciently long period to observe the socioeconomic characteristics of the parental household and

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to link these data with the children’s socioeconomic characteristics when becoming members of other family units. Thus, the data can be used to draw inferences about the effects of being exposed to redistribution policy measures in the parental household on the income situation as young adults. The data do not allow identifying parents–child relations exactly. The underlying analysis considers adults, whose marital status is ‘‘married,’’ or ‘‘living with partner’’ and who are living in households with persons with the status ‘‘child’’ as parents–child pairs. The sample includes persons aged 10–18 years, co-resident with their parents in 1980 (United States) or 1984 (Germany). We do not consider children older than 18 years to avoid the overrepresentation of children staying at home until a late age (Kolodinsky & Shirey, 2000). The children are at least 24 years old when we analyze their income situation in the years 1999–2005 (Germany) or 1994–2003 (USA). The selection process leads to a sample of 1,430 individuals in Germany and of 2,569 individuals in the United States. Due to sample selection design we observe West-German persons only. We follow the standard conventions and assume that income is shared within families and thus household income is arguably a better measure of the economic and social status than individual income variables. The study is based on the equivalent pre- and post-government household income. The pre-government household income includes labor earnings (wages, salaries, nonwage compensations, income from on the job training, self-employed income, or bonus), and the cash income from private sources (property, pensions, alimony, or child support) of all household members. The postgovernment household income equals the pre-government household income plus household public transfers (social beneﬁts: dwellings, child or family allowances, unemployment compensation, assistance, and other welfare beneﬁts), plus household security pensions (age, disability, widowhood), deducting household total family taxes (mandatory social security contributions, income taxes, or mandatory employee contributions). To consider the family structure we adopt the OECD-equivalence scale to arrive at the equivalent pre-government household income and the equivalent post-government household income. The household income variables are deﬂated with the national CPI (2001 ¼ 100) to reﬂect constant prices. To exclude the impact of transitory shocks and cross-section measurement errors we use 4 year averages of the equivalent household income variables. The equivalent household income variables 1984–1987 (Germany) or 1980–1983 (USA) capture the income situation of the persons when living in the parental household. The economic and social situation of

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these persons living in their own households is observed in the periods 1999–2002, 2000–2003, 2001–2004, and 2002–2005 (Germany), or 1996–1999, 1997–2001, and 1999–2003 (USA).

METHODOLOGY The change of the overall level of inequality and poverty between market income and disposable income is an important dimension of the well-being of the population in a country and focuses on the functioning of the redistribution policy. To evaluate the extent of income inequality we employ two inequality measures, differing concerning their sensitivity to changes at the bottom or the top of the income distribution. The (generalized) Gini coefﬁcient measures n-times the surface between the Lorenz curve, which maps the cumulative income share on the vertical axis against the distribution of the population on the horizontal axis, and the line of equal distribution. The easiest mathematical expression of the Gini coefﬁcient is based on the covariance between the income of an individual (yi) and the rank (F) the individual occupies in the income distribution. The rank takes a value between 0 for the poorest and 1 for the richest. Denoting y as the mean income, the Gini coefﬁcient then is deﬁned The parameter n is used to emphasize various as GiniðnÞ ¼ ncovðy; F n1 Þ=y. parts of the income distribution, the higher the weight, the more emphasis is placed on the bottom part of the income distribution. The parameter n ¼ 2 characterizes the standard Gini coefﬁcient Gini ¼

2covðY; FÞ y

(1)

which is more sensitive to changes at the bottom of the income distribution and ranges from 0 (perfect equality) to 1 (total inequality). The Generalized Entropy inequality measure GEð2Þ ¼

n 1 X 2 ðyi yÞ 2 2ny i¼1

(2)

with n the number of individuals, y the arithmetic mean of the individual incomes yi, belongs to the inequality measures of the Generalized Entropy P a 1. These inequality measures class GEðaÞ ¼ 1=ða2 aÞ½1=n ni¼1 ðyi =yÞ are sensitive to changes at the lower end of the income distribution for a close to zero, equally sensitive to changes across the distribution for a equal

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to one (Theil index), and sensitive to changes at the upper end of the distribution for higher values of a. We use the GE(2) inequality measure which is sensitive to changes at the higher end of the income distribution. The higher the value of GE(2), the higher the degree of inequality. We decompose the GE(2) inequality measure to evaluate the contribution of labor income and public transfers to the total income inequality. The total inequality, P I, sums up Sk absolute contributions of the (k) income sources I ¼ k S k , which can be written as S k ¼ sk GEð2Þ ¼ rk wk

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ GEð2ÞGEð2Þk

(3)

The sk=Sk/I indicates the proportional contribution of income component k to total inequality, rk is the correlation between source k and total the share of source k in total income, and GE(2) and income, wk ¼ yk =y, GE(2)k, the inequality measures of total income and the kth income source, respectively. A large value of Sk suggests that income source k is an important source of income inequality. If Sk>0, the kth income source provides a disequalizing effect, if Sko0 the income source k provides an equalizing effect. To quantify the poverty intensity and the extent of inequality below the poverty line we employ the distance measures used in poverty analysis (Foster et al., 1984). The poverty threshold is deﬁned as half the median income variable (z). The headcount ratio (FGT(0)) indicates the amount of poor persons in the total population. The average normalized poverty gap (FGT(1)) expresses the relative average distance between the poverty line (z) and the income of individual i (yi) FGTð1Þ ¼

zd z

(4)

P with d ¼ 1=n ni¼1 z yi the absolute value of the arithmetic mean of the distance between the poverty line (z) and the individual income (yi), and n the number of the individuals. Table 1 presents the empirical speciﬁcation of the models of the intergenerational income elasticity and the relative risk of intergenerational poverty persistence. The most common approach to quantify how economic (dis)advantages are transmitted across generations is to estimate the intergenerational income elasticity applying ordinary least squares (OLS) to the regression of a logarithmic measure of the children’s income variable (yc) on a logarithmic measure of the income variable of the parental

164

Table 1.

Empirical Speciﬁcation of the Models to Quantify Intergenerational Income Elasticity and the Relative Risk of Dynastic Poverty Persistence.

Model

Dependent Variables

Intergenerational income elasticity:

Explanatory Variables

yc

yc ¼ b 0 þ b 1 yp þ c

yc ¼ b 0 þ b 1 yp þ

n P

yp

Xc

bc X c þ c

c¼2

GEN EDU

EMP

Pn Probðpov ¼ 1Þ ¼ eB0 þ c¼2 Bc X c Probðpov ¼ 0Þ

pov Xc

GEN EDU

EMP

Logarithmic measure of the permanent income variables (4-year averages), measured in 1999 and 2002 (Germany) or 1994 and 1997 (USA) Logarithmic measure of the permanent income variables (4-year averages) of the parental household, measured in 1984 (Germany) or 1980 (USA) Gender of the individual: 1, male; 0, female Educational attainment of the individual, measured in school years in 1999 and 2002 (Germany) or 1994 and 1997 (USA) Annual working hours of the individual, measured in 1999 and 2002 (Germany) or 1994 and 1997 (USA) Intergenerational poverty transition: 1, poverty persistence; 0, transition out of poverty Gender of the individual: 1, male; 0, female Educational attainment of the individual, measured in school years in 1999 and 2002 (Germany) or 1994 and 1997 (USA) Annual working hours of the individual, measured in 1999 and 2002 (Germany) or 1994 and 1997 (USA)

VERONIKA V. EBERHARTER

Relative risk of dynastic poverty persistence:

Description

Intergenerational Income Inequality and Dynastic Poverty Persistence

165

household (yp) yc ¼ b0 þ b1 yp þ c y c ¼ b 0 þ b1 y p þ

n X

bc X c þ c

(5a)

(5b)

c¼2

The constant term b0 represents the change in the economic status common to the children’s generation. The slope coefﬁcient, b1, is the elasticity of the children’s income variable with respect to the parents’ income situation. The larger b1 the more likely an individual as an adult will inhabit the same income position as her parents, which implies a greater persistence of the intergenerational economic status. The closer b1 is to zero the higher is the intergenerational income mobility. The random error component ec is usually assumed to be distributed N(0, s2). The inclusion of a set of control variables (Xc) in (5b) allows to account for the individual characteristics of the children which partly express the indirect effects of the parental income on the children’s income. To the extent that these variables lower the coefﬁcient b1 compared to (5a) these other effects ‘‘account for’’ the raw intergenerational income elasticity. We control for human capital and include the years of education (EDU). In the case of missing values the educational attainment is set equal to the amount reported in the next year, for it is possible to increase human capital by education but impossible to decrease it. The higher the income of the parents the higher their investment in the education of the children, which in turn causes a higher income of the children. The labor market engagement (EMP) is considered with the annual working hours of the persons. Finally, the gender dummy (GEN 1 male, 0 female) controls for gender differences in the effects of redistribution policies on intergenerational income elasticity. The control variables (Xc) are observed in 1999 or 2002 for the German sample, and 1994 or 1997 for the US sample. To evaluate the extent to which gender, educational attainment, and labor market presence determine the probability of intergenerational poverty persistence we employ a binomial logit model. Intergenerational poverty persists if a person is positioned in the ﬁrst or in the second income quintiles given that her parents are positioned in the ﬁrst or second income quintiles. The dependent variable (pov) indicates the intergenerational poverty transition of the equivalent household income variables and take the value 1 if the person experienced an intergenerational poverty persistence, and take the value 0 if a person experienced an intergenerational transition out of poverty. The probability of intergenerational poverty persistence is

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estimated to be Probðpov ¼ 1Þ ¼

eZ 1 þ eZ

(6)

P The Z characterizes the linear combination Z ¼ B0 þ nc¼2 Bc X c with Xc, the independent variables and Bc, the regression coefﬁcients. In general, if the probability is greater than 0.5, we predict dynastic poverty, and if the probability is less than 0.5, we predict an intergenerational transition out of poverty. The interpretation of the regression coefﬁcients Bc is based on the odds, that is, the ratio of the probability of intergenerational poverty persistence and the probability of an intergenerational transition out of poverty Pn Probðpov ¼ 1Þ ¼ eB0 þ c¼2 Bc X c (7) Probðpov ¼ 0Þ The exp(Bc) are the factors by which the odds change when the cth independent variable increases by one unit, e.g., this value expresses the relative risk of intergenerational poverty persistence with a one-unit change in the cth independent variable. For the underlying analysis, the Xc include the years of education (EDU) used to control for human capital. The labor market engagement (EMP) and the gender dummy (GEN) are used as controls. The (Xc) control variables are observed in 1999 or 2002 for the German sample, and 1994 or 1997 for the US sample.

EMPIRICAL RESULTS For both Germany and the United States, the Gini coefﬁcient and the Generalized Entropy inequality measure (GE(2)) indicate that the tax and beneﬁt system reduces the income inequality. In Germany, the Gini coefﬁcient indicates an increasing pre-government income inequality between 0.30 and 0.37 for different age groups. Redistribution signiﬁcantly lowers the income inequality for all age groups by about 10 percent points. In the United States, the Gini coefﬁcient showed a higher pre-government income inequality than in Germany, which does not signiﬁcantly differ by age group. The tax and beneﬁt system reduces the income inequality by about 7 percent points. The GE(2) income inequality measure, which is sensitive to changes at the higher end of the distribution, reveals a signiﬁcantly higher inequality of both the permanent income variables in the

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Intergenerational Income Inequality and Dynastic Poverty Persistence

United States. In both the countries, the GE(2) inequality measure showed a higher pre-government income inequality for persons aged 10–18 years and living in the parental household than for other age groups. In Germany, the redistribution policy bisects the income inequality, in the United States taxes and transfers reduce the extent of income inequality by about one-third, which corroborates the working of a more progressive redistribution scheme in Germany (Fig. 1). country Germany

USA

0.40000

0.20000

%

0.10000

0.00000 0.40000 pre-government income

0.30000

0.20000

0.10000

income source

post-government income

0.30000

0.00000 102526272818years 33years 34years 35years 36years

102526272818years 33years 34years 35years 36years

age Gini Coefficient

GE(2)

Fig. 1. Pre-Government Income and Post-Government Income Inequality for Different Age Groups in Germany and the USA. Source: PSID-GSOEP 1980–2006 and own calculations.

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VERONIKA V. EBERHARTER

For both the countries, the decomposition of the Generalized Entropy (GE(2)) inequality measure corroborates the results of Fields (2004) that labor income is the most important source of income inequality in industrialized countries. In Germany, labor income accounts for between 84 percent (persons living with their parents) and 90 percent (persons aged 28–36 years) of total income inequality. In the United States, labor income accounts for 64 and 80 percent of total income inequality. The equalizing effect of public transfers on total income inequality differs by age group and country, and reveals the different redistribution schemes in both the countries: in Germany, public transfers equalize total income inequality by 1 percent for persons living in the parental household up to 4 percent for persons aged 28–36 years. In contrast, in the United States the equalizing effect of public transfers is highest for persons living with their parents and then decreases in the life course (Fig. 2). For the German sample, the pre-government income poverty rate (FGT(0)) of persons living in the parental household amounts about 10 percent, and then increases up to 23 percent for persons aged between 28 and 36 years. In Germany, redistribution policy downsizes the poverty rate to between 3 percent (persons living with their parents) and 12 percent (persons aged 28–36 years). In the United States, redistribution policy reduces the poverty rate onto a lower extent by about 6 percent points. In Germany, the average normalized poverty gap (FGT(1)) based on the pregovernment income widens with increasing age indicating a growing dispersion in the lower tail of the income distribution. The results corroborate empirical ﬁndings (Jantti & Danziger, 1994; Smeeding, 2005) that labor income disparities signiﬁcantly determine the extent of poverty. In both the countries, redistribution policy reduces the dispersion of the incomes below the poverty line to below 5 percent (Fig. 3). The results of the OLS regression indicate a higher intergenerational income persistence before redistribution in the United States than in Germany. Redistribution policy slightly increases the intergenerational income persistence from 0.478 to 0.492 for persons aged 25–33 years and from 0.444 to 0.489 for persons aged 28–36 years. In Germany, we ﬁnd a higher income mobility before redistribution than in the United States, redistribution policy lowers the intergenerational income mobility from 0.140 to 0.237 (persons aged 25–33 years) and from 0.160 to 0.407 (persons aged 28–36 years). Controlling for the individual characteristics the regression results reveal that gender, educational attainment, and labor market engagement signiﬁcantly contribute to the intergenerational income mobility by more than 10 percent points for the US sample. In Germany,

Intergenerational Income Inequality and Dynastic Poverty Persistence

169

country Germany

USA

102526272818years 33years 34years 35years 36years

102526272818years 33years 34years 35years 36years

0.3

percent

0.2

0.1

0.0

age total inequality GE(2)

contribution labor income

contribution public transfers

Fig. 2. Contribution of Labor Income and Public Transfers to Total Income Inequality (GE(2)). Source: PSID-GSOEP 1980–2006 and own calculations.

only labor market presence signiﬁcantly determines the intergenerational income mobility. Possible explanations may be that the public transfer system and the various welfare-state programs in Germany (Federal Childcare Payment, Parental Leave Act (Bundeserziehungsgeldgesetz, BErzGG, 2001)) ‘‘overrule’’ the inﬂuence of individual characteristics on intergenerational income mobility. Another explanation could be that labor

170

VERONIKA V. EBERHARTER country Germany

USA

102526272818years 33years 34years 35years 36years

102526272818years 33years 34years 35years 36years

0.25000

0.15000 0.10000

percent

0.05000 0.00000 0.25000

pre-government income

0.20000 0.15000 0.10000 0.05000

income source

post-government income

0.20000

0.00000 age FGT(0)

FGT(1)

Fig. 3. Poverty Rate (FGT(0)) and Average Normalized Poverty Gap (FGT(1)) for Different Age Cohorts. Source: PSID-GSOEP 1980–2006 and own calculations.

market segregation and wage discrimination partly diminishes the importance of gender and educational attainment (Table 2). Table 3 presents the results of the binomial logit model – the dependent variables, the estimated regression coefﬁcients (Bc) for each of the explanatory variables Xc, the estimated standard errors (SEBc) in parentheses, and the z-value and the signiﬁcance level. In Germany, gender, educational attainment, and the labor market presence signiﬁcantly determine the probability of intergenerational poverty persistence of both

Table 2. Model Speciﬁcation

Estimated Independent Coefﬁcients Variables

yc ¼ b0 þ b1 yp þ c

b0 b1

Yp_pre

b1

Yp_post

b0 y c ¼ b0 þ b 1 y p þ

n P

bc X c þ c

b1

Yp_post

b2

X2

b3 b4

X3 X4

c¼2

Intergenerational Income Elasticity.

Dependent Variables

Constant Pre-government income, parents Post-government income, parents R2-adj RMSE LL Constant Post-government income, parents Gender: 1, male; 0, female Years of education Employment hours R2-adj RMSE LL

Germany

USA

Pregovernment income yc_1999

Pregovernment income yc_2002

Postgovernment income yc_1999

Postgovernment income yc_2002

Pregovernment income yc_1994

Pregovernment income yc_1997

Postgovernment income yc_1994

Postgovernment income yc_1997

8.050 0.140

7.910 0.160

7.000

5.550

5.060 0.478

5.540 0.444

4.820

4.940

0.237

0.407

0.492

0.489

0.0277 0.495 216 6.730

0.0997 0.417 147 5.470

0.244 0.480 725 4.630

0.238 0.471 401 4.890

0.274

0.377

0.385

0.361

0.100

0.0833

0.114

0.114

0.0178 0.232 0.0546 0.488 211

0.00735 0.251 0.165 0.402 136

0.0759 0.177 0.351 0.445 642

0.071 0.324 0.379 0.425 338

0.0159 0.886 391

0.0154 0.984 379

0.229 0.814 1265

0.242 0.735 916

Note: Signiﬁcance levels in a two-tailed test po0.05; po0.01; po0.001. Source: GSOEP-PSID 1980–2006 and own calculations.

172

Table 3.

Relative Risk of Intergenerational Poverty Persistence.

Explanatory Variables

z

P>|z|

Bc (SEBc)

z

P>|z|

3.24 6.60

0.001 0.000

1.2228 (0.3251) 0.14093 (0.02333)

3.76 6.04

0.000 0.000

Employment hours N LR w2 (3) ProbWw2 LL Pseudo R2

1.1844 (0.3660) 0.1890 (0.02863) 0.0004 (0.0002) 609 627.09 0.000 108.579 0.7428

1.95

0.052

0.0004 (0.0002) 609 557.65 0.000 143.304 0.6605

2.43

0.015

Gender: 1, male; 0, female Years of education Employment hours N LR w2 (3) ProbWw2 LL Pseudo R2

0.6275 (0.2443) 0.1737 (0.0151) 0.0004 (0.0001) 1,785 1,831.95 0.000 321.293 0.7403

2.57 11.49 3.08

0.010 0.000 0.002

0.9442 (0.2702) 0.2031 (0.0169) 0.0001 (0.0001) 1,785 1,888.07 0.000 293.231 0.7630

3.49 12.00 0.97

0.000 0.000 0.330

Source: PSID-GSOEP 1980–2006 and own calculations.

VERONIKA V. EBERHARTER

(b) USA X2 X3 X4

povPost-government income

income

Bc (SEBc) Dependent Variables (a) Germany Gender: 1, male; 0, female X2 Years of education X3 X4

povPre-government

Intergenerational Income Inequality and Dynastic Poverty Persistence

173

the permanent income variables. Each additional year of education and each additional working hour signiﬁcantly reduce the risk to experience intergenerational poverty persistence. In general, women have a lower relative risk of dynastic poverty persistence than men. Probably the family background conditions in their own families are more inﬂuential than that of their parents. In the United States, the results point out that labor market attainment does not signiﬁcantly inﬂuence the relative risk of intergenerational poverty persistence.

CONCLUSIONS The analysis of the effects of redistribution policy on income inequality, poverty intensity, intergenerational income mobility, and the relative risk of intergenerational poverty persistence in Germany and the United States partly corroborates that countries with a forced redistribution policy succeed in attenuating income inequality and poverty intensity, but on the other hand scale down the intergenerational income mobility and elevate the risk of dynastic poverty persistence. In Germany, redistribution policy reduces income inequality, and poverty intensity to a greater extent than in the United States. In Germany, the equalizing effect of transfers on total inequality increases with age, in the United States it decreases with age. In both the countries, redistribution policy reduces the dispersion of the incomes below the poverty line to a comparable degree. In the United States, the intergenerational income persistence before redistribution is higher than in Germany, redistribution only slightly elevates the intergenerational income persistence. In both the countries, redistribution policy increases the intergenerational income persistence. Gender, educational attainment, and labor market engagement signiﬁcantly lower the intergenerational income mobility in the United States, whereas in Germany only labor market presence signiﬁcantly contributes to the intergenerational income mobility. In Germany, gender, educational attainment, and labor market engagement signiﬁcantly determine the relative risk of intergenerational poverty persistence. In the United States, labor market engagement does not signiﬁcantly reduce the relative risk of dynastic poverty persistence. Given the signiﬁcant inﬂuence of educational attainment on intergenerational income mobility and the relative risk of dynastic poverty persistence, suggest the potential of education to be a means to advance the social ladder

174

VERONIKA V. EBERHARTER

regardless of the social policy regime. However, we have to bear in mind that both the countries not only differ concerning their redistribution and social policy regime but also concerning labor market institutions, and family role models. Therefore, the results call for broader thinking on the mechanisms and causes of income inequality and poverty, the structural features of the labor markets and its institutions, and how families, labor markets and social policy interact in determining the degree of intergenerational income mobility and dynastic poverty persistence.

NOTE 1. For a detailed description of the databases see Burkhauser, Butrica, Daly, and Lillard (2001).

ACKNOWLEDGMENTS The author wishes to thank Professor John Bishop, and an anonymous referee for helping in clarifying the exposition in several points, and the participants of the Second ECINEQ Conference, July 12–14, 2007 in Berlin for valuable comments and discussions. The shortcomings and errors remain the author’s as usual.

REFERENCES An, C., Haveman, R., & Wolfe, B. (1993). Teen out-of-wedlock births and welfare receipt: The role of childhood events and economic circumstances. Review of Economics and Statistics, 72, 195–208. Bane, M. J., & Ellwood, D. (1994). Welfare realities: From rhetoric to reform. Cambridge, MA: Harvard University Press. Burkhauser, R. V., Butrica, B. A., Daly, M. C., & Lillard, D. R. (2001). The cross-national equivalent ﬁle: A product of cross-national research. In: I. Becker, N. Ott & G. Rolf (Eds), Soziale Sicherung in einer dynamischen Gesellschaft. Festschrift fu¨r Richard Hauser zum 65. Geburtstag (pp. 354–376). Frankfurt: Campus. Corak, M., & Heisz, A. (1999). The intergenerational earnings and income mobility of Canadian men: Evidence from longitudinal income tax data. Journal of Human Resources, 34, 504–533. Corak, M. (2006). Do poor children become poor adults? Lessons from a cross-country comparison of generational earnings mobility. In: J. Creedy & G. Kalb (Eds), Research on economic inequality, Vol. 13: Dynamics of inequality and poverty (pp. 143–188). Amsterdam: Elsevier.

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Eberharter, V. V. (2008). Parental background and intergenerational occupational mobility – Germany and the United States compared. Journal of Income Distribution, 17(2), 1–22. Ellwood, D., & Summers, L. (1986). Is welfare really a problem? The Public Interest, 83, 57–78. Fields, G. (2004). Dualism in the labor market: A perspective on the Lewis model after half a century. Manchester School, 72, 724–735. Foster, J. E., Greer, J., & Thorbecke, E. (1984). A class of decomposable poverty measures. Econometrica, 52, 761–766. Gottschalk, P. (1992). The intergenerational transmission of welfare participation: Facts and possible causes. Journal of Policy Analysis and Management, 11, 254–272. Hertz, T. (2005). Rags, riches and race: The intergenerational economic mobility of black and white families in the United States. In: S. Bowles, H. Gintis & M. Osborne (Eds), Unequal chances: Family background and economic success (pp. 165–191). New York: Russel Sage and Princeton University Press. Hyson, R. (2003). Differences in intergenerational mobility across the earnings distribution. Working Paper no. 364, U.S. Bureau of Labor Statistics. Jantti, M., & Danziger, S. (1994). Child poverty in Sweden and the United States: The effect of social transfers and parental labor force participation. Industrial and Labor Relations Review, 48, 48–64. Kolodinsky, J., & Shirey, L. (2000). The impact of living with an elder parent on adult daughter’s labor supply and hours of work. Journal of Family and Economic Issues, 21, 149–174. Mayer, S. E. (1997). What money can’t buy: Family income and children’s life chances. Cambridge: Harvard University Press. Page, M. E. (2004). New evidence on the intergenerational correlation in welfare participation. In: M. Corak (Ed.), Generational income mobility in North America and Europe (pp. 226–244). Cambridge: Cambridge University Press. Pepper, J. V. (2000). The intergenerational transmission of welfare receipt: A nonparametric bounds analysis. The Review of Economics and Statistics, 82, 472–488. Smeeding, T., Rainwater, L., & Burtless, G. (2001). United States poverty in a cross-national context. In: S. H. Danziger & R. H. Haveman (Eds), Understanding poverty (pp. 162–189). New York: Russel Sage Foundation, Cambridge: Harvard University Press. Smeeding, T. (2005). Public policy, economic inequality, and poverty: The United States in comparative perspective. Social Science Quarterly, 86, 955–983. Vartanian, T. P. (1999). Childhood conditions and adult welfare use: Examining neighborhood factors. Journal of Marriage and The Family, 61, 225–237. Vartanian, T. P., & MacNamara, J. M. (2000). Work and economic outcomes after welfare. Journal of Sociology and Social Welfare, 27, 41–78. Vartanian, T. P., & MacNamara, J. M. (2004). The welfare myth: Disentangling the long-term effects of poverty and welfare receipt for young single mothers. Journal of Sociology and Social Welfare, 31, 105–140.

MEASURING INEQUALITY WITH ORDINAL DATA: A NOTE Buhong Zheng ABSTRACT This note formally investigates the applicability of stochastic dominance (Lorenz dominance) to ordinal data such as self-reported health status. We conﬁrm that for ordinal data distributions, stochastic dominance has limited applicability in ranking social welfare, while it has no applicability in ranking inequality.

1. INTRODUCTION Understanding the variabilities of natural phenomena and social activities is important in the process of human development. Through analyzing the variabilities, we deduce the laws governing these activities/phenomena. An important step in the analysis is to quantify the variability of an activity, so that we can say whether the variability is increased or decreased over time or across space. In natural sciences, the variance and the coefﬁcient of variation are commonly used to measure variability, while in social sciences the issue becomes much more complex. The most comprehensively developed measurement of variability is in the area of income distributions where ‘‘variability’’ is referred to as ‘‘income

Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 177–188 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16008-2

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BUHONG ZHENG

inequality.’’ The measurement of income inequality concerns how unequally incomes are distributed among the recipients of a social group. Although the issue of measuring income inequality is age-old and various axioms and measures had been introduced, the literature took a giant step forward only after Atkinson’s classical contribution that laid the welfare-economics foundation for income inequality measurement. Built upon Rothschild and Stiglitz’s (1970) seminar work on measuring risk-aversion, Atkinson (1970) established the important connection between Lorenz inequality dominance and the Pigou-Dalton principle of transfers which, at that time, both had been in the literature for more than half century. With some 30 more years of further research, we now have a much better understanding of the various issues in income inequality measurement and the effects of unequal income distributions on socioeconomic activities. Inequality measures such as Lorenz curve and the Atkinson inequality indices are now standard tools for research and economic policy analyses. In recent years, the measures developed for income inequality have been increasingly applied to measure inequality of non-income subjects. Most noticeably, they have been used to measure health inequality and health care inequity. In fact, there is now a large literature devoted to measuring unequal distribution of health outcomes and health care provisions using income inequality measures such as Lorenz curve and the associated Gini coefﬁcient [see Wagstaff, Paci, and van Doorslaer (1991) for a review of the early literature]. In measuring health inequality, however, the health value is frequently the ‘‘self-reported health status,’’ which is ordinal or categorical rather than ratio. For example, both the US National Health Interview Survey (NHIS) and the Canadian National Population Health Survey (NPHS) adopt a ﬁve-category health rating: poor, fair, good, very good, and excellent. To compute an inequality index or apply an inequality dominance criterion, it is necessary to assign a cardinal number to each health category. Although sophisticated procedures are sometimes developed to convert ordinal data into cardinal values (e.g., Wagstaff & van Doorslaer, 1994), most studies measuring health inequality assign arbitrary numbers to the ﬁve categories. For example, in investigating aging and inequality in health, Deaton and Paxson (1998) calculated health inequality using the variance by assigning values 1 through 5 for the ﬁve health levels. Clearly, we have to be very careful in accepting any conclusion that is based on a set of arbitrary health values. In fact, Allison and Foster (2004) illustrated that the variance is sensitive to the cardinal values assigned: different sets of values for the ﬁve health categories can rank either health distribution as more unequal than the other.

Measuring Inequality with Ordinal Data

179

Although the problem with using ordinal data in inequality measurement is generally known to researchers, the problem has largely been documented through numerical illustrations and for some summary inequality indices. One purpose of this note is to formally prove an impossibility result in measuring inequality with ordinal data. More broadly, this note investigates the applicability of the well-established stochastic dominance approach to ordinal data. We show that stochastic dominance approach has limited applicability in ranking social welfare of ordinal data distributions: only ﬁrst-order stochastic dominance (rank dominance) can be used; all higher orders of dominance, such as generalized Lorenz dominance, will have no additional power in ranking distributions. For inequality orderings, we show that two ordinal distributions can be rank ordered by Lorenz dominance for all possible cardinal values if and only if they are identical. It follows that Lorenz dominance cannot be used to rank inequality with ordinal data. In the remainder of the note, for ease of presentation, we use self-reported health status as an example of ordinal data. The results, of course, are valid for all ordinal variables.

2. RANKING HEALTH DISTRIBUTIONS Consider a society having m health categories with 2 mo1. Denote hi the cardinal value of health level i, i ¼ 1; 2; . . . ; m. Assume health statuses are listed in increasing order, i.e., from the poorest to the best health and h1 h2 hm . Also assume that all health values are positive, i.e., 1 h1 40, which may be interpreted as an assumption of ‘‘life is worth Pm living.’’ The proportion of people in each health class is denoted pi with i¼1 pi ¼ 1. A health distribution is deﬁnedPas p ¼ ð p1 ; p2 ; . . . ; pm Þ and the cumulative population proportion is p~i ¼ ij¼1 pj . A health distribution is degenerate if pi=0 for some i; a nondegenerate distribution has people ﬁlling into each and every health category. The average health level of distribution p is ¼ Pm p hi . For all societies of interest, the health denoted as hðpÞ i¼1 i classiﬁcations are the same.

2.1. Generalized, Relative, and Absolute Lorenz Dominances: Deﬁnitions In the literature of income distributions, welfare comparisons between distributions are often carried out using generalized Lorenz dominance,

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BUHONG ZHENG

while inequality comparisons are performed with either relative or absolute Lorenz dominance.2 Generalized Lorenz dominance compares the generalized Lorenz curves between two health distributions. For each cumulative population share, p~l , the corresponding generalized Lorenz ordinate is the (normalized) cumulated total health level, i.e., GLðp; p~l Þ ¼

l X

pi hi

i¼1

The generalized Lorenz curve is the linear segment of the m Lorenz coordinates and the origin (0, 0). It follows that for a cumulative population share p that is not in f0; p~1 ; p~2 ; . . . ; 1g, its generalized Lorenz ordinate is simply an extrapolation from the nearest generalized Lorenz ordinate below p. For example, if p~1 opop~2 , then GLðp; pÞ ¼ p1 h1 þ ðp p1 Þh2 Distribution p generalized Lorenz dominates distribution q if and only if the generalized Lorenz curve of p lies nowhere below and somewhere strictly above that of q. Relative and absolute Lorenz dominances, respectively, compare relative and absolute Lorenz curves. The difference between the two approaches lies in the different ways that health distributions are normalized to have equal mean-health levels. Relative Lorenz curve remains unchanged to an equalproportional change in all health levels (e.g., if all hi’s are increased by 10%), while absolute Lorenz curve remains the same when all health values are increased by an equal-absolute amount (e.g., if all hi’s are increased by 0.1).3 Relative Lorenz ordinates fRLðp; p~l Þg for distribution p are deﬁned by Pl p hi RLðp; p~l Þ ¼ i¼1 i hðpÞ and distribution p relative Lorenz dominates q if and only if the relative Lorenz curve of p lies nowhere below and somewhere strictly above that of q. Absolute Lorenz ordinates fALðp; p~l Þg for distribution p are deﬁned by ! l l X X p hi p hðpÞ ALðp; p~ Þ ¼ l

i

i¼1

i

i¼1

and distribution p absolute Lorenz dominates q if and only if the absolute

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Measuring Inequality with Ordinal Data

Lorenz curve of p lies nowhere below and somewhere strictly above that of q. Note that both relative and absolute Lorenz curves are derived from generalized Lorenz curve: relative Lorenz curve is obtained by scaling down generalized Lorenz curve by hðpÞ, while absolute Lorenz curve is obtained by shifting down generalized Lorenz curve by hðpÞ. Clearly, the location of any Lorenz curve (generalized, relative, or absolute) depends on the values that hj’s are assigned and the dominance evaluation may hinge on the speciﬁc values used. Given the arbitrariness in assigning such values, it is important to investigate whether there are conditions under which dominance relations hold with respect to all possible values of health levels. In what follows, we explore the possibility and impossibility for generalized, relative, and absolute Lorenz dominances.

2.2. Generalized Lorenz Dominance: Limited Possibility For two health distributions p and q, p generalized Lorenz dominates q if and only if GLðp; rÞ GLðq; rÞ for all r 2 ½0; 1 with the inequality hold strictly for some r 2 ð0; 1. A necessary condition is GLðp; 1Þ GLðq; 1Þ – the two ending points of the two generalized Lorenz curves – or m X i¼1

pi h i

m X

qi hi

(1)

i¼1

Using Abel’s partial summation formula (Rudin, 1976, p. 70), Eq. (1) can be written as ( ) ( ) m m m m X X X X pj ðhi hi1 Þ qj ðhi hi1 Þ (1a) i¼1

j¼i

i¼1

j¼i

where h0 0. Abel’s lemma states that for any 0oh1 h2 hm , a sufﬁcient condition for Eq. (1) is4 m X j¼i

pj

m X

qj

for

i ¼ 1; 2; . . . ; m

(2)

j¼i

Condition (2) Pm also necessary to maintain Eq. (1a) or (1). This Pism obviously p o is because if j¼l j j¼l qj for some l ð1 l mÞ, then by choosing

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BUHONG ZHENG

h1 ¼ ¼ hl1 , hl ¼ ¼ hm , and hl 4hl1 , we will have the desired contradiction. Pm P Since m j¼1 pj ¼ j¼1 qj ¼ 1, Eq. (2) is equivalent to p~j ¼

j X i¼1

pi

j X

qi ¼ q~j

for

j ¼ 1; 2; . . . ; m 1

(2a)

i¼1

i.e., the cumulative frequency of health status is no greater in p than in q. Condition (2a) also ensures that the entire generalized Lorenz curve of p lies nowhere below that of q. This assertion can be directly veriﬁed as follows (though it will also become apparent shortly). Since the generalized Lorenz curve of p is a linear segment of (0, 0), it follows that GLðp; rÞ ½ p~1 ; GLðp; p~1 Þ, ½ p~2 ; GLðp; p~2 Þ; . . . , and ½1; hðpÞ, GLðq; rÞ for all r 2 ð0; 1Þ holds if and only if GLðp; rÞ GLðq; rÞ holds for . . . ; q~m1 ; 1g. Suppose r 2 f p~1 ; p~2 ; . . . ; p~m1 g, say r ¼ p~l , r ¼ f p~1 ; q~1 ; p~2 ; q~2 ;P then GLðp; rÞ ¼ li¼1 pi hi . If p~l lies between q~k1 and q~k such that q~k1 op~l q~k , then GLðq; rÞ ¼

k1 X

qi hi þ ðp~l q~k1 Þhk

i¼1

Since p~l q~l by (2a), we must have l k; otherwise l k 1 and thus GLðq; rÞ. p~l q~k1 – a contradiction. Abel’s lemma then entails GLðp; rÞ P Suppose now r 2 fq~1 ; q~2 ; . . . ; q~m1 g, say r ¼ q~l , then GLðq; rÞ ¼ li¼1 qi hi . If q~l lies between p~k and p~kþ1 such that p~k q~l op~kþ1 , then GLðp; rÞ ¼

k X

pi hi þ ðq~l p~k Þhkþ1

i¼1

Since p~l q~l by (2a), we must have k l; otherwise k þ 1 l and thus p~kþ1 q~l – also a contradiction. Abel’s lemma again entails GLðp; rÞ GLðq; rÞ. To ensure that the generalized Lorenz curve of p lies somewhere strictly above that of q, i.e., GLðp; rÞ4GLðq; rÞ for some r, we need further requirement of the inequality in (2a), which holds strictly for some j ¼ 1; 2; . . . ; m 1. The above derivations and discussions can be summarized formally into the following proposition. Proposition 1. For any two health distributions p and q, p generalized Lorenz dominates q if and only if condition (2a) holds for all j ¼ 1; 2; . . . ; m and holds strictly for some j ¼ 1; 2; . . . ; m 1.

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Condition (2a), however, is essentially the ﬁrst-order stochastic dominance condition as derived in Allison and Foster (2004, Theorem 1). While we consider a second-order requirement – generalized Lorenz dominance is equivalent to second-order stochastic dominance – the requirement simply collapses to the ﬁrst-order condition if it has to hold for all possible values of hi’s. This observation holds for any higher-order stochastic dominance condition because such a condition would require a dominance of the mean between the two distributions, which leads precisely to condition (2a).5 On the other hand, from the literature of stochastic dominance, Eq. (2a) leads to any higher-order stochastic dominance. In this sense, it is futile to consider any higher-order stochastic dominance than ﬁrst-order in measuring welfare of health distributions. The following corollary then suggests that Allison and Foster’s ﬁrst-order condition cannot be generated to any higher orders; ﬁrst-order stochastic dominance is the only tool for welfare rankings of health distributions. Corollary 1. For any two health distributions p and q, p welfare dominates q in any degree of stochastic dominance if and only if condition (2a) holds for all j ¼ 1; 2; . . . ; m and holds strictly for some j ¼ 1; 2; . . . ; m 1. 2.3. Lorenz Dominances: Impossibility For two health distributions p and q, p relative Lorenz dominates q if and only if RLðp; rÞ RLðq; rÞ for all r 2 ½0; 1 with the strict inequality holding for some r 2 ð0; 1Þ. Distribution p absolute Lorenz dominates q if and only if ALðp; rÞ ALðq; rÞ for all r 2 ½0; 1 with the strict inequality holding for some r 2 ð0; 1Þ. In what follows, we ﬁrst consider relative Lorenz dominance. The impossibility result for relative Lorenz dominance stems from the following lemma in which only two health statuses are considered. Lemma 1. Suppose there are only two health statuses (e.g., healthy and unhealthy) with 0oh1 oh2 . Then for any nondegenerate health distributions p and q, the two relative Lorenz curves must be either cross or identical; there is no possibility of dominance.

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Proof. Since there are only two health statuses, p ¼ ðp1 ; p2 Þ and q ¼ ðq1 ; q2 Þ. In order for the Lorenz curve of p to dominate that of q, it must be the case that RLðp; rÞ RLðq; rÞ at r ¼ fp1 ; q1 g. If p1 oq1 , then RLðp; rÞ RLðq; rÞ at r ¼ p1 means p1 h1 p1 h1 p 1 h 1 þ p2 h 2 q 1 h 1 þ q2 h 2 which requires p1 h1 þ p2 h2 q1 h1 þ q2 h2 since p1 h1 40 by the nondegenerate assumption. But p1 oq1 implies p2 4q2 and, consequently, we must have p1 h1 þ p2 h2 ¼ h1 þ p2 ðh2 h1 Þ4h1 þ q2 ðh2 h1 Þ ¼ q1 h1 þ q2 h2 for any h1 and h2 satisfying 0oh1 oh2 – a contradiction. It follows that we can only have p1 q1 . Condition RLðp; rÞ RLðq; rÞ at r=p1 becomes p1 h1 q h1 þ ðp1 q1 Þh2 1 p 1 h 1 þ p2 h 2 q 1 h 1 þ q2 h 2 Expanding the inequality, we arrive at ðp1 q1 Þp2 h1 ðp1 q1 Þp2 h2 Clearly, for 0oh1 oh2 , the above weak inequality can hold only as an equality and only when either p1=q1 or p2=0. But p2=0 means a degenerate distribution, i.e., only one health status for all individuals and hence is excluded. It follows that the only case remains is p1=q1 and thus p=q. This means that the only case where two distributions do not cross is when they are identical. The lemma is of interest in its own right: if a population is classiﬁed dichotomously into healthy and unhealthy groups, then there is no way to rank any distributions using relative Lorenz dominance if none of the distributions is degenerate. In the case where two distributions are not identical and neither one is degenerate, the two relative Lorenz curves will always cross. For a health-evaluating system with more than two categories, however, the impossibility presented in this lemma will disappear. A simple example can verify that this is indeed the case. In fact, many empirical studies have demonstrated the possibility. But when the nonuniqueness of the health value is considered, the possibility again turns into impossibility. This result is formally stated and proved as follows.6

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Proposition 2. For a given health-evaluating system with m health statuses and two nondegenerate health distributions p and q, the relative Lorenz curve of p lies nowhere below that of q for all possible values of health statuses satisfying 0oh1 h2 hm if and only if p=q. Proof. The key to the proof is to group all m health classes into just two groups and then apply Lemma 1. We ﬁrst consider the lowest-health group versus the rest of P the population by choosing h2 ¼ ¼ hm . The distribution p becomes ðp1 ; m i¼2 pi Þ that is nondegenerate. Do the same for distribution q and the distribution becomes P ðq1 ; m i¼2 qi Þ. Let h2 4h1 , Lemma 1 then implies that if the relative Lorenz curve of p lies nowhere below that of q, then p1=q1. In general, if pi=qi for i ¼ 1; 2; . . . ; s, then we must have psþ1 ¼ qsþ1 . This is proved by choosing h1 ¼ h2 ¼ ¼ hsþ1 ohsþ2 ¼ ¼ hm and grouping the ﬁrst s+1 health classes together against the remaining P Psþ1 pi ¼ sþ1 m s 1 classes. Again by applying Lemma 1, we have i¼1 i¼1 qi Ps Ps that, together with i¼1 pi ¼ i¼1 qi , implies psþ1 ¼ qsþ1 .7 Thus, we have pi=qi for i ¼ 1; 2; . . . ; m or p=q. The sufﬁciency of the proposition is obvious. For absolute Lorenz dominance, similar results can be established with similar proofs. Lemma 2. Suppose there are only two health statuses (e.g., healthy and unhealthy) and 0oh1 oh2 . Then for any two nondegenerate health distributions p and q, the two absolute Lorenz curves must be either cross or identical; there is no possibility of dominance. Proposition 3. For any health-evaluating system with m health statuses and two nondegenerate health distributions p and q, the absolute Lorenz curve of p lies nowhere below that of q for all possible values of health statuses h1 ; h2 ; . . . ; hm satisfying 0oh1 h2 hm if and only if p=q. Why (limited) possibility for generalized Lorenz dominance but impossibility for relative and absolute Lorenz dominances? In other words, why it is possible to compare welfare across health distributions but not to compare inequality? The key point here is that generalized Lorenz curve does not hold the average health level constant, but both relative and absolute Lorenz curves employ some normalization process so that the mean health levels become the same across different distributions. It is well known that if two distributions have the same mean, then generalized and relative (absolute) Lorenz dominances become the same. If we require equal

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mean between the two distributions, say p and q, then ¼ hðpÞ

m X i¼1

pi hi ¼

m X

qi hi ¼ hðqÞ

i¼1

Clearly, for the above equality to hold for all possible values of hi’s, p and q must be identical.

3. CONCLUDING REMARKS The demonstrated difﬁculties in applying conventional tools of inequality measurement to ordinal data have promoted researchers to develop alternative ways to evaluate distributions of ordinal data. A notable approach is to measure the disparity of ordinal data as ‘‘polarization.’’ Blair and Lacy (2000) claimed to have developed a ﬁrst set of such measures. They noted that for an ordinal data distribution, there are multiple references of equal distribution; for example, for the ﬁve-category health status, there are ﬁve equal distributions (one for each health category). This makes the conventional approach of inequality measurement unadaptable: which of the ﬁve equal distributions should be used as the reference to deﬁne inequality? To get around the issue, they suggested to use the most unequal distribution as the reference distribution. They argued that the most unequal (polarized) distribution would be that half of the people in the lowest category (the poorest health) and half of the people in the highest category (the best health). The deﬁned measures are the (transformations) of the distance of the distribution in question to this ‘‘most unequal’’ reference distribution. Their measures, as they pointed out, inter alia, do not satisfy Pigou-Dalton’s principle of transfers. Following Blair and Lacy (2000), Apouey (2007) further investigated the measurement issue within the context of health disparity. By incorporating the axioms from the newly developed literature of polarization measurement (e.g., Wolfson, 1994; Wang & Tsui, 2000; Duclos, Esteban, & Ray, 2004), Apouey characterized a class of single-parameter measures of health polarization. Recently, Allison and Foster (2004) also proposed to measure health inequality as bipolarization and derived a median-based dominance condition for ranking health distributions. All these new polarization measures are unquestionably useful in measuring the disparity of ordinal data. But since these measures concern only about how concentrated the data points are distributed toward the two

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ends, they may not provide a complete picture of data distribution disparity. We hope that future research will shed more light on the issue of disparity measurement with ordinal data.

NOTES 1. For welfare rankings, this assumption is not needed, but for inequality orderings all health levels must be strictly positive in order to have a meaningful dominance condition. 2. For a recent and thorough introduction to the various Lorenz dominances and the literature of income distributions in general, see Lambert (2001). 3. It is well-known that the two types of Lorenz dominance reﬂect two value judgments in measuring inequality. It is possible to consider a ‘‘balanced’’ or ‘‘intermediate’’ value judgment between the relative and the absolute views. In a recent paper, Zheng (2007) showed that under certain reasonable conditions, the only intermediate Lorenz curve is a weighted geometric mean between the relative and the absolute relative Lorenz curves. All results derived in this section for relative and absolute Lorenz dominances remain valid for an intermediate Lorenz dominance. 4. The result in Eq. (2) has been proved by Allison and Foster (2004, Theorem 1). We furnish a proof here since Abel’s partial summation and Abel’s lemma will be used again in the rest of the paper. 5. For a detailed and up-to-date exposition on stochastic dominance, one may consult Levy (2006). In this paper, due to space limitation, we are unable to expand the discussions on stochastic dominance. 6. A general result can be established for all health distributions (including degenerate distributions). Since in reality a health distribution is unlikely to be degenerate (otherwise an entire health status is null; none of the health distributions that we have come across is degenerate), we only include in the paper the result for nondegenerate distributions. The general result, however, is available from the author upon request. 7. If one desires strict inequality among the health values, i.e., h1 oh2 o ohm , then the described grouping process cannot be done. In this case, for a given s, we can ðkÞ ðkÞ ðkÞ ðkÞ . . . ; hðkÞ construct a sequence of health values fhðkÞ m g such that h1 ok2 o ohm 1 ; k2 ;ðkÞ ðkÞ ðkÞ ðkÞ with limk!1 h1 ¼ limk!1 hsþ1 and limk!1 hsþ2 ¼ limk!1 hm . Thus, we can group the population into two groups in the limit. The continuity of Lorenz curve in health values then carries through the proof.

REFERENCES Allison, R. A., & Foster, J. E. (2004). Measuring health inequality using qualitative data. Journal of Health Economics, 23, 505–524.

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Apouey, B. (2007). Measuring health polarization with self-assessed health data. Health Economics, 16, 875–894. Atkinson, A. B. (1970). On the measurement of inequality. Journal of Economic Theory, 2, 244–263. Blair, J., & Lacy, M. G. (2000). Statistics for ordinal variation. Sociological Methods and Research, 28, 251–279. Deaton, A., & Paxson, C. (1998). Aging and inequality in income and health. American Economic Review (Papers and Proceedings), 88, 248–253. Duclos, J., Esteban, J., & Ray, D. (2004). Polarization: Concepts, measurement, estimation. Econometrica, 72, 1737–1772. Lambert, P. (2001). The distribution and redistribution of income (3rd ed.). Manchester: The Manchester University Press. Levy, H. (2006). Stochastic dominance. Berlin: Springer-Verlag. Rothschild, M., & Stiglitz, J. E. (1970). Increasing risk: I. Deﬁnition. Journal of Economic Theory, 2, 225–243. Rudin, W. (1976). Principles of mathematical analysis (3rd ed.). New York: McGraw-Hill. Wagstaff, A., Paci, P., & van Doorslaer, E. (1991). On the measurement of inequalities in health. Social Science and Medicine, 33, 545–557. Wagstaff, A., & van Doorslaer, E. (1994). Measuring inequalities in health in the presence of multiple-category morbidity indicators. Health Economics, 3, 281–291. Wang, Y., & Tsui, K. (2000). Polarization orderings and new classes of polarization indices. Journal of Public Economic Theory, 2, 349–363. Wolfson, M. (1994). When inequality diverges. American Economic Review (Papers and Proceedings), 84, 353–358. Zheng, B. (2007). Inequality orderings and unit consistency. Social Choice and Welfare, 29, 515–538.

MULTIDIMENSIONAL UNIT- AND SUBGROUP-CONSISTENT INEQUALITY AND POVERTY MEASURES: SOME CHARACTERIZATIONS Henar Dı´ ez, Ma Casilda Lasso de la Vega and Ana Urrutia ABSTRACT Purpose: Most of the characterizations of inequality or poverty indices assume some invariance condition, be that scale, translation, or intermediate, which imposes value judgments on the measurement. In the unidimensional approach, Zheng (2007a, 2007b) suggests replacing all these properties with the unit-consistency axiom, which requires that the inequality or poverty rankings, rather than their cardinal values, are not altered when income is measured in different monetary units. The aim of this paper is to introduce a multidimensional generalization of this axiom and characterize classes of multidimensional inequality and poverty measures that are unit consistent. Design/methodology/approach: Zheng (2007a, 2007b) characterizes families of inequality and poverty measures that fulﬁl the unit-consistency Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 189–211 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16009-4

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axiom. Tsui (1999, 2002), in turn, derives families of the multidimensional relative inequality and poverty measures. Both of these contributions are the background taken to achieve our characterization results. Findings: This paper merges these two generalizations to identify the canonical forms of all the multidimensional subgroup- and unit-consistent inequality and poverty measures. The inequality families we derive are generalizations of both the Zheng and Tsui inequality families. The poverty indices presented are generalizations of Tsui’s relative poverty families as well as the families identiﬁed by Zheng. Originality/value: The inequality and poverty families characterized in this paper are unit and subgroup consistent, both of them being appropriate requirements in empirical applications in which inequality or poverty in a population split into groups is measured. Then, in empirical applications, it makes sense to choose measures from the families we derive.

INTRODUCTION This work takes as a reference two recent contributions concerning inequality and poverty measurement. The ﬁrst one is Zheng (2007a, 2007b), who introduces a new unit-consistency axiom into the unidimensional context and characterizes families of inequality and poverty measures that fulﬁl this axiom. The other starting point is Tsui (1999, 2002), who derives classes of multidimensional inequality and poverty measures. Zheng (2007a, 2007b) accurately argues that all the invariance conditions – be that scale, translation, or intermediate, usually invoked as axioms to characterize most inequality and poverty indices – impose value judgments on the measurement, and there is no justiﬁcation for any of them being assumed to characterize an inequality or poverty measure. However, it is true that it makes no sense that inequality or poverty comparisons vary when the units in which income is measured change. Zheng introduces a new principle, the unit-consistency axiom, which requires that the inequality or poverty rankings, rather than the inequality or poverty levels, not be affected by the units in which incomes are expressed. On the other hand, after the seminal articles by Kolm (1977) and Atkinson and Bourguignon (1982), researchers have become aware that

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191

inequality and poverty are multidimensional concepts; therefore, other attributes related to health or education should also be taken into consideration in their measurement. Many efforts have been made to develop multidimensional indices.1 This paper proposes a straightforward extension of the unit-consistency axiom to the multidimensional setting, and we characterize the classes of multidimensional subgroup-consistent inequality and poverty measures that are unit consistent. The paper is structured as follows. The ﬁrst two sections below are devoted to inequality and poverty measures, respectively. Each of these sections begins with a brief introduction of the notation and basic deﬁnitions, then the generalization of the unit-consistency axiom to the multidimensional setting is introduced in each ﬁeld, and ﬁnally our characterization results are presented. The paper ﬁnishes with some concluding remarks. Most of the proofs of our paper follow both Zheng (2005, 2007a, 2007b) and Tsui’s (1999, 2002) papers and the relevant results by Shorrocks (1984) and Foster and Shorrocks (1991) as well.

MULTIDIMENSIONAL UNIT-CONSISTENT INEQUALITY MEASURES Notations and Basic Axioms of Multidimensional Inequality Measures We consider a population consisting of nZ2 individuals endowed with a bundle of kZ1 attributes, such as income, health, education, and so on. An n k real matrix X represents a multidimensional distribution among the population. The ijth entry of X, denoted xij, represents the ith individual’s amount of the jth attribute. The ith row is denoted xi and the jth column is denoted x j . For each attribute j, mj(X) represents the mean value of the jth attribute, and mðXÞ ¼ ðm1 ðXÞ; . . . ; mk ðXÞÞ is the vector of the means of the attributes. When measuring inequality, we assume that the attributes should be positive. The set of all the n k matrices over the positive real elements is denoted Mþþ(n, k), and Dþþ is the set of all such matrices. In this paper, a multidimensional inequality index is deﬁned as a function I : Dþþ ! R that possesses the following four properties, straightforward generalizations of their familiar one-dimensional equivalents: Anonymity: IðXÞ ¼ IðPXÞ for any X 2 M þþ ðn; kÞ and for any n n permutation matrix P.

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Normalization: I(X ) ¼ 0 if all the rows of the matrix X are identical, i.e., all the individuals have exactly the same bundle of attributes. Replication Invariance: I(Y )=I(X ) if Y is obtained from X by a replication. Continuity: I is a continuous function in any individual’s attributes. The above standard axioms are insufﬁcient to guarantee that function I be able to capture the essence of multidimensional inequality. In the unidimensional setting, the well-known Pigou-Dalton transfer principle is the basic axiom for doing so. This principle has a number of equivalent formulations (Hardy, Littlewood, & Po´lya, 1934; Marshall & Olkin, 1979) that are used to generalize this axiom to the multidimensional approach. The following, proposed by Kolm (1977),2 is used in this paper. Uniform Principle (UM): A multidimensional inequality measure I satisﬁes UM if I(BX )oI(X ) for any X 2 M þþ ðn; kÞ and for any n n bistochastic matrix B that is not a permutation matrix of the rows of X. This criterion establishes that multidimensional inequality should be a function of the uniform inequality of a multivariate distribution of attributes across people. If the matrix X is multiplied by a bistochastic matrix B, then, for each individual, the attributes of the resulting distribution are linear convex combinations of the attributes of the previous distribution and intuitively display less inequality. On the other hand, Atkinson and Bourguignon (1982) and Walzer (1983) point out that a multidimensional inequality measure should also be sensitive to the cross-correlation between inequalities in different dimensions. This idea is captured by Tsui (1999), who introduces a new criterion based on the concept of arrangement-increasing transfer introduced by Boland and Proschan (1988). Deﬁnition. A distribution Y may be derived from a distribution X by a correlation-increasing transfer if there exist row indices p and q such that (i) yp ¼ ðminfxp1 ; xq1 g; . . . ; minfxpk ; xqk gÞ, (ii) yq ¼ ðmaxfxp1 ; xq1 g; . . . ; max fxpk ; xqk gÞ, and (iii) ym ¼ xm 8map; q. A correlation-increasing transfer is strict whenever yp axp . Correlation-Increasing Principle (CIM): A multidimensional inequality measure I satisﬁes CIM if I(X )oI(Y ) whenever Y may be derived from X by a permutation of rows and a ﬁnite sequence of correlation-increasing transfers, at least one of which is strict. CIM has an intuitive interpretation. We may imagine the situation in which the ﬁrst individual in the society receives the lowest amount of each

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193

attribute; the second individual is endowed with the second lowest amount, up to the individual n who receives the greatest amount of each attribute. CIM ensures that this distribution is the most unequal in the sense that any other distribution matrix of the same amount of attributes is more equal.3 If the population in which we want to measure inequality is split into groups according to characteristics such as age, gender, race, or area of residence, an appropriate requirement is to demand that if inequality in one group increases, the overall inequality should also increase. This property proposed by Shorrocks (1984) in the unidimensional framework is generalized for multidimensional distributions in the following way: Subgroup-Consistency Principle: A multidimensional inequality measure I is subgroup consistent if there exists a function A such that " #! X1 ¼ AðIðX 1 Þ; mðX 1 Þ; n1 ; IðX 2 Þ; mðX 2 Þ; n2 Þ I X2 for all X 1 ; X 2 2 D, and A is a continuous and strictly increasing function in the index values I(X1) and I(X2). In the unidimensional setting, the additive decomposition condition (Bourguignon, 1979; Cowell, 1980; Shorrocks, 1980) allows us to decompose overall inequality as the sum of the inequality level of a hypothetical distribution in which each person’s income is replaced by the mean income of his/her group and a weighted sum of the group inequality levels. This property can be generalized to the multidimensional setting in the following way: Decomposition Property: If a population is classiﬁed in G nonempty subgroups, the inequality index I meets the decomposition property if the following relationship holds: IðXÞ ¼ IðX 1 ; X 2 ; . . . ; X G Þ ¼

G X

wg ðmðX g Þ; ng Þ IðX g Þ þ IðA1 ; . . . ; AG Þ

g¼1

where I (Xg) is the inequality of subgroup g, wg the weight attached to subgroup g, and Ag an ng k matrix with Agij ¼ mj ðX g Þ for g ¼ 1; . . . ; G. It is clear that if an inequality measure satisﬁes the decomposition property, it also fulﬁls the subgroup-consistency principle. In the literature on inequality indices, invariance properties are often invoked. A relative inequality index remains unchanged with proportional

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changes in any attribute for all the individuals, whereas an absolute inequality index remains unchanged if a common amount is added to any attribute for all the individuals. Scale Invariance Principle: A multidimensional inequality measure I is scale invariant if IðXÞ ¼ IðXLÞ, for all k k diagonal matrices L with lj 40. Relative inequality indices are those that are scale invariant. Translation Invariance Principle: A multidimensional inequality measure I is translation invariant if IðXÞ ¼ IðX þ AÞ, for all matrices A with identical rows a ¼ ða1 ; a2 ; . . . ; ak Þ and aj 0. Absolute inequality indices are those that are translation invariant. Unit-Consistency Axiom for Multidimensional Inequality Measures The above section ends with two possible answers as to how to distribute a given amount of attributes among all the individuals, without altering the inequality level. In the unidimensional framework, Zheng (2007a) has analyzed in depth the value judgments involved in the different ways in which this problem is faced and proposed a new axiom of unit consistency that requires that the inequality ranking between two distributions should not be affected by the unit in which income is expressed. This axiom has a straightforward generalization to the multidimensional framework, allowing several attributes to be measured in different units without changing the inequality rankings of the multidimensional distributions.4 The natural generalization of the unit-consistency axiom to the multidimensional framework that we propose is the following: Unit-Consistency Axiom: A multidimensional inequality measure I is unit consistent if, for any two distributions X; Y 2 M þþ ðn; kÞ such that I (X )o I(Y ), IðXLÞoIðYLÞ for any k k diagonal matrix L with lj 40. It is clear that the scale invariance principle implies unit consistency; hence, every relative multidimensional inequality measure is unit consistent. Unfortunately, none of the rest of the multidimensional indices traditionally used in the literature fulﬁls this property. Speciﬁcally, for the members that are not relative derived by Tsui (1995), Maasoumi (1999) and Bourguignon (1999), it can be proved that none of them meets the unit-consistency axiom. Consider for instance the

Unit- and Subgroup-Consistent Inequality and Poverty Measures

following two distributions: 6 4 X¼ 7 6

and

Y¼

40

0

0

1

6

4

8

4

195

and the diagonal matrix L¼

With respect to the multidimensional generalization of the absolute Akinson–Kolm–Sen index proposed by Tsui (1995), if we calculate Eq. (22) according to the parameter values c1 ¼ c2 ¼ 0.5, we ﬁnd I A ðXÞ ¼ 1:2874 I A ðYÞ ¼ 1:238, whereas I A ðXLÞ ¼ 10oI A ðYLÞ ¼ 20. As regards the Maasoumi class, taking the parameter values b ¼ 2 and a1 ¼ 0.2, a2 ¼ 0.8 [in order to deﬁne Si in Eq. (9)], and g ¼ 0 [Eq. (13)], we obtain M 0 ðXÞ ¼ 0:00574M 0 ðYÞ ¼ 0:00082, but M 0 ðXLÞ ¼ 0:0012oM 0 ðYLÞ ¼ 0:0044. Similarly, choosing the parameter values b ¼ 0.5 and g ¼ 0.5 for the Bourguignon indices [Eq. (46)], we have I D ðXÞ ¼ 0:00274I D ðYÞ ¼ 0:0014, but I D ðXLÞ ¼ 0:0012oI D ðYLÞ ¼ 0:0022. In order to prove that the multidimensional generalized Gini indices (Gajdos & Weymark, 2005) may also violate this principle, let’s consider the two distributions: 6 7 2 5 8 2 and Y ¼ X¼ 2 7 2 5 6 2 and the matrix

2

1 63 A¼6 42 3 taking

2

0:1 0:9

3 0:4 7 7 5 0:6

3

40

0

0

6 L¼4 0 0

1

7 05

0

1

If we compute the measures derived by Gajdos and Weymark (2005) according to the parameter values r ¼ 2, g1 ¼ g2 ¼ g3 ¼ 1=3 [Eq. (22)],

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we get I R ðXÞ ¼ 0:0360oI R ðYÞ ¼ 0:0701, whereas I R ðXLÞ ¼ 0:16624I R ðYLÞ ¼ 0:0001. As regards Eq. (29) with r ¼ 0, we obtain I A ðXÞ ¼ 0:2222oI A ðYÞ ¼ 0:2666, whereas I A ðXLÞ ¼ 8:88884I A ðYLÞ ¼ 0:2666. Moreover, it is not difﬁcult to prove in a similar way that all the rest of the members of these families may also violate the unit-consistency property.

Multidimensional Unit-Consistent Inequality Measures In this section, we characterize unit-consistent inequality measures. As already mentioned, most of the proofs in this section follow those in Zheng (2005, 2007a), Tsui (1999), and Shorrocks (1984) and are not included. All of them are available on request. Before going on to our characterization theorems, we explore the implications of the unit-consistency axiom ﬁrst on a general multidimensional index, Proposition 1, and then on a decomposable one, Proposition 2. The results we obtain are straightforward generalizations of those established by Zheng (2007a) in Propositions 1 and 3. Proposition 1. A multidimensional inequality index I : Dþþ ! R is unit consistent if and only if, for any distribution X 2 M þþ ðn; kÞ and for any k k diagonal matrix L with lj 40, there exists a continuous function f : Rkþþ R ! R increasing in the last argument such that IðXLÞ ¼ f ðl1 ; l2 ; . . . ; lk ; IðXÞÞ

(1)

Proof. This proof is an extension to the multidimensional setting of that of Proposition 1 in Zheng (2007a). Proposition 2. If a multidimensional inequality measure I : Dþþ ! R satisﬁes UM, the decomposition property, and the unit-consistency axiom, then Y t l IðXÞ (2) IðXLÞ ¼ j 1 jk for any X 2 M þþ ðn; kÞ and for all k k diagonal matrices L with lj 40, and some constant t 2 R. Moreover, I is a homogenous function of degree kt. Proof. The proof is straightforward following that of Proposition 3 in Zheng (2007a).

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Proposition 1 reveals that if any changes in the attribute units have no inﬂuence on inequality rankings, both the unit change matrix L and the inequality value I(X ) must enter into I(XL) independently. If the axioms are supplemented with UM and the decomposition property, then I must be a homogeneous function. Our main characterization results are presented in Theorems 2 and 4. In Theorem 2, we identify the multidimensional subgroup-consistent inequality measures that are unit consistent. In fact, this theorem is a generalization of Theorem 3 in Tsui (1999) replacing the scale invariance property by the unit-consistency axiom. Then in Theorem 4, we supplement CIM, deriving a generalization of the family that Tsui (1999) obtains in his Theorem 4. In order to prove these theorems, we follow two steps. First, we characterize, in Theorems 1 and 3, the subfamily of decomposable measures that are unit consistent. Then, following the equivalent unidimensional results, we show that, in this framework also, every subgroup-consistent measure can be expressed as an increasing transformation of some decomposable. Theorem 1. A multidimensional inequality measure I : Dþþ ! R satisﬁes UM, the decomposition property, and the unit-consistency axiom if and only if it is a positive multiple of the form " # n k k Y X Y r aj aj ðxij Þ ðmj Þ (3) IðXÞ ¼ Qk aj t n j¼1 mj i¼1 j¼1 j¼1 where t 2 R and the parameters aj and r have to be chosen such that the function fðxi Þ ¼ rP1 jk ðxij =mj Þaj is strictly convex, or !# " X n k X 1 xim xij IðXÞ ¼ Qk amj log (4) mm mj n j¼1 mt i¼1 j¼1 j where m 2 f1; 2; . . . ; kg, t 2 R, andP the parameters amj have to be chosen such that the function fðxi Þ ¼ kj¼1 ðxim amj =um Þ logðxij =mj Þ is strictly convex, or n X k X mj 1 (5) dj log IðXÞ ¼ Qk t xij n j¼1 mj i¼1 j¼1 where t 2 R and dj 40 for all j.

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Proof. We are going to provide just a sketch of the proof: (i) Generalizing the proof of Proposition 4 in Zheng (2005), it may be P shown that IðXÞ ¼ ð1=nlðmÞÞ ni¼1 ðfðxi Þ fðmÞÞ, for some continuous function lð:Þ. (ii) Deﬁne GðXÞ ¼ IðXÞPkj¼1 mt j . It is easy to verify that G(X ) is a decomposable and therefore subgroup-consistent, relative, inequality measure. Then G(X ) is an increasing transformation of some of the functional forms identiﬁed by Theorem 3 of Tsui (1999). Following an analogous discussion of that of Section II of Zheng (2005), it is possible to obtain the functional forms of this theorem. (iii) The sufﬁciency of the theorem is straightforward, taking the appropriate weights for the different functional forms. Replacing the decomposition property by the subgroup-consistency axiom, we just get increasing transformations of the functional forms obtained in the above theorem. Theorem 2. A multidimensional inequality measure I : Dþþ ! R satisﬁes UM, the subgroup-consistency principle, and the unit-consistency axiom if and only if there exists a continuous increasing transformation F : R ! Rþ , with F(0) ¼ 0, such that for any X 2 M þþ ðn; kÞ, JðXÞ ¼ FðIðXÞÞ is another multidimensional inequality measure satisfying UM, the decomposition property, and the unit-consistency axiom. Proof. One can easily generalize the results in Shorrocks (1984) to show that, for any continuous subgroup-consistency multidimensional inequality index I, there exists a decomposable multidimensional index J and a continuous strictly increasing function F : R ! R, with F(0) ¼ 0, such that JðXÞ ¼ FðIðXÞÞ. If I satisﬁes UM, since F is a strictly increasing function, then J also satisﬁes UM. Moreover, if I is unit consistent, the same holds for J. Indeed, if IðXÞoIðYÞ, i.e., FðIðXÞÞoFðIðYÞÞ since F is a strictly increasing function, then JðXÞoJðYÞ. As a consequence, for any X 2 M þþ ðn; kÞ and any k k diagonal matrix L with lj 40, we have IðXLÞoIðYLÞ and then FðI ðXLÞÞo FðIðYLÞÞ, i.e., JðXLÞoJðYLÞ, concluding that J is a unit-consistent multidimensional inequality index. Therefore J belongs to the class characterized in Theorem 1. The sufﬁciency of this theorem is straightforward, simply considering the function G ¼ F 1 and J satisfying UM, the decomposition property, and the unit-consistency axiom.

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If the compelling axiom CIM is also assumed, then only the ﬁrst of the expressions of Theorem 1 remains with additional conditions upon the coefﬁcients. Theorem 3. A multidimensional inequality measure I : Dþþ ! R satisﬁes UM, CIM, the decomposition property, and the unit-consistency axiom if and only if it is a positive multiple of the form " # n k k X Y Y r aj aj ðxij Þ ðmj Þ (6) IðXÞ ¼ Qk aj t n j¼1 mj i¼1 j¼1 j¼1 where t 2 R, r40, and aj o0; j ¼ 1; 2; . . . ; k. Proof. Following Tsui (1999, Theorem 4) it can be proved that the last two functional forms given by Eqs. (4) and (5) are incompatible with CIM, and moreover, the restrictions on the parameters of Eq. (3) can be derived. Theorem 4. A multidimensional inequality measure I : Dþþ ! R satisﬁes UM, CIM, the subgroup-consistency principle and the unit-consistency axiom if and only if there exists a continuous increasing transformation F : R ! Rþ , with F(0) ¼ 0, such that for any X 2 M þþ ðn; kÞ, JðXÞ ¼ FðIðXÞÞ is another multidimensional inequality measure satisfying UM, CIM, the decomposition property, and the unit-consistency axiom. Proof. If I in Theorem 2 also satisﬁes CIM, since F is strictly increasing, then J satisﬁes CIM. Thus, J is a multidimensional index that belongs to the class characterized in Theorem 3. This proves the necessity of the theorem. Once again, the sufﬁciency of this theorem is straightforward simply considering the function G ¼ F 1 and J satisfying UM, CIM, the decomposition property, and the unit-consistency axiom. Some remarks about the families derived in Theorems 1–4. (i) Assuming the most usual criteria, we have derived the family of subgroup- and unit-consistent multidimensional inequality measures. As already mentioned, unit consistency is an appropriate requirement in the sense that it only demands that inequality orderings are not altered when the units in which attributes are measured change. On the other hand, if the population is split into groups, the subgroup consistency is also a suitable requirement that only demands that overall inequality should increase if inequality in one group increases.

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(ii)

(iii)

(iv)

(v)

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Then in empirical applications, it makes sense to choose measures from these families. If only one attribute is taken into consideration, the families characterized in Theorems 1 and 2 coincide with the families identiﬁed by Zheng (2005). Hence, the families derived in this section are multidimensional generalizations of the unit-consistency measures identiﬁed by Zheng. Moreover, if t ¼ 0, the family identiﬁed in Theorem 2 coincides, up to a constant, with the generalized entropy family, and interestingly enough, the subfamily fulﬁlling CIM, Theorem 4, corresponds to the tail of this family, which meets the transfer-sensitive principle according to Shorrocks and Foster (1987). As usual, the decomposable measures identiﬁed in Theorems 1 and 3 can be considered ‘‘canonical forms’’ of the subgroup- and unitconsistent measures. As shown in Proposition 2 for these canonical t forms, IðXLÞ ¼ P1ik li IðXÞ holds. As a consequence, they are relative measures if and only if t ¼ 0. These cases correspond exactly with the two families that Tsui (1999) characterizes in Theorems 3 and 4. In other words, the families obtained in this paper are extensions of the two respective classes derived by Tsui (1999). In addition, maintaining t ¼ 0 and taking a suitable increasing function F in Theorem 2, we obtain the multidimensional generalization of the relative Akinson–Kolm–Sen index (Tsui, 1995). On the other hand, when tW0, inequality increases when any attribute is increased for all people in the same proportion. These measures represent points of view designated as ‘‘variable views’’ according to Amiel and Cowell (1997) since the value judgments represented by these measures can vary from the intermediate to the extreme leftist, depending on different distributions. In contrast, an extreme rightist view holds when to0 since, in these same situations, inequality decreases. As regards absolute measures, it can be proved that none of the members identiﬁed in Theorem 4 fulﬁls the translation invariance principle, even if only one of the attributes is affected by an absolute change.5 In other words, in empirical applications, if researchers consider attributes for which it makes sense to have relative changes that do not alter inequality rankings, they should be aware that, in these cases, it is not possible to take into consideration at the same time categorical variables for which absolute changes are bound to alter inequality values.

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MULTIDIMENSIONAL UNIT-CONSISTENT POVERTY MEASURES Notations and Basic Axioms of Multidimensional Poverty Measures Let’s consider a population consisting of nZ2 individuals endowed with a bundle of kZ1 attributes. A multidimensional distribution among the population is represented by an n k matrix X, whose ijth entry is xij, representing the ith individual’s amount of the jth attribute. The ith row is xi and the jth column is x j . In the poverty ﬁeld, contrary to the inequality one, it makes sense to assume negative values for any attribute; in particular, it is not unusual to ﬁnd negative income in poor households.6 Thus, if the attributes may take any real value, we denote by M(n, k) the class of n k matrices and let D ¼ [n2Nþ [k2Nþ Mðn; kÞ. Then, if the attributes should be positive, the class of matrices is restricted to Dþþ ¼ [n2Nþ [k2Nþ M þþ ðn; kÞ where Mþþ(n, k) is the class of n k matrices such that xij 40. The main difference between inequality and poverty lies in their focus. Whereas inequality measurement involves the whole population, the concern of poverty is focused on those who are worse off. Thus, the ﬁrst step in measuring poverty is the identiﬁcation of the poor through the speciﬁcation of a poverty line. For doing so, let’s consider zj 40 to be the minimum level of subsistence of the jth attribute and z ¼ ðz1 ; z2 ; . . . ; zk Þ 2 Rkþþ the k-vector of thresholds. For any X 2 D and z 2 Rkþþ , we denote by QðX; zÞ ¼ fi=xij zj for some jg the set of poor persons of cardinality q.7 We deﬁne XðzÞ 2 Mðq; kÞ as the matrix derived from X by selecting only rows xi such that i 2 QðX; zÞ. Once the poverty line is drawn and the poor people identiﬁed, any poverty measure, which according to Sen (1976) should be sensitive to the number of poor people, the intensity of the poverty, and the inequality among the poor,8 becomes somehow an inequality measure. In this paper, we assume that a multidimensional poverty index is a nonconstant function P : D Rkþþ ! R, which possesses the following basic properties: Anonymity: PðX; zÞ ¼ PðPX; zÞ for all n n permutation matrices P. Normalization: PðX; zÞ ¼ 0 if and only if xij zj for all i and j. Replication Invariance: PðY; zÞ ¼ PðX; zÞ if Y is obtained from X by a replication. Monotonicity: PðY; zÞ PðX; zÞ whenever Y is derived from X by increasing any one attribute with respect to a person who is poor.

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Focus: P remains unchanged if any attribute such that xij zj is increased for a person i. Restricted Continuity: P is a continuous function in any poor individual’s attributes and in the poverty line. Regarding properties assuring that the function P is sensitive to the inequality within the poor, the following introduced by Tsui (2002) is assumed: Poverty-Nonincreasing Minimal Transfer Axiom with Respect to the Uniform Majorization Principle (PNTUM) : A multidimensional poverty measure P satisﬁes PNTUM if PðY; zÞ PðX; zÞ whenever Y is derived from X by redistributing the attributes of the poor, using a bistochastic transformation (and not permutation) matrix. Intuitively, if it is possible to go from X to Y by simply redistributing the attributes of the poor through a bistochastic matrix, then each poor person’s row of Y is a linear convex combination of the rows corresponding to the poor of X, and clearly, the distribution Y becomes more equal within the poor than X and thus contains less poverty. The following property, also suggested by Tsui (2002), takes into account the correlation between the attributes of the poor:9 Poverty-Nondecreasing Rearrangement (PNR): A multidimensional poverty measure P satisﬁes PNR if PðX; zÞ PðY; zÞ whenever Y is derived from X by a permutation of rows and a ﬁnite sequence of correlationincreasing transfers among the poor with no one becoming nonpoor due to the transfers. The interpretation of this principle is similar to the inequality framework. Intuitively, if a poor individual having less of one attribute also has less of the rest of the attributes after the transfer, then inequality within the poor and consequently poverty should increase or at least be nondecreasing after the process.10 In empirical applications, if the population is split into groups according to certain socioeconomic and demographic characteristics, a desirable condition is that overall poverty be sensitive to poverty in any subgroup. This property proposed by Foster and Shorrocks (1991) in the unidimensional framework may be easily extended to the multidimensional setting: Subgroup-Consistency Axiom: A multidimensional poverty measure P is subgroup consistent if for any " # " # Y1 X1 and Y ¼ X¼ X2 Y2

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where X 1 ; Y 1 2 Mðn1 ; kÞ and X 2 ; Y 2 2 Mðn2 ; kÞ, PðX; zÞ4PðY; zÞ whenever PðX 1 ; zÞ4PðY 1 ; zÞ and PðX 2 ; zÞ ¼ PðY 2 ; zÞ. Moreover, poverty measures that fulﬁl the following decomposability property allow us to calculate the contribution of each subgroup to overall poverty: Subgroup Decomposability: If a population is classiﬁed in G nonempty subgroups, the multidimensional poverty measure P is subgroup decomposable ifP the following relationship holds: PðX; zÞ ¼ PððX 1 ; X 2 ; . . . ; X G Þ; zÞ ¼ G g¼1 ðng =nÞPðX g ; zÞ, where PðX g ; zÞ is the poverty of subgroup g for g ¼ 1; . . . ; G. A subgroup-decomposable poverty measure is clearly subgroup consistent. Generally, poverty comparisons involve distributions that may use different units of measures. The following property ensures that the poverty levels do not change when different units are used:11 Scale Invariance Principle: A multidimensional poverty measure P is scale invariant if for any X 2 D and z 2 Rkþþ PðX; zÞ ¼ PðXL; z LÞ for all k k diagonal matrices L with lj 40. Relative indices are those that are scale invariant. Following Tsui’s (2002) axiomatic framework, we also assume the following property concerning the robustness of the poverty line: Poverty Criteria Invariance (PCI): A multidimensional poverty measure P satisﬁes PCI if let z and z0 be such that z az0 , then PðX; zÞ PðY; zÞ if and only if PðX; z0 Þ PðY; z0 Þ whenever XðzÞ ¼ Xðz0 Þ and YðzÞ ¼ Yðz0 Þ. This property ensures that changes in the poverty line that do not alter the distribution of the poor do not change poverty rankings.12 The following notation is going to be used: Deﬁnition. For any k-vectors u and v, the two operators 4 and 3 are deﬁned as follows: u ^ v ¼ ðminfu1 ; v1 g; . . . ; minfuk ; vk gÞ and u _ v ¼ ðmaxfu1 ; v1 g; . . . ; maxfuk ; vk gÞ. Unit-Consistency Axiom for Multidimensional Poverty Measures As already mentioned, in the unidimensional setting, Zheng (2007b) has proposed a new axiom of unit consistency that requires that the poverty ranking between two distributions should not be affected by the unit in which the income and the poverty line are expressed. This axiom has a

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straightforward generalization to the multidimensional framework, allowing several attributes and poverty lines to be measured in different units without changing the poverty orderings. The natural generalization of this axiom, which we propose, is the following: Unit-Consistency Axiom: A multidimensional poverty measure P is unit consistent if, for any two distributions X; Y 2 Mðn; kÞ and two given vectors of thresholds z; z0 2 Rkþþ , if PðX; zÞoPðY; z0 Þ, then PðXL; z LÞo PðYL; z0 LÞ for all k k diagonal matrices L with lj 40. Clearly, the scale invariance principle implies unit consistency, and hence, all relative multidimensional poverty measures are unit consistent. However, none of the non relative multidimensional poverty measures traditionally used in the literature fulﬁls this property. In fact, examples can be found to prove that the unit-consistency axiom is not fulﬁlled by, among others, the members that are not relative in the Maasoumi and Lugo (2008) poverty family and the absolute poverty measures proposed by Tsui (2002). For doing so, let’s consider the following pair of distributions: X¼

6

4

7

6

and

Y¼

40

0

0

1

2

16

15

6

and the diagonal matrix L¼

taking the vector of thresholds z ¼ ð10; 10Þ. As regards Maasoumi and Lugo-mentioned family, let’s choose the parameter values y ¼ 1 and w1 ¼ w2 ¼ 0.5 [in order to compute Si in Eq. (2.2)] and a ¼ 1 [Eq. (2.6)]. Then we get PðXÞ ¼ 0:854PðYÞ ¼ 0:1, whereas PðXLÞ ¼ 0:70oPðYLÞ ¼ 0:76. With respect to Tsui’s absolute poverty measures, taking r1 ¼ c1 ¼ 0.5 and r2 ¼ c2 ¼ 0.5, we obtain [according to Eq. (6a)] P4 ðXÞ ¼ 89:764 P4 ðYÞ ¼ 29:99, whereas P4 ðXLÞ ¼ 5:5 1035 oP4 ðYLÞ ¼ 1:5 1069 and [Eq. (6b)] P5 ðXÞ ¼ 4:254P5 ðYÞ ¼ 3, but P5 ðXLÞ ¼ 72:5oP5 ðYLÞ ¼ 120. Once again, it is not difﬁcult to prove that all the rest of the members of these families may also violate the unit-consistency property.

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Multidimensional Unit-Consistent Poverty Measures In this section, we characterize classes of multidimensional poverty measures that are unit consistent. Most of the proofs in this section follow Zheng (2005, 2007a), Tsui (2002), and Foster and Shorrocks (1991) and are not included. All of them are available on request. First of all, the implications of unit consistency on a multidimensional poverty measure are explored in Propositions 3 and 4. The results we obtain are generalizations of those established by Zheng (2007b) in the unidimensional setting and analogous to the results obtained in the previous section for inequality measures. Proposition 3. A multidimensional poverty index P : D Rkþþ ! R is unit consistent if and only if for any distribution X 2 Mðn; kÞ, z 2 Rkþþ , and k k diagonal matrix L with lj 40, there exists a continuous function f : Rkþþ R ! R increasing in the last argument such that PðXL; z LÞ ¼ f ðl1 ; l2 ; . . . ; lk ; PðX; zÞÞ

(7)

Proof. This proof is an extension to the multidimensional setting of that of Proposition 1 of Zheng (2007b). Proposition 4. A multidimensional subgroup-decomposable poverty measure satisﬁes the unit-consistency axiom for any distribution X 2 Mðn; kÞ, for all z 2 Rkþþ , if and only if for any k k diagonal matrix L with lj 40, PðXL; z LÞ ¼ ðl1 l2 . . . lk Þt PðX; zÞ

(8)

for some constant t 2 R. Moreover, P is a homogenous function of degree kt. Proof. It is a generalization of the proof of Proposition 6 of Zheng (2007b). In Theorems 5 and 6, we consider the case in which all the attributes should be positive. In Theorem 6, we prove that any subgroup-consistent poverty measure is ordinally equivalent to a subgroup-decomposable measure, whose functional forms are identiﬁed in Theorem 5. In fact, this result is a generalization of Proposition 5 in Tsui (2002), replacing the scale invariance principle by the unit-consistency axiom. In turn, the indices derived can also be viewed as multidimensional generalizations of the

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unit-consistent Dalton–Hagenaars indices obtained by Zheng (2007b, Proposition 7). Theorem 5. A multidimensional poverty index P : Dþþ Rkþþ ! R satisﬁes the subgroup decomposability axiom, PNTUM, PNR, PCI, and the unit-consistency axiom if and only if, for any distribution X 2 M þþ ðn; kÞ and z 2 Rkþþ , it is a positive multiple of the form aj n Y X 1 zj PðX; zÞ ¼ Q 1 (9) 1 jk x ^ z n 1 jk zt ij j j i¼1 where t 2 R, aj 0, j ¼ 1,2,y,k, and the parameters aj have to be chosen such that the function fðyÞ ¼ P1 jk ðyj Þaj , yj 2 ð0; 1 is convex, or n X k X 1 zj (10) d ln PðX; zÞ ¼ Q j xij ^ zj n 1 jk zt j i¼1 j¼1 where t 2 R and dj 0, j ¼ 1,2,y,k. Proof. It is a particularization to the poverty ﬁeld of the proof of Proposition 4 of Zheng (2005), applying results of Tsui (2002). Theorem 6. A multidimensional poverty index P : Dþþ Rkþþ ! R satisﬁes the subgroup consistency axiom, PNTUM, PNR, PCI, and the unit-consistency axiom if and only if there exists a strictly increasing transformation F : R ! Rþ , with Fð0Þ ¼ 0, such that for any distribution X 2 M þþ ðn; kÞ and z 2 Rkþþ , P0 ðX; zÞ ¼ FðPðX; zÞÞ is another multidimensional poverty measure satisfying the subgroup decomposability axiom, PNTUM, PNR, PCI, and the unit-consistency axiom. Proof. It is straightforward following Foster and Shorrocks (1991) and Zheng (2007a). As already mentioned in the poverty ﬁeld, it makes sense to assume negative values for any attribute. In the following theorem, this possibility is taken into account. The result we obtain is a generalization of Proposition 6 in Tsui (2002), replacing once again the scale invariance condition by the unit-consistency axiom. Theorem 7. A multidimensional poverty index P : D Rkþþ ! R satisﬁes the subgroup consistency axiom, PNTUM, PNR, PCI, and the unitconsistency axiom if only if, for any distribution X 2 Mðn; kÞ and

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z 2 Rkþþ , there is at most one attribute, say, the mth attribute, and an increasing transformation F : R ! Rþ , with Fð0Þ ¼ 0, such that " 8 # n > 1 1X xim ^ zm rm > > Q ; xim 40 > t 1 n > zm < 1jk zj i¼1 " (11) FðPðX; zÞÞ ¼

rm # n

X >

1 a x > im >

Q > 1 ; a 1; xim 0 > : 1jk zt n i¼1 zm

j for some m 2 f1; 2; . . . ; kg, rm 40, and t 2 R. Proof. In a similar way to the proofs of Theorems 5 and 6, we can derive the functional form Eq. (11), considering the functional form by Tsui (2002, Proposition 6). Some remarks about the families derived in Theorem 6 and 7. (i) Similarly to the previous section, in this case, the indices derived in Theorems 5 and 6 are subgroup and unit consistent, both of them being minimal requirements in empirical applications in which poverty in a population split into groups is measured. (ii) The measures obtained in Theorem 6 are increasing transformations of the indices derived in Theorem 5, which can be considered ‘‘canonical forms’’ of the subgroup- and unit-consistent poverty measures . After Proposition 4, these indices are relative if and only if t ¼ 0, corresponding to the relative poverty indices derived by Tsui (2002). In other words, the families obtained in this section are extensions of the respective ones characterized by Tsui (2002). (iii) With respect to the measures derived in Theorem 7, assuming the attributes may be even negative, they depend on at most one attribute, as corresponds to those derived in Tsui (2002). (iv) Similarly to the inequality framework, an extreme rightist view holds when to0 since, in these cases, poverty decreases when any attribute and the respective poverty line are increased by the same proportion. On the contrary, when t40, the same transformations increase poverty. Thus, the value judgments can vary from the intermediate to the extreme leftist, depending on different distributions. (v) Any increasing transformation of the functional form given by Eq. (10) is a multidimensional generalization of the poverty index of Watts (1968). These functional forms satisfy CIM in the weakness sense: the value of the measure remains without change after a

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correlation-increasing transfer. If poverty should strictly increase, then Eq. (10) should be eliminated.

CONCLUDING REMARKS Most of the characterization results in the multidimensional framework assume the scale invariance principle in both the inequality and poverty ﬁelds. In the unidimensional setting, B. Zheng has introduced a weaker axiom, the unit-consistency principle, and generalized many of the previous results. In this paper, we have proposed a multidimensional generalization of the unit-consistency principle, and replacing the scale invariance property by this new proposal, we have derived generalizations of measures already existing in the literature. Speciﬁcally, we have characterized inequality and poverty multidimensional measures that are subgroup consistent and fulﬁl unit consistency. In empirical applications concerned with the measure of inequality or poverty in a population classiﬁed into groups, both the subgroup- and unitconsistency axioms are appropriate requirements for any measure. The families identiﬁed in this paper meet these two properties and allow us to adopt different value judgments in the measurement. We hope that our paper will be a contribution to this ﬁeld.

NOTES 1. For surveys of the literature on multidimensional inequality, see Weymark (2006), Savaglio (2006), Lugo (2005), and Maasoumi (1999) and on multidimensional poverty, see Maasoumi and Lugo (2008), Bibi (2003), Garcia (2003), and Chakravarty, Mukherjee, and Ranade (1998). 2. Apart from Kolm (1977), other generalizations of the Pigou–Dalton transfer principle to the multidimensional setting can be found in Marshall and Olkin (1979), Koshevoy and Mosler (2007), Fleurbaey and Trannoy (2003), Savaglio (2006), and Diez, Lasso de la Vega, Sarachu, and Urrutia (2007). 3. Bourguignon and Chakravarty (2003) make some objections to this axiom, arguing that CIM is not sensitive to individual preferences and somehow implies that the attributes are substitutable. In turn, Tsui (1999, 2002) highlights what CIM really means in the context of both inequality and poverty. 4. Actually, properties of this kind have already been proposed in the literature as regards the social welfare functions that underlie the multidimensional relative indices (Tsui (1995) and Gajdos and Weymark (2005), for instance). 5. In fact, it can be proved that given any n k matrix A with identical rows a ¼ ða1 ; a2 ; . . . ; ak Þ and aj 0, then @IðX þ AÞ=@al jal ¼0 ¼ 0 if and only if

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P a a tP1 jk mj j ¼ ð1=nÞ 1in P1 jk xijj ðml al xil ðal tÞÞ=xil , but this is impossible since the right-side term, taking into account that aj o0, tends to N when xil tends to 0, whereas the left-side term is a constant. 6. Zheng (1997) has already discussed this problem in the unidimensional setting. 7. This deﬁnition corresponds to measures that take into account the union of the various aspects of deprivation as those considered by Tsui (2002) and Bourguignon and Chakravarty (2003), among others. In contrast, Chakravarty et al. (1998) opt for measures based on the intersection. Nevertheless, all the results of this work can be established with any other deﬁnition of the poor. 8. These are the well-known three ‘I’s of poverty according to Jenkins and Lambert (1998a, 1998b). 9. The deﬁnition of a correlation-increasing transfer has been presented in Notations and Basic Axioms of Multidimensional Inequality Measures. 10. Objections to this axiom are similar to those pointed out with respect to CIM in the inequality section. 11. In the multidimensional framework, only Tsui (2002) uses another invariance property, the translation invariance principle, analogous to the axiom assumed in the inequality ﬁeld, to characterize absolute poverty measures. 12. Tsui (2002) argues that the multidimensional generalization of the class of Foster–Greer–Thorbecke poverty measures proposed by Bourguignon and Chakravarty (2003) is incompatible with PCI.

ACKNOWLEDGMENTS We would like to thank Professor Peter Lambert for having introduced us to Zheng’s work and for his useful comments. We are also grateful to Professor Buhong Zheng for his encouragement and advice. Preliminary versions of this paper were presented in both the Second Meeting of ECINEQ (Berlin, 2007) and the 5th International Conference on Logic, Game Theory and Social Choice (Bilbao, 2007). We wish to thank the participants in these two conferences for their suggestions and comments. This research has been partially supported by the Spanish Ministerio de Educacio´n y Ciencia under the project SEJ2006-05455, cofunded by FEDER, by the University of the Basque Country under the project UPV05/117, and by the Basque Departamento de Educacio´n Universidades e Investigacio´n under the project GIC07/146-IT-377-07.

REFERENCES Amiel, Y., & Cowell, F. (1997). Income transformation and income inequality. Distributional analysis. WP 24 STICERD. School of Economics, London.

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Atkinson, A. B., & Bourguignon, F. (1982). The comparison of multi-dimensioned distributions of economic status. The Review of Economic Studies, 49, 183–201. Bibi, S. (2003). Measuring poverty in a multidimensional perspective: A review of literature. WP PMMA 2005-07. Faculte´ des Sciences E´conomiques et de Gestion de Tunis, CIRPE´E, Universite´ Laval, Que´bec, Canada. Boland, P. J., & Proschan, F. (1988). Multivariate arrangement increasing functions with applications improbability and statistic. Journal Multivariate Analysis, 25, 286–298. Bourguignon, F. (1979). Decomposable income inequality measures. Econometrica, 47, 901–920. Bourguignon, F. (1999). Comment to ‘‘Multidimensioned approaches to welfare analysis’’ by Maausoumi, E. In: J. Silber (Ed.), Handbook on income inequality measurement (pp. 477–484). Boston: Kluwer Academic Publishers. Bourguignon, F., & Chakravarty, S. (2003). The measurement of multidimensional poverty. Journal of Economic Inequality, 1, 25–49. Chakravarty, S. R., Mukherjee, D., & Ranade, R. R. (1998). On the family of subgroup and factor decomposable measures of multidimensional poverty. Research on Economic Inequality, 8, 175–194. Cowell, F. A. (1980). Generalized entropy and the measurement of distributional change. European Economic Review, 13, 147–159. Diez, H., Lasso de la Vega, C., Sarachu, A., & Urrutia, A. (2007). A consistent multidimensional generalization of the Pigou-Dalton transfer principle: An analysis. The B.E. Journal of Theoretical Economics, 7(1), article 45. Available at: http:// www.bepress.com/bejte/vol7/iss1/art45 Fleurbaey, M., & Trannoy, A. (2003). The impossibility of a Paretian egalitarian. Social Choice and Welfare, 21, 243–263. Foster, J., & Shorrocks, A. (1991). Subgroup consistent poverty indices. Econometrica, 59, 687–709. Gajdos, T., & Weymark, J. A. (2005). Multidimensional generalized Gini indices. Economic Theory, 26, 471–496. Garcia, R. (2003). Multidimensional poverty. WP YO10 5DD. Department of Economics and Related Studies, University of York. Hardy, G. H., Littlewood, J. E., & Po´lya, G. (1934). Inequalities. Cambridge: Cambridge University Press. Jenkins, S., & Lambert, P. (1998a). Ranking poverty gap distributions: Further TIPs for poverty analysis. Research on Economic Inequality, 8, 31–38. Jenkins, S., & Lambert, P. (1998b). Three ‘I’s of poverty curves and poverty dominance: TIPs for poverty analysis. Research on Economic Inequality, 8, 39–56. Kolm, S. Ch. (1977). Multidimensional egalitarianisms. The Quarterly Journal of Economics, 91, 1–13. Koshevoy, G. A., & Mosler, K. (2007). Multivariate Lorenz dominance based on zonoids. Advances in Statistical Analysis, 21(1), 57–76. Lugo, M. A. (2005). Comparing multidimensional indices of inequality: Methods and application. WP. Department of Economics, University of Oxford. Maasoumi, E. (1999). Multidimensioned approaches to welfare analysis. In: J. Silber (Ed.), Handbook on income inequality measurement (pp. 437–477). Boston: Kluwer Academic Publishers.

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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. Palgrave Macmillan. Marshall, A., & Olkin, I. (1979). Inequalities: Theory of majorization and its applications. New York: Academic Press. Savaglio, E. (2006). Three approaches to the analysis of multidimensional inequality. In: F. Farina & E. Savaglio (Eds), Inequality and economic integration. London: Routledge. Sen, A. K. (1976). Poverty: An ordinal approach to measurement. Econometrica, 44, 219–231. Shorrocks, A. F. (1980). The class of additively decomposable inequality measures. Econometrica, 48, 613–625. Shorrocks, A. F. (1984). Inequality decomposition by population subgroups. Econometrica, 52, 1369–1385. Shorrocks, A. F., & Foster, J. E. (1987). Transfer sensitive inequality measures. The Review of Economic Studies LIV, 1, 485–497. Tsui, K. Y. (1995). Multidimensional generalizations of the relative and absolute inequality index: The Atkinson-Kolm-Sen approach. Journal of Economic Theory, 67, 251–265. Tsui, K. Y. (1999). Multidimensional inequality and multidimensional generalized entropy measures: An axiomatic derivation. Social Choice and Welfare, 16, 145–157. Tsui, K. Y. (2002). Multidimensional poverty indices. Social Choice and Welfare, 19, 69–93. Walzer, M. (1983). Spheres of justice. New York: Basic Books. Watts, H. W. (1968). An economic deﬁnition of poverty. In: D. P. Moynihan (Ed.), On understanding poverty. New York: Basic Books. Weymark, J. A. (2006). The normative approach to the measurement of multidimensional inequality. In: F. Farina & E. Savaglio (Eds), Inequality and economic integration. London: Routledge. Zheng, B. (1997). Aggregate poverty measures. Journal of Economic Surveys, 11, 123–162. Zheng, B. (2005). Unit-consistent decomposable inequality measures: Some extensions. WP 2005-2. Department of Economics, University of Colorado at Denver and HSC. Zheng, B. (2007a). Unit-consistent decomposable inequality measures. Economica, 74, 97–111. Zheng, B. (2007b). Unit-consistent poverty indices. Economic Theory, 31, 113–142.

RANKINGS OF INCOME DISTRIBUTIONS: A NOTE ON INTERMEDIATE INEQUALITY INDICES Coral del Rı´ o and Olga Alonso-Villar ABSTRACT The purpose of this paper is to analyze the advantages and disadvantages of several intermediate inequality measures, paying special attention to the unit-consistency axiom proposed by Zheng (2007). First, we demonstrate why one of the most referenced intermediate indices, proposed by Bossert and Pﬁngsten (1990), is not unit-consistent. Second, we explain why the invariance criterion proposed by Del Rı´o and RuizCastillo (2000), recently generalized by Del Rı´o and Alonso-Villar (2008), leads instead to inequality measures that are unaffected by the currency unit. Third, we show that the intermediate measures proposed by Kolm (1976) may also violate unit-consistency. Finally, we reﬂect on the concept of intermediateness behind the above notions together with that proposed by Krtscha (1994). Special attention is paid to the geometric interpretations of our results.

Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 213–229 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16010-0

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INTRODUCTION There is a wide consensus in the literature about the properties an inequality measure has to satisfy when using it to compare income distributions having the same mean. Basically, we must invoke the symmetry axiom – which guarantees anonymity – and the Pigou–Dalton principle of transfers – which requires a transfer of income from a richer to a poorer person to decrease inequality.1 However, if we are interested in comparing two income distributions that have different means, we need to specify the type of mean-invariance property we want our inequality indices to satisfy. This requires introducing another judgment value into the analysis, and no agreement has been reached among scholars with respect to this matter. Some opt to invoke the scale invariance axiom, according to which the inequality of a distribution remains unaffected when all incomes increase (or decrease) by the same proportion. This is the approach followed by the relative inequality indices. Others prefer, instead, to call on the translation invariance axiom, under which inequality remains unaltered if all incomes are augmented (or diminished) by the same amount, thereby giving rise to the absolute inequality measures. However, as Kolm (1976) pointed out, some people may prefer an intermediate invariance approach between these two extreme views. He labeled such an inequality attitude as ‘‘centrist,’’ against the ‘‘rightist’’ and ‘‘leftist’’ labels he used to term the aforementioned relative and absolute notions, respectively. So far, the intermediate and absolute inequality indices have rarely been applied to ranking income distributions since these measures are cardinally affected by the currency unit in which incomes are expressed. In a recent paper, Zheng (2007) invoked a new axiom, the unit-consistency axiom, requiring that inequality rankings between income distributions remain unchanged when all incomes are multiplied by a (positive) scalar.2 In this new scenario, not only relative measures but also absolute and intermediate measures that satisfy the unit-consistency axiom appear as plausible options for empirical research. The purpose of this paper is to analyze the advantages and disadvantages of several intermediate inequality measures, paying special attention to the unit-consistency axiom. First, we demonstrate why one of the most referenced intermediate indices, proposed by Bossert and Pﬁngsten (1990) (B-P hereafter), is not unit-consistent. A geometric interpretation of this result is also given. This analysis reveals that the problem lies in the isoinequality criteria behind that index, which helps to explain why the

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decomposable intermediate inequality measures a` la B-P proposed by Chakravarty and Tyagarupananda (2008) do not satisfy unit-consistency either, as shown by Zheng (2007). Second, we explain both analytically and graphically why the invariance criterion proposed by Del Rı´ o and Ruiz-Castillo (2000), recently generalized by Del Rı´ o and Alonso-Villar (2008), leads instead to inequality measures that are unaffected by the currency unit. Third, we demonstrate that the intermediate measures proposed by Kolm (1976) may also violate unit-consistency. Finally, we reﬂect on the concept of intermediateness behind the above notions together with the ‘‘fair compromise’’ notion proposed by Krtscha (1994), closely examining their geometric interpretations.

UNIT-CONSISTENCY AND INTERMEDIATE INEQUALITY MEASURES In order to ensure independence of the unit of measurement without imposing scale invariance, Zheng (2007) introduced the following property into inequality measures:3 Unit Consistency For any two distributions x; y 2
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for any nZ2, x 2
As shown in Fig. 1 for n ¼ 2 and m ¼ 0.25, for a given income distribution x 2 0.015 ¼ I0.5(y), but I0.5(10x) ¼ 0.122o0.148 ¼ I0.5(10y), where I m ðxÞ ¼ ð1 þ sÞ½1 Pni¼1 ððxi þ sÞ=ðx þ sÞÞ1=n ; s ð1 mÞ=m and x represents the mean of distribution x.

x2

- Invariance line

y

x

x + (1 - )1n

2

(1, 1) 0

Fig. 1.

2

Invariance in B-P (n ¼ 2, m ¼ 0.25).

x1

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In Fig. 2, we illustrate why this popular intermediate inequality equivalence criterion leads to measures that do not satisfy the unitconsistency axiom. A formal proof is given in the appendix. Thicker dash lines represent the two m-invariance lines passing through points x and 2x, that is, the set of distributions equivalent to x and 2x, respectively. Vector y represents an income distribution that is equivalent to x, since it is located on the invariance line of the latter. It is easy to see that any distribution between y and z has a larger inequality level than x because of the Pigou–Dalton transfer axiom. However, distributions resulting from doubling their individual incomes (which are located between 2y and 2z) would have instead a lower inequality level than distribution 2x.

2z x2

2y

2x z y

x

2 (1, 1) 0

(

(− 1- ,− 1-

2

Fig. 2. Unit Consistency in B-P (n ¼ 2, m ¼ 0.25).

x1

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Therefore, changes in the currency unit do affect rankings between income distributions. The above graphical analysis permits us to illustrate that the aforementioned ﬁve-dimensional example was not an isolated one. We have shown that, even in a two-dimension space, for any given income distribution it is possible to ﬁnd an interval of distributions that violate the axiom when comparing them with the former distribution. The explanation of this behavior relies on the notion of inequality equivalence proposed by B-P. The slope of the inequality-invariance line given by direction mx þ (1 m)1n does depend on the total income of distribution x. In fact, keeping the relative inequality as constant, the larger the total income, the larger this slope (the slope of the invariance line corresponding to 2x is larger than that of x, as shown in Fig. 2). This means, ﬁrst, that m represents a different intermediate inequality attitude depending on the distribution on which the index is evaluated. Since the invariance lines are, therefore, not parallel, it is impossible to state that m-inequality rankings are not affected by changes in the scale when comparing any two distributions.7 Thus, we have shown that the heart of this equivalence criterion is incompatible with the unit-consistency axiom, so that any measure based on this notion violates this axiom. Second, the m-inequality concept approaches the ‘‘rightist’’ view of inequality (the invariance line becomes closer to the relative ray) when aggregate income rises (see Fig. 2).8 This means that results obtained by using this intermediate concept can be quite close to those obtained with relative measures, which can be seen as unsuitable for a ‘‘centrist’’ measure. Moreover, in Fig. 3, which shows the m-isoinequality contours corresponding to distribution x ¼ (20, 80) for two m values, we see that the invariance line corresponding to m ¼ 0.5 is roughly indistinguishable from the ray passing by x (which deﬁnes the isoinequality line of relative measures). In fact, to obtain an isoinequality contour closer to the ‘‘leftist’’ view (i.e., closer to the absolute ray), it would be necessary to choose a parameter value extraordinary low (for example m ¼ 0.005).9 This suggests that parameter m, even though it takes a value between 0 and 1, has not a clear economic interpretation, since its value does not give us any idea of the invariance line location. In particular, in the above example, m ¼ 0.5 does not represent an equidistant position between the relative and absolute rays, but is instead a position close to the ‘‘rightist’’ view. If one is interested in deﬁning a linear ‘‘centrist’’ measure as a convex combination of a relative and an absolute ray, one could ﬁx not only parameter m, but also the reference distribution that gives rise to the

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Absolute ray

Relative ray

B-P (Mu = 0.005)

B-P (Mu = 0.5)

DR-AV (Pi = 0.5, v = (0.2, 0.8))

DR-AV (Pi = 0.75, v = (0.2,0.8))

600

DR-AV (Pi = 0.25, v = (0.2, 0.8))

Fig. 3.

Isoinequality Contours Corresponding to Distribution (20, 80): Linear Cases.

‘‘rightist’’ and ‘‘leftist’’ views. In this regard, Del Rı´ o and Ruiz-Castillo (2000) (DR-RC hereafter) proposed the (v, p)-inequality, where v is a vector belonging to the n-dimensional simplex, and pA[0, 1]. The ﬁrst component ﬁxes the distribution of reference, while the second refers to the convex combination of the relative and absolute rays associated to v.10 Thus, if we chose a value of p close to 1, the notion represents value judgments rather ‘‘rightist,’’ while if p is close to 0, the inequality attitude is rather ‘‘leftist,’’ as compared to the distribution of reference. Once these two components are ﬁxed, we can calculate the n-dimensional simplex vector a ¼ pv þ (1 p)(1/n)1n (see Fig. 4), which deﬁnes the direction of the inequality equivalence ray, and the set of income distributions Gu(a) for which a represents an intermediate attitude.11 This set can be expressed as follows: Gu(a) ¼ {xAD: pxvx þ (1 px)(1n/n) ¼ a, for some pxA[0, 1]}, where D is the set of all possible ordered income

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P(v,) (x) = {y ∈ D : y = x + , ∈ }

X2 x

45°-line

1 v vx

X1

1

Fig. 4.

Invariance in DR-RC (n ¼ 2, p ¼ 0.25).

distributions, and vx represents the vector of income shares associated to x (it therefore belongs to the n-dimensional simplex). If we want vector a to represent an intermediate notion, it must be Lorenz-dominated by the egalitarian distribution. On the other hand, this vector can only be used for income distributions that are (weakly) Lorenz-dominated by it. Thus, an intermediate inequality index is deﬁned as (v, p)-invariant in the set of income distributions Gu(a) if for any xAGu(a) the following expression holds: I ðv;pÞ ðxÞ ¼ I ðv;pÞ ðyÞ;

for any y 2 Pðv;pÞ ðxÞ

where Pðv;pÞ ðxÞ ¼ y 2 D : y ¼ x þ tðpv þ ð1 pÞð1n =nÞÞ; t 2 < represents the inequality-invariance line. Note that this line is obtained as the convex combination, given by p, of the ‘‘leftist’’ and ‘‘rightist’’ views associated with vector v. On the other hand, since xAGu(a), the invariance line can also be obtained as a convex combination of the ‘‘leftist’’ and ‘‘rightist’’ views associated to vector vx by using px. The distribution of reference, v, plays a very important role in this approach. Note that vector v does not necessarily have to coincide with vector vx (as shown in Fig. 4). However, in comparing any two distributions

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x and y (which can be assumed to have a higher mean without loss of generality), vector v could be chosen as the income shares of x, i.e., v ¼ vx. By using this benchmark, together with the parameter p reﬂecting the inequality-invariance value judgments of society, it would be possible to determine whether y has a lower inequality than the distribution reached if p 100% of the income gap had been distributed according to income shares in x and (1 p) 100% in equal amounts among individuals. Note that, in doing so, the same vector of reference (v ¼ vx) has to be used for both distributions x and y. It would not be possible to use v ¼ vx for measuring the inequality level corresponding to x, while using v ¼ vy in the case of distribution y, since that would imply that different inequality attitudes would be used for each distribution.12 In other words, once v and p are chosen, they cannot be changed: the same intermediate notion must be used when comparing any two income distributions.13 Therefore, when studying the evolution of an economy over time, this approach allows the possibility of taking into account the starting point.14 As opposed to B-P’s approach, the same vector a is now used for obtaining the isoinequality lines, which implies that the invariance lines passing through distributions x and 2x are parallel (see Fig. 5). Therefore, the (v, p)-invariance notion does not have the problem shown in Fig. 2. In other words, if distributions x and y are in the same invariance line, distributions 2x and 2y also share a common inequality level (see Fig. 5). In the appendix, we formally prove that according to the (v, p)-invariance concept, inequality rankings are unaffected by the monetary unit in which incomes are expressed. This can help explain why the family of indices based on this approach, proposed by Del Rı´ o and Alonso-Villar (2008) (DR-AV hereafter), does satisfy the unit-consistency axiom. The invariance lines corresponding to three of these indices are shown in Fig. 3, where pA{0.25, 0.5, 0.75} and v ¼ (0.2, 0.8). We can see that p ¼ 0.5 leads to an isoinequality contour that is ‘‘equidistant’’ from the ‘‘rightist’’ and ‘‘leftist’’ views of distribution (20, 80), when choosing the vector of reference v ¼ (0.2, 0.8). Nonlinear Invariance Criteria An alternative to the above intermediate notions is to assume that the isoinequality contours are not straight lines. In this regard, Krtscha (1994) proposes an adaptive intermediate notion that gives rise to parabolas. According to his ‘‘fair compromise’’ notion, to keep inequality unaltered, any extra income should be allocated among individuals in the following way. The ﬁrst extra dollar of income should be distributed so that 50 cents

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X2 P(v,) (2x) = {y ∈ D : y = 2x + , ∈ᑬ}

2y

2x

P(v,)(x) = {y ∈ D : y = x + , ∈ ᑬ}

y x

1

45°-line

v

1

Fig. 5.

X1

Invariance in DR-RC (n ¼ 2, p ¼ 0.25).

goes to the individuals in proportion to the initial income shares, and 50 cents goes in equal absolute amounts. The second extra dollar should be allocated in the same manner, starting now from the distribution reached after the ﬁrst dollar allocation, and so on. According to this invariance notion, two income p distributions, x and y, have the same inequality level ﬃﬃﬃ where y ¼ tx and the bar represents the so long as y y ¼ tðx xÞ, average of the corresponding distribution. The ‘‘fair compromise’’ index [and the generalizations proposed by Zheng (2007)] does satisfy unit-consistency and decomposability, as shown by the latter. However, this ‘‘centrist’’ attitude is rather challenging since it approaches the absolute view rather soon when income increases, which makes it difﬁcult for inequality to decrease when analyzing an economy over

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time. In Fig. 6, we can see that, according to Krstcha’s index, inequality would remain unaltered with respect to distribution (20, 80) if the poorer reached an income of 400 and the richer of 590, which would imply income shares of 40 and 60%, approximately. If we continued our previous simulation and plotted the invariance curve for larger income levels (which are not shown in Fig. 6), we would reach distribution (81,920, 84,367), which represents income shares of 49.3 and 50.7%, respectively. This proximity to the absolute view does not contradict, however, the tendency of this index to a relative inequality measure when income increases to inﬁnity, while keeping inequality constant, as shown by Zheng (2004). He proved that when moving repeatedly along an isoinequality 600

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Fig. 6.

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Absolute ray

Relative ray

Kolm (Xi = 10, Epsilon = 2)

Kolm (Xi = 0, Epsilon = 2)

Kolm (Xi = 10, Epsilon = 10)

Kolm (Xi = 0, Epsilon = 0.75)

Krtscha "fair compromise"

Zheng (Alpha = 2, Beta = 1.5)

Zheng (Alpha = 2, Beta = 0.5)

Isoinequality Contours Corresponding to Distribution (20, 80): Nonlinear Cases.

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contour, the curve becomes eventually a straight line, so that the intermediate notion becomes relative. This does not mean, however, that the isoinequality contour is close to the ‘‘rightist’’ view. A relative ray can be as close as wanted to the line representing total equity. In fact, when repeatedly moving along the invariance line, Zheng (2004) showed that pﬃﬃﬃ =xðkÞ since xiðkþ1Þ ¼ txðkÞ limk!1 ðxðkþ1Þ i Þ ¼ t. On thepﬃﬃﬃother hand, i þ pﬃﬃﬃ iðkÞ p ﬃﬃ ﬃ ðkÞ ðkÞ ðkÞ ðkÞ ðt tÞx , then ðxðkþ1Þ =x Þ ¼ t ½x þ ð t 1Þ x =x It is easy to i i i i prove that the limit of the above quotient tends to t if and only if limk!1 ðx ðkÞ =xðkÞ i Þ ¼ 1. Therefore, when moving repeatedly along the invariance line, distribution x(k) tends to the egalitarian distribution x ðkÞ . Kolm’s (1976) ‘‘centrist’’ measures also lead to isoinequality contours that are not straight lines.15 However, as opposed to Zheng’s family of indices, that of Kolm does not cover the whole intermediate space since, as shown in Fig. 6,16 ‘‘centrist’’ attitudes are close to the ‘‘leftist’’ view, while those near the ‘‘rightist’’ view are not permitted for any parameter value.17 On the other hand, Kolm’s ‘‘centrist’’ measures may violate the unit-consistency axiom when x 6¼ 0 (if x ¼ 0 the index is homogeneous of degree 1 and, therefore, it does satisfy the axiom). In this regard, if x ¼ 10 and e ¼ 10, for distributions x ¼ (2, 2, 6, 7, 7) and y ¼ (2, 2, 3, 8, 8), it follows I(10,10)(2x) ¼ 4.13W3.94 ¼ that I(10,10)(x) ¼ 1.63o1.66 ¼ I(10,10) (y) while 1=ð1Þ P . Therefore, I(10,10)(2y), where I ðx;Þ ðxÞ ¼ x þ x ð1=nÞ ni¼1 ðxi þ xÞ1 (x, e)-inequality rankings may be affected by currency units.

FINAL REMARKS The unit-consistency axiom, recently invoked by Zheng (2007), requires that inequality rankings between income distributions remain unaffected by the unit in which incomes are expressed. This axiom does not impose such strong value judgments on inequality measurement as the scale invariance condition, and therefore, intermediate indices satisfying it appear to be plausible options for empirical research. Intermediate measures are especially useful when comparing two income distributions, x and y, where (at the same time) y has a higher absolute inequality level and a lower relative inequality level than x, according to the absolute and the relative Lorenz criterion, respectively. We have revised the ‘‘centrist’’ measures offered by the literature in order to verify whether they are unit-consistent. We have shown that both the class of intermediate inequality indices proposed by Bossert-Pﬁngsten (1990) and Kolm (1976) are affected by the currency unit. Therefore, only the

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‘‘fair compromise’’ index proposed by Krtscha (1994), the generalizations proposed by Zheng (2007), and the indices proposed by Del Rı´ o and Alonso-Villar (2008) – which, as opposed to the others, are ray invariant – are intermediate inequality measures satisfying unit-consistency. One advantage of both the ‘‘fair compromise’’ index and the family of measures proposed by Zheng (2007) is that they are decomposable, which is helpful for empirical analysis. Krtscha’s index also has a clear economic interpretation, but the ﬂexibility of this approach, imposing an intermediate notion based on marginal changes with respect to each distribution, entails a challenging ‘‘centrist’’ attitude. The fact that the invariance set is a parabola approaching the absolute view makes it rather difﬁcult for inequality to decrease when analyzing an economy over time. The family of indices proposed by Del Rı´ o and Alonso-Villar (2008), which is based on the invariance notion put forward by Del Rı´ o and RuizCastillo (2000), also brings a clear economic interpretation of intermediateness while proposing a more conservative approach. Since the isoinequality contours are straight lines, the ‘‘centrist’’ attitude remains constant when income increases. Therefore, this approach does not allow any change in individuals’ value judgments regarding inequality when varying aggregate income, which seems plausible for analysis in the short and medium run, bringing a complementary perspective to the former.

NOTES 1. Properties such as normalization, continuity, differentiability, and replication invariance are also commonly invoked, but they are of a more technical nature. 2. Zheng (2007) characterized the entire class of unit-consistent decomposable inequality measures without imposing any invariance condition, so that relative, absolute, and intermediate measures are included in this class. Zheng (2005) also characterized theses measures when relaxing some assumptions (the differentiability assumption is replaced with continuity and decomposability is replaced with either aggregability or subgroup consistency). 3. Zoli (2003) also proposed an analogous property, named ‘‘weak currency independence,’’ in a context of inequality equivalence criteria. 4. The proposal by Zheng (2007) also includes extreme-rightist and extreme-leftist unit-consistent measures. 5. See Besley and Preston (1988), Chakravarty (1988), Ebert and Moyes (2000), and Lambert (1993), among others. 6. Note that the invariance line drawn in Fig. 1 should actually ﬁnish at x1 ¼ 0, but we have enlarged it in order to make the picture clearer. The same considerations apply to Fig. 2.

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7. This characteristic of the B-P invariance lines does not contradict the path independence axiom, since these lines do not intersect in the space of positive income distributions (the invariance lines cut at point ðð1 mÞ=m; ð1 mÞ=mÞ, as shown by B-P, see Fig. 2). Therefore, the distribution reached after allocating extra income among individuals, without altering inequality, remains the same whether the extra income has taken place in only one step or in several, which allows building indices based on this notion. However, if two lines cut in the ﬁrst quadrant, it would not be possible to construct inequality indices based on the invariance notion. 8. This tendency to the relative ray was initially pointed by Seidl and Pﬁngsten (1997) and Del Rı´ o and Ruiz-Castillo (2000). More recently, Zheng (2004) offered a formal proof. 9. This explains why Atkinson and Brandolini (2004) found similar empirical results either by using B-P’s index or relative indices, even when considering extremely low m-values (m ¼ 0.00273x ¼ 365 dollars). 10. We have changed their original notation in order to make it clearer. In particular, we have switched vector x by simplex vector v, since only the income shares of the distribution of reference are required to obtain the invariance ray. 11. DR-RC can be considered as a special case of Seidl and Pﬁngsten (1997) since the latter previously proposed a a-invariance concept according to which any extra income should be distributed in ﬁxed proportions, given by a, in order to keep inequality unaltered. Their vector does not have, however, a clear economic interpretation. On the other hand, their invariance concept does not satisfy horizontal equity, as discussed by Zoli (2003). 12. Certainly, the same applies if comparing three or more distributions. 13. Perhaps the notation employed by DR-RC led both Zoli (2003) and Zheng (2004) to the interpretation that v depends on distribution x, which is not the case. This confusion led to the former to conclude that DR-RC invariance notion does not satisfy the path independence axiom, when it really does (see Del Rı´ o & Alonso-Villar, 2008). It also led to the latter to conclude that DR-RC’s proposal tends to an absolute notion when a given transformation that keeps inequality unaltered is performed repeatedly, while it does not. When the linear transformation x þ ta is applied repeatedly to distribution x, its easy to see that in each iteration the difference between one distribution and the next one is equal to ta rather than t. 14. This method allowed Del Rı´ o and Ruiz-Castillo (2001) to compare income distributions in Spain between 1980 and 1990. By choosing v as the income shares corresponding to 1980’s distribution, they concluded that for those people whose opinions are closer to the relative inequality notion (that is, if pA[0.87, 1]), inequality would have decreased in Spain during that decade. However, for people more skewed towards the left side of the political spectrum (that is, if pA[0, 0.71]), it would be the opposite. 15. As in previous cases, Zheng (2004) proved that these curves become straight lines in the limit. 16. Kolm’s family of indices has isoinequality contours that monotonically approach the absolute ray as either x or e increases (if e>1). However, when eA[0, 1], there is no monotonicity with respect to this parameter. In this example, the contour closer to the relative ray (‘‘rightist’’ view) is that corresponding to x ¼ 0, e ¼ 0.75.

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17. Recent empirical evidence obtained by Atkinson and Brandolini (2004, p.13) seems to support this idea: ‘‘Kolm’s centrist measure basically conﬁrms the pattern shown by Kolm’s absolute measure.’’ 18. Distribution w exists since distributions x and y are in the same plane (which is deﬁned by the former and the egalitarian distribution), and the straight line and the invariance representing distributions in that plane having a common mean x, line corresponding to distribution x always intersect. 19. Another requirement is that one of the distributions has to belong to the intermediate space of the other (see deﬁnition of Gu(a) in Section Unit-consistency and intermediate inequality measures).

ACKNOWLEDGMENTS Financial support from the Ministerio de Educacio´n y Ciencia (grants SEJ2005-07637-C02-01/ECON and SEJ2007-67911-C03-01/ECON), from the Xunta de Galicia (PGIDIP05PXIC30001PN and Programa de Estruturacio´n de Unidades de Investigacio´n en Humanidades e Ciencias Sociais 2006/33) and from FEDER is gratefully acknowledged.

REFERENCES Atkinson, A., & Brandolini, A. (2004). Global world inequality: Absolute, relative or intermediate? Paper presented at the 28th General Conference of the International Association in Income and Wealth, Ireland. Besley, T., & Preston, I. (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. (1988). On quasi-orderings of income proﬁles. In: B. Fuchssteiner, T. Lengauer and H. Skala (Eds), Methods of operations research (Vol. 60, pp. 455–473). XIII Symposium of Operations Research, University of Paderborn. Chakravarty, S., & Tyagarupananda, S. (2008). The subgroup decomposable absolute and intermediate indices of inequality. Spanish Economic Review (Forthcoming). Del Rı´ o, C., & Alonso-Villar, O. (2008). New unit-consistent intermediate inequality indices. Economic Theory (Forthcoming). Del Rı´ o, C., & Ruiz-Castillo, J. (2000). Intermediate inequality and welfare. Social Choice and Welfare, 17, 223–239. Del Rı´ o, C., & Ruiz-Castillo, J. (2001). Intermediate inequality and welfare: The case of Spain 1980–81 to 1990–91. Review of Income and Wealth, 47, 221–237. Ebert, U., & Moyes, P. (2000). Consistent income tax structures when households are heterogeneous. Journal of Economic Theory, 90, 116–150. Kolm, S. C. (1976). Unequal inequalities I. Journal of Economic Theory, 12, 416–442.

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Krtscha, M. (1994). A new compromise measure of inequality. In: W. Eichorn (Ed.), Models and measurement of welfare and inequality (pp. 111–120). Heidelberg: Springer-Verlag. Lambert, P. (1993). The distribution and redistribution of income. A mathematical analysis. Oxford: Manchester University Press. Seidl, C., & Pﬁngsten, A. (1997). Ray invariant inequality measures. In: S. Zandvakili & D. Slotje (Eds), Research on taxation and inequality (pp. 107–129). Greenwich: JAI Press. Zoli, C. (2003). Characterizing inequality equivalence criteria. Mimeo, University of Nottingham. Zheng, B. (2004). On intermediate measures of inequality. Research on Economic Inequality, 12, 135–157. Zheng, B. (2005). Unit-consistent decomposable inequality measures: Some extensions. Working Paper Series No. 05-02, Department of Economics, University of Colorado. Zheng, B. (2007). Unit-consistent decomposable inequality measures. Economica, 74, 97–111.

APPENDIX Result #1. According to the inequality-invariance notion proposed by B-P, inequality rankings are affected by the monetary unit in which incomes are expressed. Proof. Assume that y is an income distribution that belongs to the isoinequality contour corresponding to x. Therefore, y is located in the plane deﬁned by distributions x and 1n since it can be written as y ¼ x þ ty ½mx þ ð1 mÞ1n ¼ x þ ty ax

(A.1)

where ax ¼ mx þ (1 m)1n represents the direction of the invariance line. In what follows, we will show that the unit-consistency axiom is violated by any inequality index based on the inequality-invariance notion proposed by B-P, since distribution ly has lower m-inequality than distribution lx, where lAN is higher than 1. From expression (A.1), it follows that distribution ly can be written as ax (A.2) ly ¼ lx þ lty ax ¼ lx þ lnax ty nax where a x represents the mean of the corresponding distribution and ax =nax is a simplex vector. Let us denote w as the distribution that has the same inequality level as lx and the same mean as ly.18 The ﬁrst requirement guarantees that w ¼ lx þ tw alx ¼ lx þ nalx tw ðalx =nalx Þ, where alx ¼ mlx þ (1 m)1n. By also using the second requirement, we obtain the following: alx (A.3) w ¼ lx þ lnax ty nalx

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since the mean of distribution ly is ly ¼ lx þ lnax ty ð1=nÞ and that of distribution w is w ¼ lx þ nalx tw ð1=nÞ. On the other hand, note that lax ¼ mlx þ ð1 mÞl1n ¼ alx þ ðl 1Þð1 mÞ1n . Therefore, distribution lax Lorenz-dominates distribution alx, which implies that ax also Lorenz-dominates alx. From expressions A.2 and A.3, it follows that distribution ly Lorenz-dominates distribution w. Since these two distributions have the same mean, the former must have lower m-inequality than the latter. Consequently, distribution ly has lower m-inequality than distribution lx, since w and lx are in the same invariance line. This completes the proof. Result #2. The inequality ranking between two distributions according to the invariance notion proposed by DR-RC remains unaltered when changing the currency unit. Proof. The (v, p)-inequality concept proposed by DR-RC only allows comparisons between income distributions that are in the same plane, which is deﬁned by one of the distributions and the egalitarian distribution.19 Therefore, let x and y be two income distributions in that plane. First, we assume that x and y are in the same invariance line and show that distributions lx and ly are also inequality equivalent (where l 2 <þþ ). Second, we assume that y has lower inequality than x and demonstrate that ly has lower inequality than lx. Step 1. First assume that x and y are in the same invariance line. Therefore, y ¼ x þ tya, where a ¼ pv þ (1 p)(1/n)1n. This implies that ly ¼ lx þ ltya. In other words, ly is located in the invariance line corresponding to distribution lx since this line can be written as Pðv;pÞ ðlxÞ ¼ fz 2 D : z ¼ lx þ tz a; tz 2
THE BONFERRONI, GINI, AND DE VERGOTTINI INDICES. INEQUALITY, WELFARE, AND DEPRIVATION IN THE EUROPEAN UNION IN 2000 Elena Ba´rcena and Luis J. Imedio ABSTRACT Purpose: This paper studies the Bonferroni (B) and De Vergottini (V) inequality measures, evaluating their differences and similarities, both normatively and statistically. Design: We highlight the similarities of these two indices with the wellknown Gini index (G) and use the AKS [Atkinson (1970), Kolm (1976), Sen (1973)] approach to relate social welfare functions and inequalities indices. In addition, we propose two formulations for relative deprivation, alternative to Yitzhaki (1979) and Hey and Lambert (1980) approach. Findings: The three indices belong to the same family and introduce different and, in some sense, complementary value judgments in the measurement of inequality and welfare; each of them evaluates in a different way the local inequality in the income distribution. The three indices present inequality aversion (satisfy the Pigou-Dalton Principle of Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 231–257 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16011-2

231

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Transfers). But only B satisﬁes the Principle of Positional Transfer Sensitivity. The three absolute indices are interpreted as measures of the mean social deprivation starting from different deﬁnitions of individual deprivation. Originality: The originality of this paper lies in the joint use of the three indices in the measurement of inequality, welfare, and deprivation. We apply these indices to obtain rankings of the European Union countries, using the European Community Household Panel data (2000). A sensitivity analysis of the rankings to different equivalence scales is also included.

1. INTRODUCTION Over the last 30 years, the study of income inequality has become quite important for several reasons (Chakravarty, 2005). However, there are classic indices with good properties that have not received much attention in statistics handbooks, nor in theoretical or applied papers. Two such measures are the Bonferroni (1930) and the De Vergottini (1940) indices. Nygard and Sandstrom (1981) attracted new attention to the Bonferroni index (B) as particularly suitable for the study of an important aspect of income distribution – the intensity of poverty. More recently, Giorgi (1984, 1998), Tarsitano (1990), Giorgi and Mondani (1995), Giorgi and Crescenzi (2001), Chakravarty and Muliere (2003), and Chakravarty (2005) demonstrated the usefulness of this index for income distribution analysis. References to De Vergottini index (V) are scarce; Piesch (2005) uses V in a statistical analysis, without considering its ethical properties. This paper examines these two measures of income inequality, statistically and normatively. Both are formally analogous and, at the same time, they have an evident divergence. We compare their properties with those of the Gini index (G), the most common inequality measure; these three indices belong to the same family and introduce different and, in some sense, complementary value judgments in the measurement of inequality and welfare. Yitzhaki (1979) interprets G in terms of relative deprivation. In this paper, a similar interpretation is given for the absolute indices of Bonferroni and De Vergottini. That is, B and V can also be considered as aggregate measures of the feelings of those individuals located at disadvantaged positions in the income scale following the Temkin (1986, 1993) and Cowell and Ebert (2004) approach.

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Bonferroni, Gini, and De Vergottini Indices

This paper has the following structure: we deﬁne both indices, provide a geometrical interpretation, and discuss their statistical properties in Section 2. Section 3 analyzes the indices’ normative aspects, derives their social evaluation functions, and studies their response to progressive income transfers. In Section 4, we propose two deﬁnitions of interindividual deprivation and their relationships with B and V. Previous results show that B, G, and V introduce different, although, in some sense, complementary value judgments in the measurement of inequality, deprivation, and welfare. For this reason, the joint use of these three indices can be appropriate in the analysis of income distributions. This type of analysis of the income distributions of the European Union (EU) countries in 2000, using the European Community Household Panel (ECHP), is done in Section 5. We compare the ranking of countries using different scales in order to examine the sensitivity of the inequality ranking to the equivalence scales.

2. FORMAL FRAMEWORK AND PROPERTIES The B and V are based on the comparison of the general mean of the income distribution, and the partial mean of the poorest j percent of the population or of the richest j percent of the population. In this section, we obtain different expressions for both inequality measures, for continuous and discrete distributions, and we analyze some of their statistical properties. Let us assume that the income distribution in a given population is represented by the continuous random variable X, with support [x0, xM], xMRWx0Z0, with cumulative distribution function F and mean m ¼ EðXÞ x ¼ x0M xdFðxÞ. The partial means for X over the intervals [x0, x] and [x, xM), respectively, are given by Z x 1 tdFðtÞ; x 2 ðx0 ; xM (1) mðxÞ ¼ FðxÞ x0 1 MðxÞ ¼ 1 FðxÞ

Z

xM

tdFðtÞ;

x 2 ½x0 ; xM Þ

(2)

x

For a given level of income x, the function rðxÞ ¼

m mðxÞ ; m

x 2 ðx0 ; xM

(3)

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measures the amount by which the mean income of those with income less than or equal to x falls short of the population mean. r(x) is a non-negative, decreasing, and upper-bounded function and it satisﬁes x0 rðxÞ 1 o1; x 2 ðx0 ; xM m The function RðxÞ ¼

MðxÞ m ; m

x 2 ½x0 ; xM Þ

(4)

gives, for each income x, the relative difference between the population mean income and the mean income of those with income greater than or equal to x. This function is non-negative, increasing, and satisﬁes xM 1; x 2 ½x0 ; xM Þ RðxÞ m Deﬁnition 1. (Bonferroni, 1930). The Bonferroni index of inequality of an income distribution X, B, is deﬁned as Z xM Z xM m mðxÞ dFðxÞ (5) rðxÞdFðxÞ ¼ B ¼ EðrðXÞÞ ¼ m x0 x0 Deﬁnition 2. (De Vergottini, 1940). The De Vergottini index of inequality of an income distribution X, V, is deﬁned as Z xM Z xM MðxÞ m RðxÞdFðxÞ ¼ dFðxÞ (6) V ¼ EðRðXÞÞ ¼ m x0 x0 Therefore, B and V are expected values of the relative differences represented by the functions r(x) and R(x), respectively. The following inequalities are immediate x0 xM 1 0 B 1 o1; 0 V m m B is bounded from above by 1, while the upper bound of V depends on the maximum income.R x If LðpÞ ¼ ð1=mÞ x0 tdFðtÞ; p ¼ FðxÞ; is the Lorenz curve1 for X, we can write the functions (1)–(4) as mðxÞ ¼

mLðFðxÞÞ mLðpÞ ¼ ; FðxÞ p

0 o p ¼ FðxÞ 1

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Bonferroni, Gini, and De Vergottini Indices

MðxÞ ¼

mð1 LðFðxÞÞÞ mð1 Lð pÞÞ ¼ ; 1 FðxÞ 1p

rðxÞ ¼

RðxÞ ¼

p Lð pÞ Lð pÞ ¼1 ; p p

0 p ¼ FðxÞo1

0op ¼ FðxÞ 1

1 Lð pÞ p Lð pÞ 1¼ ; 1p 1p

0 p ¼ FðxÞo1

On using the deﬁnitions of B and V [expressions (5) and (6)], we can write Z 1 Z 1 Lð pÞ p Lð pÞ B¼1 dp ¼ dp (7) p p 0 0 Z

1

V¼ 0

1 Lð pÞ dp 1 ¼ 1p

Z

1 0

p Lð pÞ dp 1p

(8)

These expressions allow us an easy geometric interpretation of each of these indices from the Bonferroni, B( p), and the De Vergottini curve, V( p). These curves are deﬁned as BðpÞ ¼

VðpÞ ¼

LðpÞ ; p

0op 1; Bð0Þ ¼

1 LðpÞ ; 1p

x0 m

0 po1; Vð1Þ ¼

xM m

Both B( p) and V( p) are nondecreasing functions for pA(0,1). In contrast to the Lorenz curve, which always is a convex function, the shapes of B( p) and V( p) are related to the shape of the underlying distribution function, F.2 Figs. 1 and 2 show these curves. From expressions (7) and (8), we get Z 1 BðpÞdp (9) B¼1 0

and Z

1

VðpÞdp 1

V¼

(10)

0

From a geometric point of view, B(V) is the area between the Bonferroni (De Vergottini) curve and the line of perfect equality. Perfect equality would

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236 B(p) (0,1)

(1, 1)

B

x0 /µ (0, 0)

Fig. 1.

(1, 0)

p

Bonferroni Curve and Bonferroni Index.

V(p) (xM /µ,1)

V

(1,1)

(0,1)

p (0, 0)

Fig. 2.

(1, 0)

De Vergottini Curve and De Vergottini Index.

result in points along the line B( p) ¼ V(p) ¼ 1, 0opo1, and B ¼ V ¼ 0. In the case corresponding to the maximum concentration, B( p) ¼ 0 and B ¼ 1, while V( p) ¼ 1/(1 p), 0opo1, and that implies that expression (8) is not a convergent integral.

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Bonferroni, Gini, and De Vergottini Indices

As it is known, the Gini index of an income distribution, G, can be interpreted as twice the area between the Lorenz curve and the line of perfect equality: Z 1 Z 1 ð p LðpÞÞdp ¼ 1 2 Lð pÞdp (11) G¼2 0

0

so that GA[0,1], G ¼ 0 if there is perfect equality, and G ¼ 1 in case of maximum concentration. This index can be expressed as Z xM Z xM Z 1 G¼2 rðxÞFðxÞdFðxÞ ¼ 2 RðxÞ ð1 FðxÞÞdFðxÞ ¼ 2 ðp LðpÞÞdp x0

0

x0

Hence, G is a weighted mean of r(x) or of R(x), whereas B and V are their simple means, respectively. It is easy to check that B, G, and V are compromise indices. They are relative indices, scale independent, while mB, mG, and mV are absolute indices, and remain unaltered when all incomes are translated. A characteristic of G is 1 G ¼ CovðX; FðXÞÞ m G can be expressed as the covariance between income and its rank in the distribution. B and V satisfy3 1 B ¼ CovðX; lnðFðXÞÞÞ; m

1 V ¼ CovðX; lnð1 FðXÞÞÞ m

B(V ) is expressed in terms of the covariance between income and the logarithm of the distribution function (survival function). The three indices belong to the class of linear measures of income inequality deﬁned by Mehran (1976). They can be obtained weighting the Lorenz differences,4 p LðpÞ, along the income distribution. From expressions (7), (8), and (11), we can write Z 1 1 ðp LðpÞÞpB ðpÞdp; pB ðpÞ ¼ ; 0op 1 B¼ p 0 Z

1

ðp LðpÞÞpV ðpÞdp;

V¼ 0

Z G¼

1 ; 1p

0 po1

1

ðp LðpÞÞpG ðpÞdp; 0

pV ðpÞ ¼

pG ðpÞ ¼ 2;

0op 1

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To obtain B the Lorenz differences are weighted with the strictly decreasing and convex function pB(p). In V the weighting scheme pV(p) is strictly increasing and convex. In G the weight is constant. For that reason, each of these three indices evaluates in a different way, the local inequality in the income distribution. B pays more attention to the inequality in the lefthand side on the distribution (low incomes), while V has the opposite behavior. G has an intermediate criterion, it attaches a constant weight to the difference p L(p).

2.1. Deﬁnition of the Indices for Discrete Distributions Consider a ﬁxed homogeneous population of n, nZ2, individuals. An income distribution in this population is represented by a non-negative . The set ill-fare-ranked vector x ¼ (x1,x2,y,xn), that is, 0rx1rx2r?rxnP of all income distributions in the society is Dn. For xADn, m ¼ 1=n ni¼1 xi is the population mean and mi ¼

i 1X xj ; i j¼1

Mi ¼

n X 1 xj ; n i þ 1 j¼i

1in

are, respectively, the mean income of the i (n i þ 1) persons with lowest (highest) incomes. The discrete expressions for Eqs. (3) and (4), are rn ðxi Þ ¼

m mi ; m

Rn ðxi Þ ¼

Mi m ; m

1in

B is the mean of the set frn ðxi Þg1in . It is Bn ¼

n n n 1X 1 X 1 X rn ðxi Þ ¼ 1 mi ¼ Z xi n i¼1 nm i¼1 nm i¼1 i

(12)

where Zi ¼ 1

n X 1 j¼i

j

;

1 Ziþ1 ¼ Zi þ ; i

n X

Zi ¼ 0

i¼1

In the same way, the mean of the set fRn ðxi Þg1in is V Vn ¼

n n n 1X 1 X 1 X Rn ðxi Þ ¼ Mi 1 ¼ x xi n i¼1 nm i¼1 nm i¼1 i

(13)

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Bonferroni, Gini, and De Vergottini Indices

where xi ¼

i X j¼1

1 ; ðn j þ 1Þ 1

xiþ1 ¼ xi þ

1 ; ni

n X

xi ¼ 0

i¼1

The expression of G as a linear combination of the ill-fare-ranked incomes is Gn ¼

n 1 X g xi nm i¼1 i

(14)

where gi ¼

2i 1 ; n1

2 giþ1 ¼ gi þ ; n

n X

gi ¼ 0

i¼1

The previous expressions allow us to obtain the three indices with micro data. This speciﬁcation helps to characterize the income-weighting scheme, showing that the weight attached to each income depends on the position of the receiver in the income distribution. The greater is the weight, the higher is the ranking in the distribution. It means that the sequences fZi g1in , fxi g1in , and fgi g1in are strictly increasing, even though they have different growth patterns. In B the weights increase at a decreasing rate, in V the opposite takes place, and in G weights increase at a constant rate. The following relationships are satisﬁed: Zi þ xniþ1 ¼ 0;

gi þ gniþ1 ¼ 0;

1in

The ﬁrst one shows that the weights attached to incomes in B and V are symmetrical in absolute values. B attaches to the minimum income, x1, a weight equal, in absolute value, to the one attached by V to the maximum income, xn; and the same happens when considering x2 and xn 1, x3 and xn 2, etc. In G the weights, in absolute value, are symmetrical respect to the median income. Fig. 3 shows the behavior of the sequences fZi g1in , fxi g1in , and fgi g1in in a population where n ¼ 100. In the case of perfect equality, x1 ¼ x2 ¼ ? ¼ xn 1 ¼ xn ¼ m, Bn ¼ Gn ¼ Vn ¼ 0. If there is maximum concentration and in the income distribution there is only one non-zero income, x1 ¼ x2 ¼ ? ¼ xn 1 ¼ 0, xn ¼ nm, the values of Bn, Gn, and Vn depend on the size of the population Bn;max ¼ Zn ¼

n1 o1 n

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V n;max ¼ xn ¼

n X j¼1

n X 1 1 ¼ ðn j þ 1Þ 1 j¼2 j

Gn;max ¼ gn ¼

n1 o1 n

Bn and Gn take values on [0, 1), while Vn has not an upper bound. For a given n, V n ¼ V n =V n;max is a normalized index with V n 2 ½0; 1. In opposite to G, B and V do not satisfy the Dalton population principle. While G is population replication invariant, B and V are not.5 This implies that B and V are not consistent with the partial ordering induced by the Lorenz curve (Foster, 1985).

5 4 3 2 1 0 0

20

40

60

80

100

-1 -2 -3 -4 -5 Ranking G

Fig. 3.

B

V

Weighting Schemes in the Indices.

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Bonferroni, Gini, and De Vergottini Indices

3. NORMATIVE ASPECTS In this paper, the relationship between social welfare functions (SWFs) and inequalities indices is based on the AKS [Atkinson (1970), Kolm (1976a, 1976b), Sen (1973)] approach through the concept of equally distributed equivalent income.6 In such a case, if I is a relative inequality index, the welfare of a distribution xADn, W(x), is WðxÞ ¼ mx ½1 IðxÞ ¼ mx mx IðxÞ

(15)

where mxI(x) becomes a measure of the loss in social welfare due to inequality. From expression (12), the SWF corresponding to B is given by W B ðxÞ ¼ mx ½1 BðxÞ ¼

n n X 1X ð1 Zi Þxi ¼ wi;B xi n i¼1 i¼1

(16)

where n 1 1X 1 ; wi;B ¼ ð1 Zi Þ ¼ n n j¼i j

1 i n;

n X

wi;B ¼ 1

i¼1

In the same way, from expression (13), the SWF corresponding to the normalized V is n n1 X 1X x 1 i xi ¼ wi;V xi (17) W V ðxÞ ¼ mx ½1 VðxÞ ¼ xn n i¼1 i¼1 where wi;V

P 1=n ni 1 xi j¼1 1=j ¼ Pn ; ¼ 1 xn n j¼2 1=j

1 i n 1;

n1 X

wi;V ¼ 1

i¼1

As shown in the previous expression, W V ðxÞ does not depend on the maximum income of the distribution, xn. The SWF corresponding to G, expression (14), is n X wi;G xi (18) W G ðxÞ ¼ mx ½1 GðxÞ ¼ i¼1

where 1 1 wi;G ¼ ð1 gi Þ ¼ 2 ð2ðn iÞ þ 1Þ; n n

1 i n;

n X i¼1

wi;G ¼ 1

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The SWFs corresponding to B, G, and V have common properties. P All of them are rank-dependent SWFs and are linear in incomes. W ¼ ni¼1 wi xi , wherePfwi g1in is a decreasing sequence of positive real numbers such that ni¼1 wi ¼ 1. But the sequences fwi;B g1in , fwi;V g1in , and fwi;G g1in have different growth patterns as a consequence of the different incomeweighting schemes of the corresponding indices. In the ﬁrst sequence, the decreasing rate decreases as the rank in the distribution increases, in the sequence corresponding to V the opposite takes place, while in the sequence corresponding to G, the decreasing rate is constant from each position to the next. Fig. 4 shows these characteristics in a population of size 100. The three SWFs are continuous, increasing, homothetic,7 and strictly S-concave. The last property implies that the inequality index (or its SWF) shows inequality aversion (or preference for equality), what is equivalent to satisfy the Pigou-Dalton Principle of Transfers (PDPT). This principle states that an income transfer from a richer to a poorer individual, that leaves their ranks in the income distribution unchanged (progressive transfer),

0,06

0,05

0,04

0,03

0,02

0,01

0,00 0

20

40

60

80

Ranking G

Fig. 4.

B

V

Weights in the SWFs.

100

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Bonferroni, Gini, and De Vergottini Indices

reduces income inequality and, when efﬁciency considerations are absent (m is ﬁxed), increases the social welfare, expression (15). On the other hand, B, G, and V do not have the same behavior when more demanding principles than the PDPT, in terms of redistribution, are required. This is the case of the Principle of Positional Transfer Sensitivity (PPTS) (Mehran, 1976; Zoli, 1999), which postulates that a progressive income transfer between two individuals with a ﬁxed difference in ranks will reduce inequality by a larger amount the lower the income of the donor is. If xADn and a progressive transfer, dW0, takes place from the donor k to the receiver h, hok, the result, xADn is xnh ¼ xh þ d;

xnk ¼ xk d;

xnj ¼ xj ;

jah; k

From expressions (12) to (14) it is Bn Bnn ¼

V n V nn ¼

k1 d X 1 nm i¼h i

k d X 1 nm i¼hþ1 n i þ 1

Gn Gnn ¼

2dðk hÞ n2 m

As Bn Bnn 40, V n V nn 40, and Gn Gnn 40, the three indices satisfy the PDPT. The previous expressions show that given a difference in ranks, k h, between the receiver and the donor, the decrement in B is greater, the poorer are the individuals. Therefore B satisﬁes the PPTS, while G and V do not. The decrement in V is greater the richer are the individuals. The change in G is proportional to the difference in ranks, i.e., it attaches an equal weight to a given transfer irrespective of whether it takes place in the upper, middle, or lower part of the distribution.

4. BONFERRONI ABSOLUTE INDEX AND DE VERGOTTINI ABSOLUTE INDEX AS DEPRIVATION MEASURES8 Deprivation arises due to inequality existing within a group. An individual feels deprived when compared with better-off individuals from his or her

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standpoint. Usually, the concept of deprivation is deﬁned in terms of income, frequently considered as an index of the individual’s ability to consume commodities. Yitzhaki (1979) proved that an appropriate index of aggregate deprivation is the absolute Gini index.9 Hey and Lambert (1980) provided an alternative motivation of Yitzhaki’s result. For these authors, the individual feeling of deprivation is the difference between the individual’s income and the income of richer individuals in the income distribution. In this section, we propose two alternative formulations of deprivation10 that allow the interpretation of the absolute indices, mB and mV, as social deprivation measures. Deﬁnition 3. (Hey & Lambert, 1980). The deprivation felt by an individual with income x in respect to an individual with income z, DHL(x,z), is given by ( z x; if z 4 x (19) DHL ðx; zÞ ¼ 0; if z x This formulation has often been justiﬁed by reference to the classical deﬁnition of relative deprivation found in Runciman (1966): ‘‘The magnitude of a relative deprivation is the extent of the difference between the desired situation [e.g., the income of the richer person] and that of the person desiring it.’’ So, an individual’s deprivation is different from zero only when he or she compares himself or herself with richer individuals in the income distribution. Then, the deprivation associated with a given level of income x, DHL(x), is Z xM DHL ðxÞ ¼ DHL ðx; zÞdFðzÞ ¼ mð1 LðFðxÞÞÞ xð1 FðxÞÞ x0

¼ ð1 FðxÞÞðMðxÞ xÞ where M(x) is the average income of the set of individuals with an income equal to or greater than x. Therefore, DHL(x) is the product of the proportion of individuals with income greater than x and the difference between the average income of such a group and x. DHL(x) is a strictly decreasing function of the level of income and DHL ðx0 Þ ¼ m x0 , DHL ðxM Þ ¼ 0. The average social deprivation is Z xM DHL ðxÞdFðxÞ ¼ mG (20) EðDHL ðXÞÞ ¼ x0

In Deﬁnition 3, individuals compare their incomes with all other incomes, taking into account the entire income distribution. But it could be that the

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Bonferroni, Gini, and De Vergottini Indices

individual feels depressed only with respect to those with incomes greater than his or her income, ignoring those with smaller incomes. It means that the individual does not look backward. This is equivalent to a left truncated distribution. Then, Deﬁnition 4 follows. Deﬁnition 4. The deprivation felt by an individual with income x in ~ zÞ, is given by respect to an individual with income z, z W x, Dðx; ~ zÞ ¼ z x; Dðx;

z4x

(21)

~ zÞ is deﬁned in the interval [x, xM]. Therefore, given x, Dðx; In order to obtain the deprivation associated with a given income level x, ~ DðxÞ, we have to consider the distribution of the income variable, X, restricted to [x, xM].11 Considering expression (2), we get Z xM 1 ~ zÞdFðzÞ ¼ MðxÞ x ~ Dðx; DðxÞ ¼ 1 FðxÞ x where M(x) is the mean income corresponding to the interval [x, xM]. Thus, ~ DðxÞ is the difference between this mean income and x. The relationship ~ between DðxÞ and DHL(x) is ~ ~ DHL ðxÞ ¼ ð1 FðxÞÞDðxÞ DðxÞ;

x0 x xM

~ 0 Þ ¼ m x0 , Dðx ~ M Þ ¼ 0. However, even though It is satisﬁed that Dðx Deﬁnition 4 considers a reasonable behavior of the individual, in general ~ DðxÞ is not a monotonic decreasing function of income. ~ The mean value of DðxÞ is the absolute De Vergottini index. From expression (8), we get Z 1 Z xM 1 LðpÞ ~ ðMðxÞ xÞdFðxÞ ¼ m dp 1 ¼ mV (22) EðDðXÞÞ ¼ 1p 0 x0 Finally, we can consider that when an individual with income x compares with those with incomes greater than x, he or she not only takes into account those with smaller incomes but also feels as a member of this group and identiﬁes himself or herself with its mean income, m(x). Then, next deﬁnition applies. Deﬁnition 5. The deprivation felt by an individual with income x, in ~~ respect to an individual with income z, D(x,z), is given by ( ~~ zÞ ¼ z mðxÞ; if z4x Dðx; (23) 0; if z x

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~~ Thus, the deprivation corresponding to income x, D(x), is Z xM ~~ ~~ zÞdFðzÞ ¼ mð1 LðFðxÞÞÞ mðxÞð1 FðxÞÞ DðxÞ ¼ Dðx; x0

¼ ð1 FðxÞÞðMðxÞ mðxÞÞ ¼ m mðxÞ ~~ DðxÞ is the difference between the population mean income and the mean ~~ income of the individuals with income smaller or equal to x, m(x). DðxÞ is ~ ~ a decreasing function of income and, as in previous deﬁnitions, Dðx0 Þ ¼ ~~ m x0 , Dðx M Þ ¼ 0. The average social deprivation is given by the absolute Bonferroni index. From expression (7), we get Z 1 Z xM LðpÞ ~~ ðm mðxÞÞdFðxÞ ¼ m 1 dp ¼ mB (24) EðDðXÞÞ ¼ p 0 x0 Therefore, the three absolute indices mB, mG, and mV can be interpreted as measures of the mean social deprivation starting from different deﬁnitions of individual deprivation.

5. AN EMPIRICAL ILLUSTRATION. EUROPEAN UNION 2000 In this section, we analyze income inequality, welfare, and deprivation across the EU countries using ECHP for wave 8 (year 2001). As shown above, B, G, and V belong to the same family of inequality measures and their welfare functions have formally analogous expressions, but each of these indices introduce a different degree of inequality aversion. B and V complement the information given by G by turning particular attention to changes that take place in the lower and upper part of the income distribution, respectively. The variable used is net money income, deﬁned as the sum of net income from work, other nonwork private income, pensions, and other social transfers. This income refers to the year previous to the survey, 2000.12 To obtain an appropriate comparable measure of individual well-being, we adjust for needs. Following the terminology in Jenkins (2000), an obvious way to write the economic measure of well-being is to use the household income-equivalent (HIE). If HIEt is the needs-adjusted

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Bonferroni, Gini, and De Vergottini Indices

household net income in year t, then Pn PK HIE t :¼

j¼1

k¼1 xjkt

mða; nÞ

where j indices individuals in the household ( j ¼ 1, 2,y, n) and k indices income source. The denominator is an equivalence scale factor depending on household size n and on a vector of household composition variables, a (ages of individuals or role within the household). So, the welfare measure HIE is the sum of all household members’ income adjusted by household needs. Since a given level of household income will correspond to a different standard of living depending on the size and composition of the household, we adjust for these differences using a variety of equivalence scales.13 We carry out a robustness analysis using different scales as suggested by Buhmann, Rainwater, Schmaus, and Smeeding (1988) to test sensitivity of the different measures to the choice of equivalence scales. We use the OECD and modiﬁed-OECD equivalence scales14 and three power-function scales (see Buhmann et al., 1988) using parameter values 0.25, 0.5, and 0.75. Our analysis is based on a panel of households. In order to describe distributions of personal incomes, the cross-sectional weight of the interviewed households has to be multiplied by the number of persons belonging to the household. We use sample weights, as it is the conventional way to mitigate potential bias introduced by potential differential nonresponse and differential attrition. First, B, G, and V are computed in order to compare the ranking of the countries in terms of inequality in income distribution. Figs. 5–7 show the ranking for the three indices. The main conclusion of each of these three ﬁgures is the robustness of the results, since signiﬁcant variations are not observed when changing the equivalence scales. If we compare the ranking of countries of the three indices, we also ﬁnd that the results are robust. We can detect ﬁve substantial groups of EU countries. These groups are similar to the ones obtained by A´lvarez-Garcı´ a, Prieto-Rodriguez, and Salas (2004) for EU countries in 1996. A ﬁrst group would be composed of Denmark, the one with the smallest degree of inequality measured by the three indices. It also shows the largest difference in the value of the inequality measure with respect to the next

Denmark

Finland

Sweden

Austria

Luxembourg

Germany Netherlands France

Belgium

Italy

Ireland

United Kingdom

Greece

Spain

Portugal

Fig. 5.

Bonferroni Ranking in 2000 [Equivalence Scales: OECD, Modiﬁed-OECD (Buhmann et al., 1988), a ¼ 0.25, 0.5, 0.75].

Denmark

Sweden

Austria

Finland Netherlands

Germany

Luxembourg

France

Belgium

Italy

Ireland

United Kingdom

Spain

Greece

Portugal

Fig. 6.

Gini Ranking in 2000 [Equivalence Scales: OECD, Modiﬁed-OECD (Buhmann et al., 1988), a ¼ 0.25, 0.5, 0.75].

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Bonferroni, Gini, and De Vergottini Indices

Denmark Austria

Finland Netherlands Sweden

Germany

Luxembourg

France

Ireland Italy

Belgium United kingdom

Greece

Spain Portugal

Fig. 7.

De Vergottini Ranking in 2000 [Equivalence Scales: OECD, ModiﬁedOECD (Buhmann et al., 1988), a ¼ 0.25, 0.5, 0.75].

country. A second group would be formed by Finland, Sweden, Austria, Luxembourg, Germany, France, and the Netherlands. Ireland, Italy, Greece, the United Kingdom, and Spain constitute the third group with greater inequality than the two previous sets of countries. Portugal presents not only the highest degree of inequality in the income distribution, but also the second largest difference in the value of the inequality measure with respect to the previous country. Therefore, Portugal could be considered the fourth set. Belgium is a peculiar country. Depending on the index applied, Belgium shows a different ranking. Italy, Ireland, and the United Kingdom are no longer dominated by Belgium if more weight is attached to incomes in the

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Luxembourg

Finland

Sweden

Denmark

Germany

Austria

Belgium

France

Ireland

Spain

Greece

Fig. 8.

United Kingdom

Netherlands

Italy

Portugal

Welfare Function Corresponding to Bonferroni Ranking in 2000.

upper part of the distribution (V). Therefore, Belgium shows more inequality relative to these countries. However, when more weight is placed in the lower part of the distribution (B) or same weight is attached to all incomes (G), then Belgium shows less inequality in relation to these countries. This suggests that Belgium presents more inequality on the righthand side of the distribution than other EU countries. Figs. 8–10 represent the dominance analysis of welfare functions corresponding to the three indices for different equivalence scales. The rankings of welfare corresponding to B and G are similar, but there are two ranking reversals. Finland and Sweden have higher welfare in

Luxembourg Denmark

Germany

Austria

Belgium

France

United Kingdom

Netherland

Ireland

Fig. 9.

Finland

Sweden

Spain

Italy

Greece

Portugal

Welfare Function Corresponding to Gini Ranking in 2000.

Luxembourg

Germany

Denmark

Austria

Belgium

United Kingdom

Ireland

France Netherland

Sweden

Finland

Spain

Italy

Greece

Fig. 10.

Portugal

Welfare Function Corresponding to De Vergottini Ranking in 2000.

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terms of the welfare function corresponding to B. For the welfare function corresponding to V, the countries at the bottom and at the top of the ordering are the same, but in the middle of the ordering, Spain and Italy are no longer dominated by Sweden. With these exceptions, the results of the ranking of countries in terms of welfare are quite robust. Luxembourg is, far and away, the country with the greatest welfare. Germany, Austria, and Denmark constitute the second group, always dominating over Belgium. The fourth group is composed of France, the United Kingdom, Ireland, the Netherlands, Spain, and Italy. As always, Greece and Portugal are the last set of countries, dominated by all the others. Finland and Sweden change rankings depending on the welfare function used. Using B they are on the top of the ranking, while for the rest they are in the middle-bottom of the ranking. Considering Figs. 11–13, we ﬁnd that we cannot rank many countries. Luxembourg is the only country clearly dominated by all other countries in terms of deprivation, independently of the deﬁnition used, but we cannot establish an unambiguous ranking among the other countries. We note that

Greece

Finland

Portugal

Italy

Spain

Sweden

Denmark

Ireland

Netherlands

Austria

France Germany

Belgium

United Kingdom Luxembourg

Fig. 11.

Deprivation Corresponding to Bonferroni Ranking in 2000.

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Bonferroni, Gini, and De Vergottini Indices

Greece

Finland

Denmark

Italy

Sweden

Neteherlands

Portugal

Ireland

Austria

Spain

France

Germany

Belgium

United Kingdom

Luxembourg

Fig. 12.

Deprivation Corresponding to Gini Ranking in 2000.

Finland

Greece

Sweden

Italy

Ireland

France

Denmark

Austria

Portugal

Netherlands

Germany

Spain

Belgium

United Kingdom

Luxembourg

Fig. 13.

Deprivation Corresponding to De Vergottini Ranking in 2000.

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Greece, Finland, and Sweden are always near the top of the ranking, and Belgium and the United Kingdom are close to the bottom. In conclusion, after analyzing inequality, welfare, and deprivation, we can ﬁnd that, for each of these concepts, rankings are very similar, independent of the index used. But the country orderings induced by deprivation, welfare, and inequality can be quite different.

NOTES 1. For p ¼ F(x), L(p) is the income share of the individuals with incomes smaller or equal to x. It satisﬁes L(0) ¼ 0, L(1) ¼ 1, and L( p)rp, 0rpr1. In case of equally distributed income L(p) ¼ p, and in case of maximum concentration L( p) ¼ 0, 0rpo1. 2. For example, if F(x) is convex (concave), B(p) is concave (convex). For a proof see Aaberge (2007). 3. Proof available upon request from the authors. 4. Note that p L(p), p ¼ F(x), is the difference between the income share of those with income less or equal to x, if income is equally distributed and the real income share in the distribution. 5. For incomes {0, a}, aW0, it is B2 ¼ 1/2, V2 ¼ 1/2, and V 2 ¼ 1. Nevertheless, for incomes {0, 0, a, a}, it is B4 ¼ 7/12, V4 ¼ 7/12, and V 4 ¼ 7=13. 6. Level of income, which if enjoyed by every individual, would make the total welfare exactly equal to the total welfare generated by the actual distribution. 7. It means that Wðcx þ a1n Þ ¼ cWðxÞ þ a, xADn, cW0, 1n ¼ ð1; 1; . . . ; 1Þ, and ‘‘a’’ a real number such that cx þ a1n is an admissible distribution. This property of the SWF characterizes compromise indices. 8. In this section, we suppose that the income distribution is represented by a continuous variable X with distribution function F(x), mean m, and Lorenz curve L( p), p ¼ F(x). 9. See Ebert and Moyes (2000) and Bossert and D’Ambrosio (2006) for an axiomatic approach. 10. Other alternative formulations are: Chakravarty and Chakraborty (1984), Berrebi and Silber (1985), Paul (1991), Podder (1996), Chakravarty and Mukherjee (1999), and Duclos (2000). Multidimensional indices of deprivation have been proposed by Bossert, D’Ambrosio, and Peragine (2007), Brandolini and D’Alessio (1998), Tsakloglou and Papadopoulos (2002), Whelan, Layte, Maitre, and Nolan (2002) among others. 11. Its income distribution function is H(z)=(F(z) F(x))/(1 F(x)), zA[x, xM]. 12. We apply purchasing power parities (PPP) to convert national currencies into PPP units, obtaining a comparable measure of income for the different countries. 13. For a survey of equivalence scales and related income distribution issues, and some comparisons of scale relativities, see Coulter, Cowell, and Jenkins (1992).

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14. Both scales differentiate among ﬁrst adult, subsequent adults, and children. The weights implied by the two scales are as follows, where a ‘‘child’’ means a person under 14 years of age:

First adult Each additional adult Each additional child

OECD

Modiﬁed-OECD

1.0 0.7 0.5

1.0 0.5 0.3

ACKNOWLEDGMENT The authors are grateful to John Bishop and one anonymous referee for their suggestions and comments.

REFERENCES Aaberge, R. (2007). Gini’s nuclear family. Journal of Economic Inequality, 5, 305–322. A´lvarez-Garcı´ a, S., Prieto-Rodriguez, J., & Salas, R. (2004). The evolution of income inequality in the European Union during the period 1993–1996. Applied Economics, 36, 1399–1408. Atkinson, A. B. (1970). On the measurement of inequality. Journal of Economic Theory, 2(3), 244–263. Berrebi, Z. M., & Silber, J. (1985). Income inequality indices and deprivation: A generalization. The Quarterly Journal of Economics, 100(3), 807–810. Bonferroni, C. E. (1930). Elementi di statistica generale. Firenze: Libreria Seber. Bossert, W., & D’Ambrosio, C. (2006). Reference groups and individual deprivation. Economics Letters, 90(3), 421–426. Bossert, W., D’Ambrosio, C., & Peragine, V. (2007). Deprivation and social exclusion. Economica, 74(296), 777–803. Brandolini, A., & D’Alessio, G. (1998). Measuring Well-Being in the Functioning Space, Vol. mimeo. Rome, Italy: Banca d’Italia. Buhmann, B., Rainwater, L., Schmaus, G., & Smeeding, T. M. (1988). Equivalence scales, wellbeing, inequality and poverty: Sensitive estimates across ten countries using the Luxembourg Income Study (LIS) database. Review of Income and Wealth, 34(2), 115–142. Chakravarty, S. R. (2005). The Bonferroni indices of inequality. Paper presented at the International Conference in Memory of C. Gini and M. O. Lorenz. Universita` degli Studi di Siena. Available at www.unisi.it/event/GiniLorenz05 Chakravarty, S. R., & Chakraborty, A. B. (1984). On indices of relative deprivation. Economics Letters, 14(2–3), 283–287. Chakravarty, S. R., & Mukherjee, D. (1999). Measures of deprivation and their meaning in terms of social satisfaction. Theory and Decision, 47(1), 89–100.

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Chakravarty, S. R., & Muliere, P. (2003). Welfare indicators: A review and new perspectives. Measurement of inequality. Metron-International Journal of Statistics, 61(3), 457–497. Coulter, F. A. E., Cowell, F. A., & Jenkins, S. P. (1992). Differences in needs and assessment of income distributions. Bulletin of Economic Research, 44(2), 77–124. Cowell, F. A., & Ebert, U. (2004). Complaints and inequality. Social Choice and Welfare, 23(1), 71–89. De Vergottini, M. (1940). Sul signifacoto di alcuni indici di concentrazione. Giornale degli economisti e annali di econ, 11, 317–347. Duclos, J. Y. (2000). Gini indices and the redistribution of income. International Tax and Public Finance, 7(2), 141–162. Ebert, U., & Moyes, P. (2000). An axiomatic characterization of Yitzhaki’s index of individual deprivation. Economics Letters, 68, 263–270. Foster, J. E. (1985). Inequality measurement. In: H. Peyton Young (Ed.), Fair Allocation (Proceedings of symposia in applied mathematics ed., pp. 31–68). Providence, RI: American Mathematical Society. Giorgi, G. M. (1984). A methodological survey of recent studies for the measurement of inequality of economic welfare carried out by some Italian statisticians. Economic Notes, 13(1), 145–157. Giorgi, G. M. (1998). Concentration index, Bonferroni. In: Encyclopedia of Statistical Sciences (pp. 141–146). New York: Wiley. Giorgi, G. M., & Crescenzi, M. (2001). A look at the Bonferroni inequality measure in a reliability framework. Statistica, 41, 571–573. Giorgi, G. M., & Mondani, R. (1995). Sampling distribution of the Bonferroni inequality index from Exponential population. The Indian Journal of Statistics. Series B, 57(1), 10–18. Hey, J. D., & Lambert, P. J. (1980). Relative deprivation and the Gini coefﬁcient: Comment. The Quarterly Journal of Economics, 95(3), 567–573. Jenkins, S. P. (2000). Modelling household income dynamics. Journal of Population Economics, 13, 529–567. Kolm, S. C. (1976a). Unequal inequalities. I. Journal of Economic Theory, 12(3), 416–442. Kolm, S. C. (1976b). Unequal inequalities. II. Journal of Economic Theory, 13(1), 82–111. Mehran, F. (1976). Linear measures of inequality. Econometrica, 44, 805–809. Nygard, F., & Sandstrom, A. (1981). Measuring income inequality. Stockholm: Almqvist and Wicksell. Paul, S. (1991). An index of relative deprivation. Economics Letters, 36(3), 337–341. Piesch, W. (2005). Bonferroni-index und De Vergottini-index. Diskussionspapiere aus dem Institut fu¨r Volkswirtschaftslehre der Universita¨t Hohenheim. Department of Economics, University of Hohenheim, Germany, Vol. 259. Podder, N. (1996). Relative deprivation, envy and economic inequality. Kyklos, 49(3), 353–376. Runciman, W. G. (1966). Relative deprivation and social justice. London: Routledge. Sen, A. K. (1973). On economic inequality. Oxford: Clarendon Press. Tarsitano, A. (1990). The Bonferroni index of income inequality. In: C. Dagum & M. Zenga (Eds), Income and wealth distribution, inequality and poverty (pp. 228–242). Berlin: Springer-Verlag. Temkin, L. S. (1993). Inequality. Oxford: Oxford University Press. Temkin, L. S. (1986). Inequality. Philosophy and Public Affairs, 15, 99–121.

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ON VARIOUS WAYS OF MEASURING UNEMPLOYMENT, WITH APPLICATIONS TO SWITZERLAND Joseph Deutsch, Yves Flu¨ckiger and Jacques Silber ABSTRACT This paper discusses ﬁrst various ways of measuring unemployment and, borrowing ideas from the poverty measurement literature, proposes four more general unemployment indices which are parallel to Sen poverty index, to its generalization by Shorrocks, to the FGT, and to the Watts poverty indices. It then presents an empirical illustration based on Swiss data at the level of the ‘‘canton.’’ More precisely, using the so-called Shapley decomposition, it computes the contribution to the difference between the value of each of these four unemployment indices in a given ‘‘canton’’ and in Switzerland as a whole, of three components measuring, respectively, the impact of differences in the traditional unemployment rate, in the average unemployment duration, and in the inequality in the unemployment durations. The paper ends by discussing the impact on the results obtained of assumptions made concerning the maximum unemployment duration.

Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 259–284 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16012-4

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INTRODUCTION Starting with the path-breaking work of Sen (1976) numerous studies have attempted during the past 30 years to measure the extent of poverty. Part of this work has been theoretical, taking either an ordinal or a cardinal approach to the measurement of poverty (see, Zheng, 1997, for a good survey of the work in this ﬁeld) but there have been also many empirical studies of the extent of poverty and these works have generally taken a look at what is known as the ‘‘Three I’s of poverty’’ (see, Jenkins & Lambert, 1997), that is, its incidence (the percentage of poor in the population), its intensity (how far the poor are from some agreed upon poverty line) and the inequality of poverty (how unequal the incomes of the poor are). Despite the numerous studies that have been published, one has to stress the fact that the most popular measure of poverty remains, for both, politicians and the public at large, the headcount ratio which gives the percentage of poor in the population. This need for a simple index explains probably also why, in another ﬁeld, unemployment, a simple measure such as the unemployment rate, remains the most popular measure of unemployment. There have been however in recent years some attempts to derive more sophisticated measures of unemployment that would take into account not only the percentage of individuals who are unemployed but also the mean duration of unemployment and even the inequality of these durations (see, e.g., the works of Sengupta, 1990; Paul, 1992; Shorrocks, 1992, 1993; Riese & Brunner, 1998, and more recently Basu & Nolen, 2008). Some of these works have stressed also the importance of the distinction between the total unemployment duration experienced by an individual and the various spells of unemployment that he experienced. But clearly the literature on unemployment measurement is much less abundant than that on income inequality or poverty measurement. The purpose of this paper is to borrow some of the ideas that have appeared in the studies that have just been cited, propose some measures of unemployment that are more sophisticated than the unemployment rate, and apply them to data on unemployment in the various Swiss cantons during the period 1993–2005. The paper is organized as follows. The Methodology section discusses various ways of measuring unemployment and, borrowing ideas from the poverty measurement literature, proposes four more general unemployment indices which are parallel to Sen Poverty Index, to its generalization by Shorrocks, to the FGT, and to the Watts poverty indices. The next section gives an empirical illustration based on

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Swiss data at the level of the ‘‘canton.’’ Using the so-called Shapley decomposition it computes the contribution to the difference between the value of each of these four unemployment indices in a given ‘‘canton’’ and in Switzerland as a whole, of three components measuring, respectively, the impact of differences in the traditional unemployment rate, in the average unemployment duration, and in the inequality in the unemployment durations. The paper ends by discussing the impact on the results obtained of assumptions made concerning the maximum unemployment duration.

THE METHODOLOGY On Various Ways of Measuring Unemployment Two indicators are commonly used to measure unemployment. The ﬁrst one is the unemployment rate which measures total unemployment as a proportion of the total labor force. This measure is obtained by asking individuals at some point of time t whether they are currently unemployed. The second indicator refers to the mean duration of unemployment. However, as stressed by Sengupta (1990) and Shorrocks (1993), there is much less agreement among economists about the way this mean duration should be measured. Several suggestions have in fact been made. The ﬁrst one is based on answers to a question like ‘‘If you are currently unemployed, for how long have you been unemployed?’’. When this type of data is taken into account to compute durations of unemployment, one looks in fact at the distribution of ‘‘interrupted spells of employment’’ (Shorrocks, 1993). A second possibility was suggested by Akerlof and Main (1980). It looks at the distribution of the completed spells of unemployment of those who are currently (at some time t) unemployed. In other words, whereas the ﬁrst approach looks ‘‘backward,’’ the second one looks ‘‘forward.’’ A third approach would also take a ‘‘backward look’’ and ask persons who are unemployed at some time t for how long they have been unemployed during a given period in the past (e.g., a year), no matter whether this unemployment duration included one or more spells of unemployment. Rather than looking at the mean duration of unemployment according to each of the three approaches previously mentioned we could also look at the

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distribution of these durations and compute some index of inequality of these durations, such as the Gini index. We can however think of a way of extending the ﬁrst approach which stresses the concept of ‘‘interrupted spell of unemployment.’’ This approach often assumes that the unemployment rate which serves as reference is that observed in December. It is also possible to base computation of the unemployment rate on data which are available for each of the 12 months and compute the expected monthly unemployment rate over a period of 12 months. Let vij be an indicator of unemployed deﬁned as follows: vij ¼ 1 if individual i was unemployed in month j and vij ¼ 0 otherwise. The expected monthly unemployment rate (U/N) during year t will then be deﬁned as ! X N 12 X U 1 1 ¼ vij (1) N 12 N i¼1 j¼1 This is in fact the way an annual unemployment rate is deﬁned. We may similarly deﬁne the mean duration of unemployment (among the unemployed) as the expected mean duration over all of the 12 months for which data are available. Let Dij denote the cumulative number of days of unemployment of individual i in month j. The expected mean duration of unemployment in year t, on the basis of the ﬁrst approach and assuming we have data for 12 months, may then be deﬁned as X 12 1 1 N X ðDij vij Þ (2) DA ¼ 12 N i¼1 j¼1 The present section has thus shown that depending on the approach adopted one may obtain very different values for the unemployment rate, the mean cumulative unemployment duration as well as for the Gini index of these cumulative unemployment durations. There is however an additional issue. We have hitherto analyzed separately three types of indicators of unemployment: the traditional unemployment rate, the mean duration of unemployment, and a measure of the inequality of these durations. The next section, using some previous work of Shorrocks (1993), will show that it is possible to construct a new measure of unemployment that will take into account all these three aspects of unemployment.

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Deriving a More Complete Measure of Unemployment As mentioned previously the data that are available often give the cumulated number of days at month j during which individual i has been unemployed without interruption. Let dij be the number of days he is unemployed in month j and let Dij denote the cumulated number of days of unemployment. We may therefore write that Dij ¼ Di; j1 þ d ij

if vij ¼ 1 and Dij ¼ 0

if vij ¼ 0

(3)

if vij ¼ 0

(4)

Let now Vij be the cumulated value of vij, that is, V ij ¼ V i; j1 þ vij

if vij ¼ 1 and V ij ¼ 0

To illustrate the inequality in these cumulated days of unemployment we can draw the following graph which has been called unemployment duration proﬁle curve by Shorrocks (1993). In Fig. 1 we plotted on the horizontal axis the cumulative values of the months of unemployment during year t of those who were unemployed Cumulative values of cumulative days of unemployment among the unemployed 1 N 12 U ∑ ∑ D = D = DLF 12N i =1 j =1 ij N A

A

M

H P

B O

U N

Q 1 Cumulative values of months of unemployment among unemployed

Fig. 1. Plot of Cumulative Values of the Months of Unemployment (Horizontal Axis) during Year t of Those Who Were Unemployed in Year t versus Cumulative Values of the Cumulative Number of Days (Vertical Axis) of Unemployment among the Unemployed.

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in year t, that is, of (1/12) (1/N) vij. This means, in fact, that we plotted the cumulative values: 1 1 V ij 12 N On the vertical axis we plotted the cumulative values of the cumulative number of days of unemployment among the unemployed, that is, of (1/12) (1/N) Dij. On both the horizontal and vertical axes the individuals are ranked by decreasing values of these cumulated durations Dij. Such a plot gives us the curve OHAM. It is easy to see, using (1) and (4), that the horizontal coordinate of A (the length OB) is equal to the annual unemployment rate (U/N) where U is the total number of individuals unemployed in year t and N the size of the labor force. The vertical ! coordinate of A (the length AB) will be equal to (1/12) (1/N) N P 12 P Dij ¼ DLF which is actually equal to the average cumulative i¼1 j¼1

duration of unemployment (in days) per individual in the labor force. It is then easy to derive that the slope of OA will be equal to the ratio ! ! N P 12 N P 12 P P Dij Dij i¼1 j¼1 i¼1 j¼1 1 1 !¼ ! (5) N P 12 N P 12 12 N P P ð1=12Þð1=NÞ vij vij i¼1 j¼1

i¼1 j¼1

which represents, in fact, the average cumulative number of days of unemployment DA among individuals who have been unemployed at some time in year t. Let now G(Dij) refer to the Gini index of the cumulative unemployment durations Dij and let OPA denote the straight line OA. The area OHAB is therefore equal to the sum of the area OHAP and of the triangle OPAB. The area of this triangle OPAB is clearly equal to (1/2) (U/N) (U/N) (DA) since the tangent of the slope AOB is equal to (AB/OB) ¼ DA. The area that lies between the curve OHA and the line OPA by construction looks like the area lying between a Lorenz curve and a diagonal and such an area is generally equal to half the Gini index of the variable whose cumulative values have been plotted. However, since this ‘‘diagonal’’ OPA does not end at a point whose coordinates are (1,1) but at point A whose coordinates are (U/N) and (U/N) DA it is easy to derive that the area between the curve OHA and the line OPA is equal to (1/2) G(Dij) (U/N) (U/N) DA ¼ (1/2) G(Dij) (U/N) DLF.

Various Ways of Measuring Unemployment

265

The sum M of the two areas OHAP and OPAB will therefore be equal to 1 U U 1 U M¼ DA ð1 þ GðDij ÞÞ ¼ DLF ð1 þ GðDij ÞÞ (6) 2 N N 2 N This indicator M may be considered as a generalized measure of unemployment. As may be observed, M is an increasing function of the probability for an individual to be unemployed during a randomly selected month (which is really the way the annual unemployment rate (U/N) is deﬁned). This indicator M increases also with the average value DLF in the whole labor force of the cumulative number of days of unemployment. Finally M will be higher, the more unequal these cumulative number of days of unemployment durations are, since it increases with G(Dij). When these cumulative durations of unemployment are standardized it turns out (see the proof in Appendix A) that this indicator M is in fact equal to half the value of the product of the unemployment rate (U/N) times ‘‘Sen’s unemployment index,’’ an index which is obtained by applying to the measurement of unemployment Sen’s (1976) index of unidimensional of poverty. This is in fact only the asymptotic value of Sen’s (1976) index, that is, it assumes that the size of the labor force is big enough. More precisely, let Eij be equal to (DM Dij) where DM ¼ {Max Dij} represents the maximal number of days of cumulative unemployment (it could correspond to one, two, or even more years). Eij represents therefore the number of days during which individual i was employed during the period under consideration (one, two, or even more years, depending on how it is deﬁned). Let now eij be equal to the ratio Eij/EM, where EM ¼ {Max Eij} is evidently equal to DM ¼ {Max Dij}. Let, also fij be equal to the ratio Dij/DM and fA to the ratio DA/DM. When applied to the analysis of unemployment, Sen’s (1976) poverty index may therefore be written as U ½ f A þ ð1 f A ÞGðeij Þ (7) SU ¼ N where G(eij) is the Gini index of the cumulative relative employment duration eij. Note that, as in the case of Sen poverty index, the formulation for SU holds only if U, the total number of unemployed individuals in year t, is big enough. Appendix A shows also that another possible measure of unemployment is twice the area lying under the curve OHAM, that is, the area OHAMQBO in Fig. 1. We then obtain (see Appendix A) what Shorrocks (1995) called

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‘‘The Revisited Sen Poverty Index’’ SUR which in the case of unemployment measurement will be expressed as " # U 2 U U fA 2 (8) ðð1 f A ÞGðeij ÞÞ þ SUR ¼ N N N Another very popular poverty index is the so-called FGT index (Foster, Greer, & Thorbecke, 1984). When applied to the measurement of unemployment this index (FGTU) will be expressed as X 12 1 1 U X FGTU ¼ ðf Þa (9) 12 N i¼1 j¼1 ij It is easy to observe that when a ¼ 0, FGTU ¼ (U/N), and that when a ¼ 1, U FGTU ¼ f N A Let us now take the case of a ¼ 2. We may then write that X 1 1 U FGTU ¼ ð f Þ2 12 N i¼1 ij ) ( X 1 U 1 U 2 ¼ ½ð f f A Þ þ f A 12 N U i¼1 ij

1 U FGTU ¼ Varð f ij Þ þ ð f A Þ2 12 N 1 U ¼ ð f A Þ2 f1 þ Coef: Var: ð f ij Þg 12 N

(10)

where Var( ) refers to the variance and Coef. Var.( ) to the coefﬁcient of variation of a variable. So in the case where a ¼ 2 we observe, as in the case of the indices SU and SUR, that the index FGTU is a function of the unemployment ratio (U/N), the average cumulative unemployment duration DA, and of a measure of the dispersion of the relative cumulative unemployment durations, in this case, the coefﬁcient of variation of these relative cumulative durations. Finally, we can also apply to the analysis of unemployment the poverty index deﬁned by Watts (1968). When applied to the measurement of

Various Ways of Measuring Unemployment

unemployment this index will be written as X 1 U EM log WU ¼ E ij N i¼1

267

(11)

Expression (11) may, however, be also written as X # "X U U U 1 EM 1 EA þ log log WU ¼ E E ij N U U A i¼1 i¼1

(12)

where EA is equal to the average of the employment durations Eij . Note, however, that the ﬁrst expression in square brackets on the R.H.S. of (12) may also be written as EM (13) W R ¼ log EA so that WR measures somehow the percentage difference between the maximum cumulative duration of employment and the average cumulative duration of employment. In other words WR indicates somehow to which percentage of the maximal employment duration the average cumulative unemployment duration corresponds. The second expression in square brackets on the R.H.S. of (12) may be written as LU ¼ logðE A Þ logðE G Þ

(14)

where EG refers to the geometric mean of the cumulative employment durations Eij. It is then easy to observe that LU measures the percentage difference between the arithmetic and the geometric means of the cumulative employment durations Eij. Since the gap between the arithmetic and a geometric mean of a variable is usually considered as an indicator of the inequality of the distribution of this variable (see, Champernowne, 1953) the indicator LU measures, in fact, the inequality of the cumulative employment durations among the individuals who were unemployed at least part of the year t. This indicator is also known under the name of Bourguignon (1979) and Theil (1967) inequality index. Combining expressions (11)–(14) we end up with U ðW R þ LU Þ (15) WU ¼ N

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Like the indices SU, SUR, and FGTU deﬁned earlier the index WU is a function of three components measuring, respectively, the unemployment rate, the gap between maximal employment duration and the average cumulative unemployment duration, and ﬁnally the inequality in the employment durations among those who were unemployed at least part of the time in year t. Comparing Unemployment Measures in Different Areas The four indices SU, SUR, FGTU, and WU that have been deﬁned previously may be computed for each area j in a given country as well as for the whole country. The difference between the value that a given index (one of the four previously mentioned) takes for the whole country and for a given area may then be decomposed, using the so-called Shapley decomposition procedure, into three components (see, Appendix B) that measure, respectively, the extent of differences between the country as a whole and a given area in the unemployment rate, in the gap between the maximal and average values of the cumulative number of days of unemployment, and ﬁnally in the inequality of the cumulative number of days of unemployment (employment) among those who were unemployed at least part of the year.

AN EMPIRICAL ILLUSTRATION The concepts that have been previously presented have been applied to data on unemployment in the 26 Swiss areas which are called ‘‘cantons’’ for the period 1993–2005. To illustrate these concepts we have used the approach where unemployment is measured via the information on the expectancy of the interrupted spells of unemployment over the whole year. But we clearly could have used one of the three other approaches (in fact computations similar to those presented in this section but based on the other three approaches are available from the authors upon request). As was just mentioned we look at the values of the cumulative durations of unemployment as they are given each month of the year for the various unemployed individuals. More precisely we work with the expectancy of these cumulative duration of unemployment on the basis of data on cumulative unemployment for each of the 12 months of the year. As maximal value for these cumulative durations we assumed again that it was equal to 365 days. In Table 1 we give data on the unemployment rate (the expectancy of the monthly unemployment rates which is also the value of the annual unemployment rate), on the average value observed during the year of the cumulative unemployment durations and ﬁnally on the Gini index of these

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Table 1. Looking at the ‘‘Expected’’ Interrupted Spells of Unemployment in 2005 [Value of the Unemployment Rate (U/N), of the Average Value DA of the Cumulative Unemployment Durations and of the Gini Index of Unemployment Durations G(Dij) for Switzerland and the Various Cantons]. Canton

ZH BE LU UR SZ OW NW GL ZG FR SO BS BL SH AR AI SG GR AG TG TI VD VS NE GE JU Switzerland as a whole

Unemployment Average Value of the Gini Index of Unemployment Rate (U/N) Unemployment Durations DA Durations (G(Dij)) 0.0402 0.0283 0.0307 0.0131 0.0231 0.0161 0.0196 0.0250 0.0315 0.0309 0.0337 0.0406 0.0330 0.0328 0.0219 0.0147 0.0297 0.0216 0.0325 0.0307 0.0486 0.0533 0.0396 0.0433 0.0737 0.0422 0.0376

173.50 153.19 166.45 123.26 154.88 126.72 130.60 147.73 183.39 158.71 165.07 179.56 177.11 174.84 196.02 155.57 168.79 119.32 168.17 164.34 182.25 209.16 134.22 203.05 234.42 192.79 179.68

0.4122 0.4373 0.4211 0.4570 0.4291 0.4679 0.4600 0.4487 0.3986 0.4313 0.4151 0.3978 0.3926 0.4084 0.3537 0.4209 0.4080 0.4696 0.4180 0.4137 0.3869 0.3432 0.4725 0.3482 0.2963 0.3715 0.3990

Note: Similar tables for the other years are available upon request from the authors.

cumulative durations of unemployment, for Switzerland as a whole and for each canton in 2005. It appears that the highest rates of unemployment are observed in the cantons of Geneva, Vaud, and Tessin and the lowest in the cantons of Uri, Appenzell (AI), and Obwalden. As far as average durations of unemployment are concerned the highest averages are observed in the cantons of Geneva and Vaud. The lowest average durations are observed in

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the cantons of Graubu¨nden, Uri, and Obwalden. Finally the highest levels of unemployment duration inequality are observed in the cantons of Valais, Graubu¨nden, and Obwalden and the lowest in the cantons of Geneva, Neuchaˆtel, and Vaud. In Table 2 we give the values in 2005, for Switzerland as a whole as well as for each canton, of the three indices of unemployment which we have deﬁned previously, the Sen index, SU, Shorrocks’ generalization of the Sen Table 2. Looking at the ‘‘Expected’’ Interrupted Spells of Unemployment in 2005 [Value of the Three Indices of Unemployment (the Sen Index SU, Shorrocks’ Extension SUR of the Sen Index, and the Index FGTU) for Switzerland and the Various Cantons]. Canton

ZH BE LU UR SZ OW NW GL ZG FR SO BS BL SH AR AI SG GR AG TG TI VD VS NE GE JU Switzerland as a whole

Sen Index SU of Unemployment

Shorrocks’ Extension SUR of the Sen’s Index of Unemployment

The Foster, Greer, and Thorbecke Index FGTU of Unemployment

26.95 17.08 19.88 6.46 14.02 8.21 10.23 14.63 22.14 19.24 21.55 27.93 22.32 22.11 15.89 8.92 19.31 10.40 21.24 19.57 33.66 41.06 21.45 32.47 61.33 30.57 25.92

37.72 23.57 27.74 8.84 19.50 11.14 13.95 20.06 31.36 26.64 30.16 39.48 31.73 31.09 23.30 12.50 27.18 14.07 29.68 27.44 47.82 60.07 28.82 47.48 92.18 43.99 36.63

13.89 8.06 9.95 2.57 6.62 3.40 4.31 6.75 11.88 9.30 10.64 14.66 11.50 11.46 8.73 4.21 9.65 4.12 10.69 9.60 17.75 23.94 9.39 18.43 38.94 16.82 13.64

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index, SUR, and the Foster, Greer and Thorbecke index FGTU. It appears that the highest values of the unemployment indices are observed in the cantons of Geneva, Vaud, Tessin, and Neuchaˆtel and the lowest in the cantons of Uri, Appenzell AI, Obwalden, and Graubu¨nden. In Tables 3–5 we give, for each of the three unemployment indices previously mentioned, the results of the Shapley decomposition of the Table 3. Looking at the ‘‘Expected’’ Interrupted Spells of Unemployment in 2005 [Shapley Decomposition of the Difference between the Value of the Sen Index SU for Switzerland as a Whole and for Each Canton]. Canton

ZH BE LU UR SZ OW NW GL ZG FR SO BS BL SH AR AI SG GR AG TG TI VD VS NE GE JU

Gap between the Contribution of Contribution of Contribution of National and Differences in the Differences in the Differences in the Cantonal Values Unemployment Average Unemployment Degree of Inequality of the Sen Index Rate (D(U/N)) Duration per Member of the Employment SU of the Labor Force Durations (DG(e)) (D(DLF)) 1.036 8.840 6.032 19.456 11.892 17.703 15.682 11.283 3.775 6.680 4.363 2.016 3.598 3.810 10.032 16.999 6.610 15.519 4.672 6.344 7.743 15.146 4.471 6.550 35.417 4.654

1.714 6.029 4.647 14.561 9.404 12.962 10.977 8.095 4.262 4.411 2.628 2.057 3.140 3.316 11.174 14.843 5.349 9.417 3.434 4.563 7.578 11.474 1.219 4.072 27.542 3.228

0.408 1.548 0.779 2.659 1.333 2.624 2.584 1.784 0.213 1.259 0.905 0.007 0.155 0.288 0.802 1.115 0.632 3.359 0.694 0.914 0.187 2.110 3.221 1.523 4.447 0.858

0.269 1.263 0.606 2.236 1.156 2.117 2.122 1.404 0.273 1.010 0.829 0.034 0.303 0.206 0.340 1.041 0.629 2.743 0.544 0.867 0.022 1.562 2.469 0.955 3.428 0.568

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Table 4. Looking at the ‘‘Expected’’ Interrupted Spells of Unemployment in 2005 [Shapley Decomposition of the Difference Between the Value of Shorrocks Generalization SUR of the Sen Index of Unemployment for Switzerland as a Whole and for Each Canton]. Canton

ZH BE LU UR SZ OW NW GL ZG FR SO BS BL SH AR AI SG GR AG TG TI VD VS NE GE JU

Gap between the Contribution of Contribution of Contribution of National and Differences in the Differences in the Differences in the Cantonal Values Unemployment Average Unemployment Degree of Inequality of the Index SUR Rate (D(U/N)) Duration per Member of the Employment of the Labor Force Durations (DG(e)) (D(DLF)) 1.091 13.057 8.896 27.793 17.133 25.487 22.685 16.567 5.269 9.987 6.469 2.848 4.900 5.542 13.329 24.133 9.448 22.562 6.952 9.191 11.186 23.437 7.808 10.853 55.546 7.358

2.382 8.338 6.458 20.041 13.058 17.786 15.086 11.164 5.966 6.108 3.656 2.872 4.404 4.626 15.943 20.715 7.468 12.881 4.775 6.356 10.596 16.261 1.661 5.782 39.401 4.547

1.281 4.676 2.417 7.683 4.038 7.636 7.533 5.358 0.687 3.844 2.783 0.023 0.485 0.909 2.604 3.385 1.959 9.595 2.157 2.805 0.591 7.103 9.373 5.033 15.938 2.788

0.010 0.042 0.021 0.069 0.037 0.065 0.066 0.046 0.010 0.035 0.030 0.001 0.011 0.007 0.011 0.032 0.021 0.086 0.019 0.030 0.001 0.073 0.096 0.039 0.207 0.023

gap between the national value of these indices and that observed in each canton. Such a breakdown allows one to identify the respective contributions to this gap of differences in the unemployment rate, in the average durations of unemployment, as well as in the inequality of unemployment (or employment, depending on the index selected) durations.

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Various Ways of Measuring Unemployment

Table 5. Looking at the ‘‘Expected’’ Interrupted Spells of Unemployment in 2005 [Shapley Decomposition of the Difference between the Value of the Foster, Greer, and Thorbecke Index FGTU of Unemployment for Switzerland as a Whole and for Each Canton]. Canton Gap between the Contribution of Contribution of Contribution of National and Differences in the Differences in the Differences in the Cantonal Values Unemployment Standardized Average Coefﬁcient of Variation of the Index Rate (D(U/N)) Unemployment of the Standardized FGTU Duration (D( fA)) Unemployment Durations (Dfij) ZH BE LU UR SZ OW NW GL ZG FR SO BS BL SH AR AI SG GR AG TG TI VD VS NE GE JU

0.243 5.589 3.692 11.071 7.028 10.244 9.331 6.892 1.760 4.344 3.007 1.013 2.140 2.187 4.911 9.432 3.997 9.521 2.955 4.043 4.103 10.295 4.252 4.788 25.293 3.174

0.893 3.016 2.387 6.842 4.704 6.170 5.256 4.012 2.266 2.230 1.342 1.081 1.636 1.732 6.015 7.429 2.746 4.422 1.769 2.323 3.993 6.375 0.592 2.230 16.057 1.738

0.637 2.182 1.174 3.256 1.892 3.273 3.271 2.457 0.350 1.825 1.347 0.012 0.243 0.453 1.371 1.588 0.958 4.014 1.053 1.354 0.302 3.914 4.136 2.717 9.470 1.464

0.013 0.390 0.131 0.973 0.432 0.801 0.804 0.422 0.155 0.289 0.318 0.056 0.261 0.003 0.266 0.415 0.293 1.084 0.133 0.366 0.192 0.006 0.708 0.159 0.234 0.028

It appears that in the four cantons with the highest positive (Geneva and Vaud) or negative (Uri and Obwalden) gap, the contribution of differences between the unemployment rate in these cantons and that in Switzerland account for 60–78% of the overall gap, depending on the index of unemployment which is used. Note, however, that for these four cantons the contribution of the two other factors (differences in the average

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unemployment durations and differences in the inequality of unemployment durations) cannot be ignored, this being specially true for the average unemployment duration.

ANALYZING THE IMPACT OF THE MAXIMUM UNEMPLOYMENT DURATION In this section we want to analyze the impact that the choice of a maximum duration of unemployment may have on the results of the ‘‘Shapley decomposition.’’ For simplicity we will only consider the case where we take a ‘‘backward looking’’ approach and measure unemployment via the information on the interrupted spells of unemployment as they are observed in the month of December. Here also we limit the analysis to the year 2005. We will compare three cases: the maximum duration is assumed to be 365 days (as was assumed previously) the maximum duration of unemployment is that actually observed in December 2005 the maximum duration of unemployment is 5,000 days, which is slightly above the greatest unemployment duration observed in all years for which data are available (1994–2005) We present the results only for the decomposition for the Sen index of unemployment SU.1 Finally in each table we give ﬁrst the actual gap between the value of the index in Switzerland and its value in a given canton and then the contributions, in percentage terms, of the three determinants of the indices of unemployment, namely the actual unemployment rate, the average duration of unemployment, and the inequality of unemployment (or employment) durations. All these results are presented in Tables 6–8. Let us ﬁrst consider the ﬁrst case where the maximum duration of unemployment is still 365 days (Table 6). It appears that although the greatest relative contribution to the unemployment indices is that of the unemployment rates, there are cases where the contribution of the average unemployment duration is quite high in comparison to that of the unemployment rate. When using the index SU this is, for example, the case of the cantons of Zurich (ZH) and of Valais (VS).

275

Various Ways of Measuring Unemployment

Table 6. Looking at the Interrupted Spells of Unemployment in 2005 [Shapley Decomposition in Percentage Terms of the Difference between the Value of the Sen Index SU for Switzerland as a Whole and for Each Canton, under the Assumption that the Maximum Duration of Unemployment is 365 Days]. Canton

ZH BE LU UR SZ OW NW GL ZG FR SO BS BL SH AR AI SG GR AG TG TI VD VS NE GE JU

Gap between the Contribution to Contribution to this Actual Values of this Gap (in Gap (in Percentage) of the Sen Index SU Percentage) of Differences in the at the National Differences in the Average Unemployment and Cantonal Unemployment Duration per Member Levels Rate (D(U/N)) of the Labor Force (D(DLF)) 1.344 7.692 5.114 18.040 10.151 15.484 13.521 10.848 4.315 6.787 5.481 2.855 3.860 4.958 8.234 13.817 6.474 14.548 4.522 5.362 11.159 12.131 4.021 4.598 32.034 1.806

43.68 63.40 81.85 79.92 83.28 67.86 64.54 55.46 116.73 47.56 53.56 49.28 94.48 48.04 120.56 85.56 82.92 56.94 78.70 72.45 101.10 74.03 148.10 88.28 73.88 127.81

33.26 20.23 9.54 11.14 8.73 17.20 18.83 24.74 8.25 28.25 22.93 27.25 2.93 25.55 12.30 7.23 8.88 24.02 10.71 12.78 0.10 15.02 146.13 9.24 14.87 8.70

Contribution to this Gap (in Percentage) of Differences in the Degree of Inequality of the Employment Durations (DG(e))

23.07 16.37 8.60 8.95 7.99 14.94 16.63 19.79 8.48 24.19 23.51 23.47 2.59 26.40 8.26 7.21 8.20 19.04 10.60 14.77 1.00 10.96 101.96 2.48 11.24 19.11

If we now take the case where the maximum unemployment duration is that actually observed in December 2005 (Table 7) we observe a somehow different picture. The cantons of Zurich (ZH) and Valais (VS) are not anymore the only ones, for the index SU, for which the contribution

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Table 7. Looking at the Interrupted Spells of Unemployment in 2005 [Shapley Decomposition in Percentage Terms of the Difference between the Value of the Sen Index SU for Switzerland as a Whole and for Each Canton, under the Assumption that the Maximum Duration of Unemployment is that Observed in December 2005]. Canton

ZH BE LU UR SZ OW NW GL ZG FR SO BS BL SH AR AI SG GR AG TG TI VD VS NE GE JU

Gap between the Contribution to Contribution to this Actual Values of this Gap (in Gap (in Percentage) of the Sen Index SU Percentage) of Differences in the at the National Differences in the Average Unemployment and Cantonal Unemployment Duration per Member Levels Rate (D(U/N)) of the Labor Force (D(DLF)) 0.127 1.049 0.689 2.081 1.334 1.884 1.601 1.256 0.499 0.691 0.713 0.047 0.594 0.455 1.012 1.700 0.854 1.768 0.661 0.772 1.123 1.931 0.535 0.969 4.924 0.462

49.61 49.86 66.62 75.11 67.07 58.55 59.34 53.18 113.20 53.48 45.30 314.89 66.78 60.35 107.72 73.47 68.50 49.75 58.55 54.40 111.22 56.21 122.85 50.57 60.22 58.75

83.46 32.51 21.63 16.05 20.26 26.27 27.11 30.89 8.60 32.03 31.56 217.02 15.85 23.35 10.78 16.06 17.80 33.18 24.36 25.91 6.32 30.07 159.18 36.64 27.78 30.24

Contribution to this Gap (in Percentage) of Differences in the Degree of Inequality of the Employment Durations (DG(e))

66.14 17.64 11.76 8.84 12.68 15.18 13.55 15.92 4.60 14.49 23.14 197.87 17.37 16.30 3.07 10.47 13.70 17.07 17.10 19.69 4.90 13.72 63.67 12.80 12.00 11.02

of the average unemployment duration (in percentage term) is important. This is now also the case for the cantons of Basel Stadt (BS), Graubu¨nden (GR), and even Jura (JU). Finally the results of the cases where we take as maximum unemployment duration 5,000 days (Table 8) are very close to those where the maximum

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Various Ways of Measuring Unemployment

Table 8. Looking at the Interrupted Spells of Unemployment in 2005 [Shapley Decomposition in Percentage Terms of the Difference between the Value of the Sen Index SU for Switzerland as a Whole and for Each Canton, Under the Assumption that the Maximum Duration of Unemployment is 5,000 Days]. Canton

ZH BE LU UR SZ OW NW GL ZG FR SO BS BL SH AR AI SG GR AG TG TI VD VS NE GE JU

Gap between the Contribution to Contribution to this Actual Values of this Gap (in Gap (in Percentage) of the Sen Index SU Percentage) of Differences in the at the National Differences in the Average Unemployment and Cantonal Unemployment Duration per Member Levels Rate (D(U/N)) of the Labor Force (D(DLF)) 1.191 9.837 6.458 19.518 2.509 17.666 15.016 11.776 4.678 6.476 6.689 0.438 5.569 4.268 9.487 15.944 8.007 16.581 6.202 7.243 10.530 18.113 5.013 9.084 46.181 4.332

49.41 49.83 66.63 75.08 67.06 58.52 59.34 53.23 113.38 53.51 45.29 316.86 66.71 60.28 107.65 73.44 68.57 49.76 58.50 54.40 111.23 56.21 122.70 50.58 60.21 58.77

83.31 32.57 21.68 16.09 20.28 26.32 27.11 30.92 8.70 32.07 31.64 218.91 15.93 23.39 10.76 16.09 17.77 33.21 24.41 25.90 6.31 30.14 159.29 36.73 27.84 30.29

Contribution to this Gap (in Percentage) of Differences in the Degree of Inequality of the Employment Durations (DG(e))

66.11 17.60 11.69 8.82 12.66 15.15 13.55 15.85 4.68 14.42 23.06 197.95 17.36 16.33 3.11 10.47 13.65 17.03 17.09 19.70 4.92 13.65 63.42 12.69 11.95 10.94

unemployment duration is that actually observed in December 2005 so that we will not analyze them separately. In all the results what however is quite striking is the growing role of the average unemployment duration when a longer maximal unemployment

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duration is selected and even to some degree the more important impact of the inequality of unemployment durations. Take, for example, the case of the canton of Geneva (GE) for which the value of the index of unemployment, whatever the index, is always the highest of all cantons. Here, we have observed that whereas 60–74% (depending on the index) of the gap between the value of the index in Switzerland as a whole and in the canton of Geneva was the consequence of differences in unemployment rates, when the maximum unemployment duration is 365 days, this impact of the unemployment rate goes down to 45–60% (depending on the index) when a longer maximal unemployment duration is selected (either the maximal duration observed or 5,000 days). Moreover, even the impact of the inequality in unemployment (or employment) durations is now greater. For the canton of Geneva, depending on the index, it varied from 0% to 11% when the maximum unemployment duration is 365 days. With a greater maximal duration the impact of this inequality in unemployment durations for the canton of Geneva varies now from 0% to 23%.

CONCLUDING COMMENTS This paper attempted to borrow some ideas from the poverty measurement literature to propose some more sophisticated measures of unemployment which take into account not only the unemployment rate but also the average duration of unemployment and the inequality in the distribution of these durations. It also applied the so-called Shapley decomposition to decompose the difference between the value of an unemployment index at the national and at the regional level into three contributions reﬂecting the three aspects of unemployment that have just been mentioned. An empirical illustration based on Swiss data for the period 1993–2005 seems to conﬁrm the usefulness of such an approach. It also showed the relative sensibility of the decomposition results to the maximum unemployment duration that has been selected.

NOTE 1. Results relative to the decomposition of Shorrocks’ generalization SUR of Sen unemployment index and to the Foster, Greer and Thorbecke (FGT) index of unemployment are available upon request from the authors.

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ACKNOWLEDGMENTS This paper was written while Jacques Silber was visiting the Fundacio´n de Estudios de Economı´ a Aplicada (FEDEA), Madrid. He wishes to thank FEDEA for its warm hospitality. All the authors acknowledge the ﬁnancial support of SECO, State Secretariat for Economic Affairs, Switzerland, for a research project entitled ‘‘Analysis of regional unemployment inequalities in Switzerland.’’

REFERENCES Akerlof, G. A., & Main, B. G. M. (1980). Unemployment spells and unemployment experience. American Economic Review, 71(5), 885–893. Basu, K. & Nolen, P. (2008). Unemployment and vulnerability: A new class of measures, its axiomatic properties and applications. In: P. Pattanaik, K. Tadenuma, Y. Xu, & N. Yoshihara (Eds), Rational choice and social welfare. Heidelberg: Springer. Bourguignon, F. (1979). Decomposable income inequality measures. Econometrica, 47(4), 901–920. Champernowne, D. G. (1953). A model of income distribution. Economic Journal, 63, 318–351. Foster, J. E., Greer, J., & Thorbecke, E. (1984). A class of decomposable poverty measures. Econometrica, 52(3), 761–766. Jenkins, S. P., & Lambert, P. (1997). Three ‘I’s of poverty curves with an analysis of UK poverty trends. Oxford Economic Papers, 49(2), 317–327. Paul, S. (1992). An illfare approach to the measurement of unemployment. Applied Economics, 24(7), 739–743. Riese, M., & Brunner, J. K. (1998). Measuring the severity of unemployment. Journal of Economics, 67(2), 167–180. Sastre, M., & Trannoy, A. (2002). Shapley inequality decomposition by factor components: Some methodological issues. Journal of Economics (Suppl. 9), 51–89. Sen, A. K. (1976). Poverty an ordinal approach to measurement. Econometrica, 44, 219–231. Sengupta, M. (1990). Unemployment duration and the measurement of unemployment. Mimeo, University of Canterbury, New Zealand. Shapley, L. S. (1953). A value for n-persons games. In: H. W. Kuhn & F. W. Tucker (Eds), Contributions to the theory of games II (Annals of mathematical studies 28) (pp. 307–317). Princeton: Princeton University Press. Shorrocks, A. F. (1992). Spell incidence, spell duration and the measurement of unemployment. Mimeo, University of Essex. Shorrocks, A. F. (1993). On the measurement of unemployment. Mimeo, University of Essex and Southern Methodist University. Shorrocks, A. F. (1995). Revisiting the Sen poverty index. Econometrica, 63(5), 1225–1230. Shorrocks, A. F. (1999). Decomposition procedures for distributional analysis: A uniﬁed framework based on the Shapley value. Mimeo, University of Essex.

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Silber, J. (1989). Factors components, population subgroups and the computation of the Gini index of inequality. The Review of Economics and Statistics, LXXI(1), 107–115. Theil, H. (1967). Economics and information theory. Amsterdam: North Holland. Watts, H. (1968). An economic deﬁnition of poverty. In: D. P. Moynihan (Ed.), On understanding poverty. New York: Basic Books. Zheng, B. (1997). Aggregate poverty measures. Journal of Economic Surveys, 11, 123–162.

APPENDIX A. ON THE LINK BETWEEN THE UNEMPLOYMENT DURATION PROFILE CURVE AND SEN’S INDEX OF POVERTY (WHEN APPLIED TO THE MEASUREMENT OF UNEMPLOYMENT) Recall (see expression (6)) that the area OHAB may be expressed as 1 U U M¼ (A.1) DA ð1 þ GðDij ÞÞ 2 N N If we now normalize the durations DA and Dij by dividing them by their maximal value DM we may write that M 1 U U DA Dij ¼ 1þG (A.2) DM DM DM 2 N N or as

M DM

1 U U ¼ ð f A Þð1 þ Gð f ij ÞÞ 2 N N

(A.3)

where fij ¼ (Dij/DM) and fA ¼ (DA/DM). But fij ¼ (1 eij) where eij ¼ (Eij/EM) ¼ (Eij/DM), we may therefore write that Gð f ij Þ ¼ Gð1 eij Þ

(A.4)

Using, however, the well-known formulas for expressing the Gini index of a sum of components (see, e.g., Silber, 1989) we may rewrite G(1 eij) as 1 eA ðPs:Gð1ÞÞ þ ðPs:Gðeij ÞÞ (A.5) Gð1 eij Þ ¼ 1 eA 1 eA

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where Ps.G(eij) refers to the pseudo-Gini and eA is equal to the average value of the standardized employment durations eij (eA ¼ (EA/EM)). Note, however, that the pseudo-Gini of a vector of the constant 1 is 0 so that the ﬁrst expression on the R.H.S. of (A.2) is 0. Since (see, Silber, 1989) in the second expression on the R.H.S. of (A.2) the pseudo-Gini of eij implies that the elements eij are ranked by decreasing values of the expressions (1 eij) we easily derive that Ps:Gðeij Þ ¼ Gðeij Þ

(A.6)

Combining (A.4) and (A.6), we obtain Gð f ij Þ ¼ Gð1 eij Þ ¼

eA eA ðGðeij ÞÞ ¼ Gðeij Þ 1 eA 1 eA

(A.7)

and since by deﬁnition eA ¼ (1 fA), we conclude that Gð f ij Þ ¼

1 fA Gðeij Þ fA

(A.8)

Combining now (A.3) and (A.8), we end up with

M DM

1 U U ¼ f ð1 þ Gð f ij ÞÞ 2 N N A 1 U U 1 fA Gðeij Þ f 1þ ¼ fA 2 N N A 1 U U 1 ðf A þ ðð1 f A ÞGðeij ÞÞÞ f ¼ 2 N N A fA 1 U U ¼ ð f A þ ðð1 f A ÞGðeij ÞÞÞ 2 N N M 1 U ¼ SU 2 DM 2 N

ðA:9Þ

where SU is the application to unemployment measurement of the Sen index of poverty (see expression (7)).

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Finally, since the area BAMQ is equal to (1 (U/N))(U/N)DA it is easy to conclude, using (A.9), that the area OHAMQBO is equal to 1 U U U U DM ð f A þ ðð1 f A ÞGðeij ÞÞÞ þ 1 f 2 N N N N A # " (" )# 1 U 2 U U U ðð1 f A ÞGðeij ÞÞ þ ¼ DM f þ2 1 2 N N A N N # " (" )# 2 1 U U U ðð1 f A ÞGðeij ÞÞ þ f 2 ðA:10Þ ¼ DM 2 N N A N Expression (A.10) is in fact the application to unemployment measurement of what Shorrocks (1995) has called ‘‘The Revisited Sen Poverty Index’’ which has better properties than Sen’s (1976) original index.

APPENDIX B. ON THE CONCEPT OF SHAPLEY DECOMPOSITION The concept of Shapley (1953) decomposition is a technique borrowed from game theory but extended to applied economics by Shorrocks (1999) and Sastre and Trannoy (2002). Let us explain it brieﬂy. Assume an indicator I is a function of three determinants a, b, c and is written as I ¼ I(a, b, c). I could be an index of inequality but more generally any function of variables, this function being linear or not. There are obviously 3! ¼ 6 ways of ordering these three determinants a, b, and c: ða; b; cÞ; ða; c; bÞ; ðb; a; cÞ; ðb; c; aÞ; ðc; a; bÞ; ðc; b; aÞ

(B.1)

Each of these three determinants may be eliminated ﬁrst, second, or third. The respective (marginal) contributions of the determinants a, b, c will hence be a function of all the possible ways in which each of these determinants may be eliminated. Let, for example, C(a) be the marginal contribution of a to the indicator I(a, b, c). If a is eliminated ﬁrst its contribution to the overall value of the indicator I will be expressed as I(a, b, c) I(b, c) where I(b, c) corresponds to the case where a is equal to zero. Since expression (B.1) indicates that there are two cases in which a appears ﬁrst and may thus be eliminated ﬁrst we will give a weight of (2/6) to this possibility.

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If a is eliminated second, it implies that another determinant has been eliminated ﬁrst (and been assumed to be equal to 0). Expression (B.1) indicates that there are two cases in which this possibility occurs, the one denoted in (B.1) as (b, a, c) and the one denoted (c, a, b). In the ﬁrst case the contribution of a will be written as I(a, c) I(c), while in the second it is expressed as I(a, b) I(b). To each of these two cases we evidently give a weight of (1/6). Finally if a is eliminated third, it implies that both b and c are assumed to be equal to 0. Expression (B.1) indicates that there are two such cases, the one denoted (b, c, a) and the one denoted (c, b, a). Since we may assume that when each of the three determinants is equal to 0, the indicator I is equal to 0, we may write that the contribution of a in this case will be equal to I(a) 0 ¼ I(a) and evidently we have to give a weight of (2/6) to such a possibility since there are two such cases. We may therefore summarize what we have just explained by stating that the marginal contribution C(a) of the determinant a to the overall value of the indicator I may be written as 2 1 1 2 CðaÞ ¼ ½Iða; b; cÞ Iðb; cÞ þ ½Iða; cÞ IðcÞ þ ½Iða; bÞ IðbÞ þ IðaÞ 6 6 6 6 (B.2) One can similarly determine the marginal contribution C(b) of b and C(c) of c and then ﬁnd out that Iða; b; cÞ ¼ CðaÞ þ CðbÞ þ CðcÞ

(B.3)

This Shapley decomposition may be also applied in a similar way to the case where one wants to understand the respective contributions to the change over time in the value of the indicator I, this change being written as DI, of the variations over time in the values of the three determinants a, b, and c, these variations being expressed as Da, Db, and Dc. In our case DI would refer, for example, to the difference between the value of the Sen index in a given canton and its value in the whole of Switzerland, Da to the difference between the unemployment rate K in the canton and in Switzerland, Db to the difference between the average unemployment duration in the canton and in Switzerland, and ﬁnally Dc to the difference between the inequality in unemployment durations in the canton and in Switzerland.

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APPENDIX C. NAMES OF THE CANTONS AND THEIR ABBREVIATION Abbreviation ZH BE LU UR SZ OW NW GL ZG FR SO BS BL SH AR AI SG GR AG TG TI VD VS NE GE JU

Full Name of Canton Zu¨rich Bern Luzern Uri Schwyz Obwalden Nidwalden Glarus Zug Freiburg Solothurn Basel Stadt Basel Land Schaffhausen Appenzell – Ausser Rhoden Appenzell – Inner Rhoden Sankt Gallen Graubu¨nden Aargau Thurgau Tessin Waadt Wallis Neuenburg Genf Jura

INCOME MOBILITY IN ARGENTINA Luis Beccaria and Fernando Groisman ABSTRACT Purpose: The paper analyzes the variability of labor incomes in Argentina from mid-1980s to 2005. The magnitude of income instability and its determinants are evaluated under different macroeconomic contexts. It also analyzes how income ﬂuctuations have inﬂuenced income distribution. Finally, the income convergence hypothesis is explored. Methodology/approach: Different quantitative procedures are employed to measure mobility from dynamic information coming from the regular household survey. Four periods are distinguished that are relatively homogeneous. Dynamic pseudo-panels are also considered. Findings: The growth in occupational instability registered since the mid1990s led to a high variability of incomes despite the macroeconomic stability enjoyed throughout the nineties. Moreover, the panorama of growing inequality in the distribution of monthly income (the usual measure employed in Argentina) is also appropriate to describe what happened with the changes in the distribution of more permanent incomes. Finally, long-term income mobility in Argentina is scarce, indicating that the income path does not converge to the general mean. Research limitations/implications (if applicable): Data refer only to Greater Buenos Aires since microdata are not available for the other areas Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 285–321 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16013-6

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covered by survey for the entire period under analysis. However, results are reasonably representative of the whole urban areas of the country. Originality/value of paper: This research identiﬁes the relative importance of labor market and macroeconomic factors in explaining income mobility. Moreover, it is for the ﬁrst time in Argentina that dynamic information coming from panel data and pseudo-panels are analyzed together.

1. INTRODUCTION The distribution of both individual and family incomes in Argentina has become steadily more concentrated since the mid-1970s. This trend persisted throughout the 1980s, which were largely years of instability and stagnation, and the following decade, despite better macroeconomic performance [see Altimir and Beccaria (2001)]. Throughout this period, there were also sharp changes in inﬂation: very high rates in the 1970s and 1980s, including hyperinﬂationary spikes towards the end of the latter decade and in the early 1990s, before substantial price stability was restored in the rest of the last decade of the 20th century. As inﬂation is a key factor in explaining the stability of real incomes, the latter ought to have worsened in the 1970s, and especially in the 1980s, and then should have improved in the following decade. Nonetheless, there is also evidence of high levels of job instability, particularly in the 1990s [see Hopenhayn (2001); Galiani and Hopenhayn (2000); and Beccaria and Maurizio (2004)], which also affects income variability at both the individual and the household levels. It is therefore worth making a more in-depth analysis of income instability in Argentina’s different macroeconomic situations, given the adverse effects of such ﬂuctuations on individual welfare levels. In particular, instability increases risk and thus diminishes the utility of a given ﬂow of resources; and it can also undermine consumption levels even when predictable. Instability may go hand in hand with mobility, which generally means changes in the relative position of individuals in the income distribution, or changes in the distances among them. Besides, mobility affects inequality; speciﬁcally, the degree of income concentration measured with crosssectional data may differ from that resulting from considering more permanent incomes. Similarly, changes in mobility may cause different evolution of more permanent and cross-sectional income inequalities.

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Given the importance of income instability experienced by Argentina during the past decades, this paper will examine some of its characteristics and effects. The panel data to be used do not arise from a speciﬁc longitudinal survey, as they are a by-product of the regular and continuous household survey and only track individuals over a relative short period. Notwithstanding that, it offers relevant evidence for discussing the characteristics and effects of income instability. They will be complemented with dynamic pseudo-panels in order to examine more long-term variables. Due to data limitations, the period going from the mid-1980s to the early 2000s will be analyzed. Despite the importance of such issue, few studies have addressed it in the past. Moreover, the few analyses that have been undertaken use a shorter time frame than the one considered here; and, in particular, they do not include periods of high inﬂation [see Albornoz and Mene´ndez (2002); Cruces and Wodon (2003); Gutie´rrez (2004); Fields and Sa´nchez Puerta (2005)]. The relationship between instability, risk, mobility, and income distribution has not been studied either. This paper analyzes some key factors determining instability and its differential intensity among individuals and household groups. It also assesses the extent to which changes in instability have affected the dynamic of income distribution. The hypothesis to be explored regarding this aspect is that the increase in concentration showed by cross-sectional data is not much different from that corresponding to average incomes. The analysis, covering the period 1988–2001, will distinguish four periods that are relatively homogeneous in terms of a set of variables that is important for the aims being pursued (see next section for an overview of the economic evolution). The ﬁrst of them covers the years 1987–1991 and is characterized by a very high inﬂation rate, a situation quite different from the other three, which were differentiated on the basis of the evolution of level of activity. The second period goes from 1991 to 1994, during which the economy stabilized and gross domestic product (GDP) grew at a high pace after the Convertibility Plan was implemented. The third phase comprises years 1995–1998, another expansionary period that followed the Mexican debt crisis (that provoked a recession in late 1994 and part of 1995). The last period, 1998–2001, is marked by deep and large reduction of aggregate output that led to the abandonment of the ﬁxed exchange rate scheme. Longitudinal data for Greater Buenos Aires will be used, since this is Argentina’s main metropolitan area and concentrates nearly one-third of the country’s population. The temporal and geographic coverage reﬂects the availability of statistical information, since longitudinal data are

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continuously available only for that region and for those years. Dynamic pseudo-panels will be also employed to test the convergence hypothesis. Section 2, which follows, contextualizes the analysis of mobility in Argentina by brieﬂy summarizing Argentine macroeconomic, labor market, and income distribution behaviors. Section 3 sets out the paper’s speciﬁc objectives while Section 4 describes the analytical methods applied. Section 5 describes the data source used. The core of the paper consists of Sections 6 and 7, which analyze the results regarding variability and mobility, respectively. The conclusions are presented in Section 8.

2. MACROECONOMIC BEHAVIOR AND INCOME DISTRIBUTION SINCE THE MID-1970S The mid-1970s marked the start of a 15-year period of macroeconomic instability and productive stagnation. GDP was broadly unchanged throughout that period, and inﬂation remained at high levels (Fig. 1). This performance was associated with an external constraint arising from the high level of external debt, which in turn was generated by the policies implemented, particularly between 1978 and 1981. The measures adopted subsequently – throughout the 1980s – were unable to successfully address a 1989: 4900% 1990: 1300% Gini per capita household income 399

Gini coefficient

0.5

0.45

Gini personal incomes

199

0.4 Annual inflation rate (CPI) 0.35

-1

0.3

Annual inflation rate (December-December) (%)

0.55

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001

Fig. 1. Greater Buenos Aires: Inequality and Inﬂation. Source: Authors’ estimates on the basis of data provided by the National Institute of Statistics and Censuses.

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number of structural aspects of the Argentine economy, such as the management of public accounts and the ‘‘high inﬂation regime’’ (although the two factors are not independent). The latter is very important for understanding both the domestic effects of external borrowing and the difﬁculties in achieving sustained stabilization. This process of macroeconomic instability culminated in the hyperinﬂationary episodes of 1989 and 1990. The new government that assumed in 1989 was initially unable to improve the situation, and it was the economic team appointed in late January 1991 that implemented a stabilization program that managed to halt inﬂation and led to an increase in GDP. The cornerstone of that program was the Convertibility Act, which ﬁxed the exchange rate, established the convertibility of all currency in circulation, and prohibited monetary issuance that was not backed by external assets. This measure, together with others regarding ﬁscal and commercial aspects, allowed for a rapid reduction in inﬂation.1 Stability allowed for an improvement in the purchasing power of wages and for an expansion of credit. These developments were associated with signiﬁcant consumption growth, particularly in the case of durable goods and construction. Investments made by privatized enterprises also contributed to the expansion of domestic demand,2 while the reduction in inﬂation made it possible to improve real tax revenues. The vigorous inﬂow of foreign capital between 1991 and 1994 boosted the growth of domestic demand. It was attracted by the greater conﬁdence generated by stability and the orientation of economic policy, but also responded to the larger supply of funds available in the international ﬁnancial market. Nonetheless, the Mexican crisis in late 1994 revealed the fragility of an economy in which expansion was based on capital inﬂows from abroad, although the Argentine recession associated with this event was brief, and the economy resumed a rapid growth path as soon as conditions on the international capital market improved. In 1998, however, when this market became more problematic again and Brazil (a major export destination) went into recession, there was a new downswing in GDP that, unlike the previous episode, lasted an uncommonly long time and triggered the abandonment of the ﬁxed exchange rate system shortly after the start of 2002. The serious macroeconomic instability experienced since the mid-1970s is one of the explanations for the signiﬁcant deterioration in the income distribution since then. Initially, the increase in inequality probably stemmed from the differential impact of the rise in inﬂation in 1975 and

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1976 on the relative wages of individuals with different skill levels. Income inequality remained high in the 1980s, with individual incomes maintaining their concentration level while family incomes became more concentrated. Despite an improvement in the macroeconomic setting and the introduction of structural reforms, inequality continued to worsen in the 1990s, except during the initial expansionary phase (1991–1994). The effects of the signiﬁcant deterioration in the labor market should be kept in mind. Unemployment rose from 6% in 1991 to 12% in 1994 and 18% in 2001. This phenomenon had its greatest effect on wages, employment possibilities, and job quality, particularly for the lower skilled.

3. THE DIFFERENT OBJECTIVES OF INCOME DYNAMICS STUDIES AND THE PURPOSE OF THIS PAPER There are a large number of studies that analyze the evolution of individual and/or family incomes using panel data. Some of them discuss the intensity of income instability, how it change over time, or the differences between groups; other investigate the impact of instability on individual and family welfare. A larger volume of research however focuses on changes in the relative position of individuals in the income distribution. Such studies reﬂect two types of concern: some investigate the magnitude and characteristics of movements and how it has changed over time, while others examine the effects of these movements on inequality. Speciﬁcally, the different aims are income mobility, the impact of mobility on inequality, changes in the intensity of mobility, and the welfare effect of income instability.

3.1. Income Mobility Many studies analyze the paths of personal or household incomes through time in order to evaluate changes in their relative position in the distribution. Changes in the ranking of income recipient units are generally referred to as income ‘‘mobility.’’ Income paths can also be tracked for the purpose of analyzing the direction and magnitude of changes, whether accompanied by alterations in the ranking or not. This is known in the specialized literature as ‘‘absolute mobility.’’ The ﬁrst of them implies the second, but not the other way round. Absolute mobility will imply mobility

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depending, partly, on the inequality that exists in the distribution of current income: when inequality is high, the absolute change in income required to cause a change in the ranking will be greater than that in a lowconcentration situation. The proportion of income recipients that change their position in the distribution is normally analyzed through matrices that show transitions from one quantile of the distribution to another, between two periods. Although this is the most common procedure in the specialized literature, it has limitations: in particular, it fails to capture changes that take place within the bounds of the selected quantiles.3 Some authors have tried to correct these shortcomings, for example, by making the boundaries of income quantiles ﬂexible (Hills, 1998). Other ways to obtain quantitative evidence of mobility are through measures of association such as the simple (Pearson) and rank (Spearman) correlation coefﬁcients (OECD, 1996). It should be kept in mind, however, that the ﬁrst of these coefﬁcients is not restricted to changes of rank. When a change of rank is not a concern, procedures that speciﬁcally quantify the magnitude of the change in incomes are generally used. A particular concern of the analyses of absolute mobility is to evaluate the presence of income convergence. This one occurs when individuals’ incomes experience changes that bring them closer to the mean income in the distribution.

3.2. The Impact of Mobility on Inequality A second type of research, closely related to the aim of mobility analysis, seeks to evaluate the impact of changes in individual incomes on income distribution. In particular, it asks whether the degree of inequality measured with cross-sectional data differs, and by how much, from that corresponding to ‘‘permanent’’ income, measured as average income over several periods. One point to highlight is that the presence of convergent income mobility is not necessarily translated into an improvement of income inequality when the latter is evaluated with cross-sectional data. Since this statement may not seem intuitive, it is worth taking a moment to consider it further. In the following scheme, we present four examples that illustrate the possible combinations of the evolution of static inequality and the convergence/ divergence of incomes. In the lower-left quadrant C, it can be observed that the variation of recipients’ incomes between observations 1 and 2 was convergent while static inequality rose. Notice that for this to happen, the

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ﬁnal income of at least some of those recipients whose incomes have improved/worsened has to be higher/lower than the incomes of the recipients who were initially in those positions. Recipients

A. Divergence and increase of static inequality

B. Divergence and reduction of static inequality

Obs. 1

Obs. 2

Average of obs. 1 and 2

Obs. 1

Obs. 2

Average of obs. 1 and 2

A B C Inequality (CV)

10 20 30 0.408

1 20 30 0.708

5.5 20 30 0.544

10 20 30 0.408

20 35 50 0.350

15 27.5 40 0.371

Recipients

C. Convergence and increase of static inequality Obs. 1 Obs. 2

A B C Inequality (CV)

10 20 30 0.408

20 9 30 0.436

D. Convergence and reduction of static inequality

Average of Obs. 1 Obs. 2 Average of obs. 1 and 2 obs. 1 and 2 15 14.5 30 0.363

10 20 30 0.408

11 20 25 0.310

10.5 20 27.5 0.360

3.3. Changes in the Intensity of Mobility Panel data contribute to a better evaluation of the dynamic of inequality when the intensity of mobility changes. But if the latter were constant, measurements of inequality using cross-sectional data would adequately reﬂect the change in the concentration of more permanent incomes. An increase in static inequality will generate greater inequity in the distribution of more permanent incomes, except if there is an offsetting increase in income mobility. As shown by Gottschalk and Danziger (1998), the variance of average incomes is a function of the average of the variances of the distributions of each observation and the average of the covariances between the different observations.

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3.4. The Welfare Effect of Income Instability A different concern from those just considered is to evaluate the intensity of the instability of individual incomes (Burgess, Gardiner, Jenkins, & Propper, 2000) insofar as this diminishes the utility of a given volume of economic resources. In particular, variability increases risk (Arrow, 1970) and, although it can be anticipated, it can also affect utility, particularly in countries with poorly developed credit markets. If two households received the same average income at the end of the year, but one of them had no income for half of that year, whereas the other received 1/12 of its annual income every month, the welfare levels of the two recipients are likely to have been very different. Among these diverse interests, this paper focuses, speciﬁcally, on the following aspects: (i) The degree of instability of real incomes, given its adverse effects on individual and family welfare: the analysis of income variability over short periods (as it will be done here due to data restrictions) is a relatively unexplored topic, probably because it is not a signiﬁcant phenomenon in the world’s leading economies. Nonetheless, in countries such as Argentina, where macroeconomic instability has been a feature throughout much of its modern history, income variability is particularly relevant, irrespective of any distributive impacts – especially, as will be seen, when it seems to persist even in situations of price stability. The paper does not only assess the level of instability and its changes over time, but also analyze differences across groups of individuals and households, and the impact of those factors that are more closely linked to instability. (ii) The evaluation of the degree of income mobility. In this case, the aspects to be analyzed will be: the intensity of mobility; its impact on the distribution of more permanent incomes; the convergence hypothesis.

4. METHODS We will discuss here the quantitative procedures to be used in order to analyze the different issues that were above indicated. Data to be used are described in the next section.

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4.1. Income Instability Two analytical approaches were used to assess income instability. The ﬁrst one measures the variability of observed current incomes (of individuals and families) around the mean, using the coefﬁcient of variation (CVi for individuals and CVh for households). qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PT i Þ2 t¼1 ðwit w (1) CV i ¼ w i where PT w i ¼

t¼1 wit

T

where wit is the earning of member i in period t and T the number of observations available in the panel, yht ¼

m X

wiht

i¼1

CV h ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PT h Þ2 t¼1 ðyht y yh

(2)

m is the number of members employed in at least one of the T observations in household h, PT y yh ¼ t¼1 ht T Mean variability of individuals and households is the average of individuals and households’ CVs. As the impact of instability was assumed to vary across types of income recipients and families (greater impact among lessskilled workers and lower-income families), disaggregated estimates were made for both cases, deﬁning groups based on individuals and head’s schooling. Steps were also taken to obtain evidence on the importance of those variables directly related to variability. For example, instability in real individual incomes is associated with changes in hourly earnings: the number of hours worked and changes in occupational status (employed/ unemployed). The intensity of the latter will change especially when job

295

Income Mobility in Argentina

mobility varies; whereas variations in nominal wages are associated, among other factors, with the degree of price stability, and are likely to be larger and more frequent in inﬂationary settings.4 To assess the impact of these variables, the effects of job instability and of the variability of monthly remuneration (which, therefore, also reﬂect changes in hours worked) are successively isolated. The effect of ‘‘pure’’ changes in remuneration is measured by calculating the coefﬁcient of variation considering only individual’s positive incomes, i.e., excluding observations corresponding to situations in which the person is not employed (CV ao i ), qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ Pni n 2 ðw w Þ it i t¼1 CV ao for wit 40 (3) i ¼ wni where ni is the number of observations in which individual i has a positive income (i.e., where witW0), Pn wit for wit 40 wni ¼ t¼1 ni To obtain an indicator of the ‘‘pure’’ effect of job instability, a coefﬁcient of variation of the individual’s income is calculated over values of the period that, when positive, correspond to the ﬁrst observation when the person was employed. qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PT nn 2 t¼1 ðAit wi Þ ar (4) CV i ¼ wnn i with ( Ait ¼

wi1 ! wit 40 0 ! wit ¼ 0

;

wnn i ¼

PT

t¼1 Ait

T

where wi1 represents remuneration in the ﬁrst observation with a positive value. Two factors are assumed to affect the variability of nominal household labor incomes: variations in the number of household’s income recipients and variability in the incomes received by them. These two factors can work in opposite directions and they may offset each other, either partially or completely; in the latter case, the resultant change in the variability of household incomes is zero. Variations in the number of household income

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LUIS BECCARIA AND FERNANDO GROISMAN

recipients may reﬂect changes in the size of the household, changes in the employment rate of a household without alterations in the number of members, or both factors together. This paper does not distinguish between the causes of such variation. The magnitude of the impact of the instability of members’ remunerations on households’ income instability is deduced from the coefﬁcient of variation of the income of each household, calculated assuming as ﬁxed over the whole period the number of employed members (CV ao h ). In this case, household members who were employed at least in one period had an income imputed for those period(s) when they were unemployed or nonactive. The value imputed is that earned in the nearest period (either before or after) where his/her remuneration was positive, adjusted for the mean variation in incomes between the two periods qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PT n nh Þ2 t¼1 ðyht y ao (5) CV h ¼ ynh where PT n

yh ¼

n

yht ¼

m X

n

t¼1 yht

T (

Biht ;

i¼1

Biht ¼

wiht ! wiht 40 w iht ! wiht ¼ 0

where w iht

wt ¼ wihs ws

where ws and wt are the average earnings over all individuals with positive earnings in the sample for period s or t. s is the period nearest to t, i.e., PQ wit wt ¼ i¼1 Q where Q is the number of members with positive earnings in the whole sample in period t. To evaluate the signiﬁcance of changes in the number of employed persons, a coefﬁcient of variation over household’s income simulated by assuming that the monthly remuneration of all employed members remain

297

Income Mobility in Argentina

ﬁxed during all periods where they had positive earning (CV ar h ); a value equal to the ﬁrst positive observation is imputed in each case qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ PT nn nn 2 ðy y Þ t¼1 ht h (6) CV ar h ¼ ynn h where ynn ht ¼

n X

Aiht

i¼1

( Aiht ¼

wih1 ! wiht 40 0 ! wiht ¼ 0

where wih1 is the remuneration in the ﬁrst period with a positive value. The second analytical approach to income instability recognizes how utility declines when income becomes more variable, using a standard, strictly concave utility function with constant relative risk aversion, to stylize the fact that risk declines with the level of income and increases with variability " #1=1r sþn 1 X 1r n y (7) yi ¼ n t¼s it where y is risk-adjusted income, y is the income of the period, i identiﬁes the household, and r is the coefﬁcient of risk aversion. The latter was assigned a value of 2 for the calculation.5 This procedure ‘‘downgrades’’ the level of average income obtained by an individual or household through time, when that average has resulted from a variable path.

4.2. Income Mobility With regard to income mobility – the second of the stated objectives – its intensity in Argentina, and particularly its variations between the phases identiﬁed, was analyzed on the basis of household movements between income quintiles that make it possible to identify different paths. Given the limitations of this method, it was complemented by analyzing correlation coefﬁcients between the household incomes obtained from the four

298

LUIS BECCARIA AND FERNANDO GROISMAN

observations. The smaller the correlation, the larger the differences between the incomes obtained by the same households in two periods of time, and, therefore, the greater the mobility of income. The Pearson and Spearman (rank) correlations were used for this. To quantify the inﬂuence of mobility on income distribution, the adjustment of inequality for mobility index was calculated (Shorrocks, 1978) RðW T Þ ¼ PT

IðwÞ

t¼1 Zt Iðwt Þ

1

(8)

where I is the inequality indicator, w average income over T periods, wt income in period t, and Zt the weighting factor deﬁned as the units’ share of total income in period t with respect to the income in the set of T periods. Fields (2004) argues that if the aim is to evaluate the extent to which mobility altered the inequality measured at a given point in time, the comparison should be made directly between I(w1) and ðw T Þ, i.e., between inequality in initial period and the inequality of average income. R tends to zero as a maximum value when there is no mobility, and it decreases as the effect of mobility on the distribution intensiﬁes.6 Finally, we examined whether income mobility implies a convergent path estimating, as usual,7 the following model using ordinary least squares (OLS) ln yi1 ¼ a þ b ln yi0 þ i

(9)

where ln yi1 is the logarithm of income of the current period and ln yi0 the logarithm of income in the previous period. There would be convergence/ divergence toward the mean, in case b was lower/higher than 1.8 Not only longitudinal data but also information coming from using pseudo-panels was employed as the latter allows for longer-period analysis.

5. THE DATABASE USED Income instability and mobility, along with their impact on levels and changes in the distribution of income, are usually analyzed employing longitudinal data, i.e., data showing the different incomes received by the same person or household through time. This type of information faces limitations, especially due to the presence of attrition and reporting errors. The former feature, i.e., the progressive loss of units of observation, is due to various reasons, such as households leaving the panel or changing address,

299

Income Mobility in Argentina

or difﬁculties arising in the ﬁeld work. It may be nonrandom and lead to a biased estimation of mobility. Reporting errors, even if also present in crosssectional data, is more important for computing income variability; speciﬁcally, they may lead to overestimation of their effect on inequality. Given these general limitations of longitudinal data and the particular limitations derived from the source to be used here (see below), besides this type of information we also analyzed pseudo-panel data in order to study households’ income mobility. This methodology consists in using cohorts of people from different surveys. It has been suggested that the information coming from this procedure – even if does not correspond to individuals but to averages of groups of individuals – would be more appropriate for the analysis of income variability (Antman & McKenzie, 2005). As the paper aims at exploring the impact of labor-market dynamics, the universe of households that was studied was limited to those headed by individuals not over 65 years of age. To obtain results in terms of the instability of purchasing power, which is the relevant concept, nominal values were corrected for variations in the consumer price index (CPI).

5.1. Longitudinal Data Although Argentina does not undertake longitudinal surveys, the permanent household survey (PHS), performed regularly by INDEC,9 provides longitudinal data. They derive from the fact that PHS’s sample panel is of a rotating type: households are interviewed on four successive occasions. Consequently, by comparing the situation of an individual in those four ‘‘waves’’, changes experienced by him/her in a number of variables, including income and employment, can be deduced. The PHS sample consists of four rotation groups, one of which enters and another exits in each of the two ‘‘waves’’ carried out each year (in May and October). On each occasion, therefore, 25% of the sample is renewed, so 75% of cases can be compared between two successive waves. Accordingly, tracking households for the maximum possible time, i.e., during the four waves in which they remain in the survey during an 18month period, would only be possible for a subset of the whole sample representing 25% of it. The proportion of households and individuals actually reinterviewed four times is even lower because of attrition due to various reasons (households leaving the panel or changing address, or difﬁculties arising in the ﬁeld work).

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LUIS BECCARIA AND FERNANDO GROISMAN

Given this sample size limitation, rotation groups that entered the sample at different points in time were combined.10 This means that individuals (and households) who responded to the survey at different times were considered simultaneously; in other words, the method aggregates changes that occurred in different periods. Data refer to Greater Buenos Aires only11 since microdata are not available for the other areas covered by the survey for the entire period under analysis. In any event, the evolution of the employment situation and income distribution in the metropolitan area has not differed from that experienced in other urban zones, so the conclusions to be reached here may reasonably be extrapolated to the whole urban areas of Argentina (Beccaria, Esquivel, & Maurizio, 2002). To analyze income paths, panel data were prepared for each of the four subperiods that were identiﬁed in the Section 1. Table 1 shows the different rotation groups considered in each case; sample size is also indicated. Comparing successive waves of the survey underestimates the number of changes that actually occurred because transitions are being identiﬁed by comparing two observations roughly 6 months apart. Accordingly, individuals could make two or more movements in the interval between the two waves (e.g., from inactivity to unemployment and vice versa), without these movements being captured. It should also be noted that the procedure only analyzes the subset of incomes obtained by household members as a result of their labor-market participation as wage earners, own-account workers, or employers. This restricted deﬁnition of income facilitates a clearer relation between the dynamics of inequality and income instability, and the labor-market factors that appear as their determinants. It also needs to be borne in mind that the household survey used here, as many others carried out in the region, does not adequately capture – and signiﬁcantly under-records – the current resources that households obtain from their ownership of capital. When studying the instability of individual labor incomes, the analysis included persons who were employed in at least one of the observations, i.e., those who registered some positive income from employment.

5.2. Pseudo-Panels The use of pseudo-panels makes it possible to follow the evolution of individuals or households of a certain cohort – usually deﬁned by age – using repeated cross-sectional data. In this case, we created ﬁctitious cohorts

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Income Mobility in Argentina

Table 1. Phases

Greater Buenos Aires: Rotation Groups Comprising the Sample in Each Phase. First Observation

Second Observation

Third Observation

Fourth Observation

High inﬂation

October 1987 May 1988 October 1988 May 1989 October 1989 May 1990 No. of individuals: 1,877 No. of households: 1,141

May 1988 October 1988 May 1989 October 1989 May 1990 October 1990

October 1988 May 1989 October 1989 May 1990 October 1990 May 1991

May 1989 October 1989 May 1990 October 1990 May 1991 October 1991

Stabilization

May 1991 October 1991 May 1992 October 1992 May 1993 No. of individuals: 1,773 No. of households: 976

October-91 May 1992 October 1992 May 1993 October 1993

May 1992 October 1992 May 1993 October 1993 May 1994

October 1992 May 1993 October 1993 May 1994 October 1994

Recovery

October 1995 May 1996 October 1996 May 1997 No. of individuals: 2,391 No. of households: 1,263

May 1996 October 1996 May 1997 October 1997

October 1996 May 1997 October 1997 May 1998

May 1997 October 1997 May 1998 October 1998

Recession

October 1998 May 1999 October 1999 May 2000 October 2000

May 1999 October 1999 May 2000 October 2000 May 2001

October 1999 May 2000 October 2000 May 2001 October 2001

May 1998 October 1998 May 1999 October 1999 May 2000 No. of individuals: 3,129 No. of households: 1,651

Source: Authors’ elaboration based on permanent household survey (PHS).

of households by age of the household head, also using data from the PHS. Hence, when the averages of two observations in the pseudo-panel are contrasted, what is actually being compared is the situation of groups of similar characteristics but not the same households. In order to obtain an appropriate number of cases, the cohorts were grouped considering 5-year periods. For example, the 1980–1984 cohort includes those households

302

LUIS BECCARIA AND FERNANDO GROISMAN

whose household heads were born between those years. The analysis was carried out grouping, i.e., making a pool of, all the possible cohorts. The study of incomes mobility was carried out for the totality of cohorts; however, for certain analyses the household cohorts were divided according to two educational levels of the household head: incomplete high school or less, and the rest.12 As it will be seen below, this procedure allows one to distinguish mobility processes between subgroups of households that are homogeneous in this variable, which constitutes a reasonable variable to stratify households. We considered household surveys since 1984 due to greater availability of cross-sectional data.13

6. INCOME INSTABILITY IN ARGENTINA IN THE 1990S This section addresses one of the paper’s two objectives, namely, to study income instability and its effects on the level of welfare. The ﬁrst part analyzes changes in the degree of variability of incomes, their sources, and the effect on different groups of workers and households. The second part reviews the impact on welfare and changes therein during the period under analysis. 6.1. Instability of Individual and Household Incomes 6.1.1. Instability of Individual Incomes As shown in Table 2, there were no signiﬁcant changes in the coefﬁcient of variation of labor incomes among individuals who were employed at some time during the four phases analyzed. This result is a curious one because, contrary to expectations, the sharp drop in inﬂation that occurred between the ﬁrst of those periods (covering the years before the Convertibility Act) and the other three did not affect the average variability of current incomes. As mentioned above, inﬂation inﬂuences the variability of an individual’s real labor income through time, via its impact on changes in the remuneration obtained in a given job. Thus, the drop in inﬂation – especially from such high rates as those recorded between 1987 and 1991– should have helped to reduce the instability of real wages. In fact, such impact can be actually identiﬁed in Table 2 as the coefﬁcient of variation of labor remuneration – considering only positive incomes and excluding observations corresponding to situations

Table 2.

Greater Buenos Aires: Coefﬁcient of Variation of Incomes of Persons Employed At Least in One Observation.

Individuals Under 65 Years of Age Who were Employed at Some Point

Total Actual Effect of variation in real remunerations (simulated controlling for job instability) Effect of occupational variation (simulated controlling for instability of remunerations) Low-education individuals Actual Effect of variation in real remunerations Effect of occupational variation Medium-education individuals Actual Effect of variation in real remunerations Effect of occupational variation High-education individuals Actual Effect of variation in real remunerations Effect of occupational variation

High Inﬂation Phase

Stabilization Phase

Recovery Phase

Recession Phase

Average Conﬁdence interval Average Conﬁdence interval Average Conﬁdence interval Average Conﬁdence interval Lower bound

Upper bound

Lower bound

Upper bound

Lower bound

Upper bound

0.562 0.280

0.540 0.271

0.583 0.288

0.326

0.301

0.606 0.285

0.558 0.211

0.533 0.202

0.582 0.219

0.565 0.194

0.543 0.187

0.587 0.202

0.351

0.389

0.362

0.417

0.421

0.397

0.579 0.274

0.634 0.296

0.605 0.212

0.574 0.202

0.636 0.223

0.641 0.204

0.374

0.342

0.406

0.441

0.406

0.476

0.525 0.263

0.481 0.247

0.568 0.280

0.489 0.206

0.444 0.191

0.294

0.243

0.345

0.317

0.346 0.281

0.308 0.259

0.384 0.302

0.079

0.038

0.120

Source: Authors’ elaboration based on PHS.

Lower bound

Upper bound

0.578 0.190

0.559 0.183

0.598 0.196

0.446

0.439

0.417

0.461

0.611 0.194

0.671 0.214

0.673 0.198

0.646 0.189

0.700 0.207

0.497

0.464

0.531

0.533

0.503

0.563

0.534 0.221

0.496 0.175

0.456 0.162

0.535 0.188

0.494 0.176

0.460 0.164

0.528 0.187

0.267

0.368

0.362

0.319

0.405

0.364

0.327

0.400

0.384 0.209

0.310 0.183

0.458 0.236

0.334 0.187

0.284 0.167

0.385 0.207

0.339 0.178

0.297 0.162

0.381 0.194

0.194

0.113

0.274

0.169

0.116

0.222

0.184

0.140

0.229

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LUIS BECCARIA AND FERNANDO GROISMAN

in which the person was not employed – declined in the second and the third period. This effect was counteracted by events in the labor market that increased job instability as the coefﬁcient of variation of incomes, controlling for changes in remuneration, rose. A least-squares model was applied to evaluate the extent to which certain individual and household variables were associated with income instability and its occupational and remuneration components (results are not shown here). It shows that status in terms of education, household headship, age, and gender (being male) had the expected signs. Only when the dependent variable is ‘‘pure’’ remuneration variability, however, is low education not signiﬁcant in the ﬁrst period, thereby indicating that the effects of inﬂation were felt by the employed population at large. This situation was repeated, i.e., during the stabilization phase, which also shows that the process would have beneﬁted all individuals, independently of other attributes. In the other two phases, however, the low-education coefﬁcient was signiﬁcant, suggesting that instability declined by less among such individuals, or even increased. Among low-skilled employed persons, income instability was greater toward the end of the period analyzed than at the beginning and remained unchanged for the other two groups (see Table 2).14 The signiﬁcance of this result, however, emerges from applying a similar model to the set of observations comprising data corresponding to the four periods but only for low-education individuals, which includes a dummy variable for each period. Only the dummy corresponding to the recessionary phase (1998–2001) was positive and signiﬁcant. In contrast, there were no signiﬁcant differences when the exercise was repeated for high-education individuals. This procedure was also used to analyze the signiﬁcance of changes in instability among different educational groups, associated either with ﬂuctuations in remuneration or with occupational status. Among lowschooling individuals, occupational variability was already increasing at the start of the 1990s, while pure income instability was not changing signiﬁcantly. In contrast, the other group did not show changes in either measure. 6.1.2. Instability of Household Incomes We now consider the variability of household incomes, which is important not only from a welfare perspective but also to evaluate to what extent it derives from the instability of individuals’ labor incomes. The relation will not necessarily be direct, since it could have been offset by the effect of other variables.

Income Mobility in Argentina

305

Table 3 shows a signiﬁcant decrease in the coefﬁcient of variation (18%) of household labor incomes between the ﬁrst and second periods (in the early 1990s), resulting from an increase in average job instability among households and a decrease in the variability of remuneration. Then, during the expansionary and recessionary phases of the second part of the decade, family income variability increased again (by 6% and 5%, respectively) basically due to rising job instability. Nonetheless, the variability of family labor incomes in the last of the periods was 9% less than the value recorded in the late 1980s; and although this aggregate result hides signiﬁcantly different experiences across strata deﬁned by the education level of household heads, on average it reﬂects a different situation than for individual income variability. The relevance of growing job instability is revealed by a persistent rise in the coefﬁcient of variation of family incomes calculated after controlling for changes in variations in the remuneration of employed household members (Table 3). Such coefﬁcient grew by 38% after price stabilization – in the early 1990s – whereas the one reﬂecting the pure real remuneration effect decreased by 22%. Considering the ﬁrst and the last phases, however, the differences between the two measures were even greater: income variability caused by job instability increased by 56%, whereas that stemming from ﬂuctuations in remunerations was 18% below the level recorded in the years of high inﬂation. Similarly to the case of individual incomes, income variability associated with job instability increased most in households headed by individuals with low levels of schooling. It should be kept in mind that the procedure used to measure variability caused by job changes also captures effects arising from the strategies deployed by household members in response to events affecting them. Speciﬁcally, substitution and complementarity mechanisms operate among active members within households, and these affect income instability through both jobs and remuneration, with the ﬁnal outcome depending on which effect prevails [see Beccaria and Groisman (2005)]. A clear example of this is the change in income that can be associated with ‘‘perfect’’ substitution of employed household members (i.e., if one member becomes unemployed, another ﬁnds a job). If the income of the new worker is different than that of the family member who becomes unemployed, household income is altered without any change in the number of employed members; this change should be attributed to the job factor and not to ﬂuctuations in remunerations, as it was the case in the method here used. Consequently, our procedure underestimates the effect of occupational instability on overall individual and household instability.

Table 3.

Greater Buenos Aires: Coefﬁcients of Variation of Real Labor Incomes of Households and Number of Employed.

Households Headed by Persons Under 65 Years of Age

High Inﬂation Phase Average

Total Employed members Household labor income Household labor income controlling for job instability Household labor income controlling for remuneration instability Low-education level Employed members Household labor income Household labor income controlling for job instability Household labor income controlling for remuneration instability Medium-education level Employed members Household labor income Household labor income controlling for job instability Household labor income controlling for remuneration instability High-education level Employed members Household labor income Household labor income controlling for job instability Household labor income controlling for remuneration instability

Conﬁdence interval Lower bound

Upper bound

0.172 0.364 0.312

0.163 0.355 0.305

0.180 0.372 0.318

0.094

0.086

0.186 0.378 0.318

Stabilization Phase Average

Conﬁdence interval Lower bound

Upper bound

0.178 0.300 0.244

0.170 0.292 0.238

0.187 0.308 0.249

0.102

0.130

0.120

0.176 0.368 0.310

0.197 0.388 0.326

0.194 0.315 0.252

0.106

0.096

0.115

0.153 0.336 0.296

0.135 0.319 0.281

0.081

Recovery Phase Average

Recession Phase

Conﬁdence interval Lower bound

Upper bound

0.217 0.317 0.259

0.208 0.309 0.252

0.225 0.326 0.266

0.140

0.127

0.119

0.184 0.306 0.245

0.204 0.325 0.259

0.245 0.348 0.282

0.141

0.130

0.153

0.171 0.354 0.311

0.131 0.263 0.229

0.115 0.245 0.217

0.062

0.100

0.100

0.078 0.290 0.286

0.061 0.275 0.272

0.096 0.305 0.300

0.017

0.008

0.026

Source: Authors’ elaboration based on PHS.

Average

Conﬁdence interval Lower bound

Upper bound

0.235 0.332 0.255

0.227 0.324 0.249

0.243 0.340 0.262

0.135

0.147

0.140

0.155

0.234 0.337 0.273

0.256 0.359 0.291

0.263 0.368 0.281

0.253 0.358 0.273

0.274 0.378 0.290

0.145

0.135

0.155

0.174

0.164

0.184

0.147 0.280 0.242

0.178 0.268 0.214

0.162 0.251 0.201

0.195 0.285 0.226

0.185 0.269 0.202

0.170 0.254 0.190

0.201 0.285 0.213

0.079

0.122

0.099

0.082

0.115

0.111

0.097

0.124

0.110 0.207 0.184

0.088 0.187 0.168

0.132 0.227 0.201

0.081 0.187 0.175

0.068 0.172 0.162

0.093 0.201 0.188

0.149 0.234 0.207

0.133 0.218 0.194

0.165 0.250 0.221

0.049

0.032

0.066

0.050

0.029

0.071

0.049

0.037

0.061

Income Mobility in Argentina

307

As it was done in the case of the instability of individual incomes, a model was estimated in order to evaluate how household instability is inﬂuenced by a series of variables (results not shown here). Their results show that it is negatively related to the education level of the head of household, and this relation strengthens as from the second expansionary phase. Other factors that had a signiﬁcant inﬂuence are age, with a negative sign, and agesquared, with a positive sign; whereas the coefﬁcient of household size and the presence of children is associated with greater variability throughout the 1990s. The same model was also estimated for the case of the number of employed household members, the pure income variability – the coefﬁcient of variation of incomes that controls changes in the number of employed family members – and the pure occupational variability. The education level of heads of household also seems to negatively affect all these types of variability. Table 3 shows that the reduction in the instability of family incomes associated with the control of inﬂation affected heads of household with different educational levels. Nonetheless, the pattern became more divergent following the post-1995 recovery, with variability increasing among households headed by individuals with low levels of schooling, whereas in other groups no changes were recorded after the reduction associated with price stabilization. This broadly reﬂects what happened with the variability of employed household members, which increased more in the ﬁrst group. Among them, the pure variability of remunerations also increased while remaining unchanged for the other groups. Toward the end of the 1990s, therefore, gaps in household income instability between socioeconomic groups (proxied by heads’ schooling) were even greater than those recorded at the beginning of the decade. Although, in the case of variability of the number of income recipients, the gap between high- and low-strata households was narrowed by the sharp increase in the former during the recessionary phase, differentials in remuneration variability widened. The foregoing analysis on individual and family income instability can be summarized by stating that it decreased in the second of the phases identiﬁed (from the early 1990s) as a result of macroeconomic stabilization. Nonetheless, in the middle of that decade, occupational paths became more unstable and, in the ﬁnal phase considered, real household incomes remained highly unstable, thereby partly losing the beneﬁt of the reduction of inﬂation. This was particularly true among households headed by loweducation individuals, in which the additional job instability fully offset the lesser instability of remunerations.

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6.2. Instability and Welfare As noted above, ﬂuctuations in the ﬂow of resources received by households are damaging because they generate uncertainty regarding future values, which may affect levels of consumption and the programming of expenses, or cause difﬁculties in cushioning the effects even when variability can be anticipated. As described in Section 4, this factor can be taken into account by the use of utility functions that estimate an income corrected for the effects of ﬂuctuations. This risk correction is also used even when variability has always been rising or falling. Nonetheless, as noted in Section 7.1, households with rising paths represent less than 5% of all cases. The results conﬁrm that adjusted income increased more than measured income between the ﬁrst and the last period due to the reduction in the variability of real incomes noted above. Nonetheless, this improvement differed in intensity across groups: among low-education households the average increase in both income measures was similar, whereas in households headed by individuals with higher levels of schooling, the riskadjusted increase rose by 52%, compared to a 29% rise in average actual incomes (Table 4).15

7. MOBILITY AND INEQUALITY This section of the paper will analyze how income instability just described was associated to income mobility and, in particular, the impact of the latter on income inequality.

7.1. The Evolution of Income Mobility As analyzed above, in the early 1990s household labor income became less variable, reﬂecting the impact of the macroeconomic stabilization program. This coincided with a reduction in levels of income concentration, which had been much accentuated during the years of high inﬂation (see Section 2). Nonetheless, and despite the maintenance of low inﬂation, household labor incomes gradually became more variable in the third and fourth phases (i.e., throughout the last half of the 1990s); and the same happened with inequality, which grew until the middle of that decade, before ﬂattening out in the downswing phase.

Table 4.

Household Labor Income: Actual Average and Risk-Adjusted Average (in 2001 pesos).

Households Headed by Persons Under 65 Years of Age

High Inﬂation Phase Average

Conﬁdence interval Lower bound

Stabilization Phase Average

Upper bound

Recovery Phase

Conﬁdence interval Lower bound

Upper bound

Average

Recession Phase

Conﬁdence interval Lower bound

Upper bound

Average

Conﬁdence interval Lower bound

Upper bound

Total Risk-adjusted Actual

597 707

564 669

630 746

874 982

823 926

925 1,037

855 958

804 911

905 1,005

851 950

806 909

895 992

Low-education level Risk-adjusted Actual

433 520

411 495

454 545

682 788

646 753

718 824

591 687

561 653

621 721

576 668

544 637

609 698

Medium- or high-education level Risk-adjusted Actual

658 942

598 868

718 1,016

870 1,235

790 1,132

948 1,338

1,014 1,273

925 1,176

1,103 1,371

1,000 1,215

930 1,137

1,070 1,293

Source: Authors’ elaboration based on PHS.

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Variability does not have to be accompanied by changes in the ranking of recipients’ incomes, or even in the differentials between them. Nonetheless, such situations are unlikely to occur, since income variability usually results in changes in the relative positions of income recipients and/or in the gaps between their incomes, particularly when labor-market events, such as an unemployment experience, are taken into account. Analysis of income mobility initially focused on the way households moved between income quintiles over the four observations. This data made it possible to identify different paths, which, following an established typology [see, e.g., Hills (1998) or Jarvis and Jenkins (1998)], were classiﬁed as ﬂat, rising, falling, blip, and zigzag. The ﬁrst included cases of households that remained in the same income quintile throughout the four observations or moved, at most, to the immediately higher or lower level than at the start (irrespective of whether they returned to the original quintile or not). Rising (falling) paths are deﬁned by households that move up (down) by at least two income quintiles with respect to the initial one, and either remain in that situation or rise (fall) further. The situation referred to as a ‘‘blip’’ included increases (decreases) of two or more quintiles from the initial one, followed by a return to the initial or even one quintile lower (higher) than at the start. Other more ﬂuctuating alternatives are classiﬁed as zigzag. This classiﬁcation procedure makes it possible to describe the patterns of household mobility across deﬁned thresholds (quintile boundaries). Between the ﬁrst and the second phases, with the stabilization of the early 1990s, the prevalence of ﬂat paths increased from 55% to 59% of households, whereas the proportion of households experiencing blips decreased from 25% to 20% (Table 5). The other categories were broadly unchanged. In the post-1995 recovery, interquintile paths reveal a substantial Table 5. Paths

Flat Rising Falling Blip Zigzag Total

Greater Buenos Aires: Mobility of Households Labor Incomes. High Inﬂation Phase (%)

Stabilization Phase (%)

Recovery Phase (%)

Recession Phase (%)

55.3 5.1 3.7 25.7 10.2 100

59.1 5.6 3.4 20.7 11.3 100

71.7 3.9 3.4 15.3 5.7 100

73.5 2.9 2.9 14.7 6.1 100

Source: Authors’ elaboration based on PHS.

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Table 6.

Greater Buenos Aires: Correlation Coefﬁcients of Households Labor Incomesa.

Households Headed by Persons Under 65 Years of Age Pearson Spearman

High Inﬂation Phase

Stabilization Phase

Recovery Phase

Recession Phase

0.695 0.703

0.715 0.731

0.817 0.782

0.875 0.791

Source: Authors’ elaboration based on PHS. a All coefﬁcients calculated between the two periods were signiﬁcant at the 1% level. The values shown in the table are simple averages of the six coefﬁcients that can be calculated between observation pairs in each phase.

change in income mobility, with ﬂat movements accounting for 72% of households while the other types of transition declined. Lastly, in the ﬁnal recessionary phase, the previous mobility pattern was maintained, with the proportion of ﬂat paths increasing again. To complement the analysis, the Pearson and Spearman correlation coefﬁcients were calculated for household incomes. Table 6 shows the average of the six coefﬁcients that can be calculated from all observation pairs for each phase.16 Both correlation coefﬁcients for household incomes increased in the last two phases (Table 6), suggesting that not only did changes in the ranking of incomes decline, but the distances between them also narrowed, which is consistent with the results of the path analysis. In fact, the difference between the third- and fourth-phase coefﬁcients was signiﬁcantly larger than that between the coefﬁcients of the ﬁrst and second phases.17 Increases were signiﬁcant in the third phase (economic expansion following the ‘‘Tequila’’ crisis), and they continued their rising trend in the ﬁnal phase (Table 6). In contrast, between the ﬁrst and the second phases, along with a steeper reduction in income variability, the income correlation was unchanged. This would reﬂect the generalization of the effects of controlling inﬂation and is compatible with the greater prevalence of ﬂat income paths mentioned above. As will be recalled, the sharp rise in income correlation that occurred between the initial stabilization and economic recovery phases was accompanied by greater variability. This result suggests the need to study the degree of mobility that accompanies instability because they do not always behave in the same way. What happened between the second and the third phases analyzed is indicative of that situation and reﬂects the fact that the differentials associated with changes in income narrowed, even as they were becoming increasingly frequent.

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The combination of evidence that arises from the procedures used in this section reveals a process in which family labor income mobility has decreased since the late 1980s, which is consistent with a consolidation of the positions occupied by households in the income distribution. This would mean increasing segmentation between households of different types; and, in particular, that it would be increasingly difﬁcult for lower-income households to move upward, either in absolute or in relative terms. This result is explained by the evolution of the labor market during the period in question. As mentioned when analyzing the instability of individual incomes, there were increases in the degree of labor turnover. This individual behavior was largely projected onto households, given their revealed inability to implement compensation mechanisms in response to ﬂuctuations in individual labor incomes.

7.2. Income Mobility and Income Distribution As described in the previous section, Argentina experienced a process of decreasing mobility of family incomes since the late 1980s and 1990s onward (until 2001). Section 2 also showed that the degree of inequality in the distribution of current incomes has intensiﬁed since the mid-1990s. These two pieces of evidence suggest that the concentration of more permanent incomes expanded faster than that of current incomes. In other words, income mobility affected the dynamic of inequality in the income distribution with decreasing intensity. To quantify this effect, the Shorrocks’ ‘‘adjustment of inequality for mobility’’ index was calculated [see Eq. (8)]; the Gini coefﬁcient was used as the inequality index.18 The adjustment to inequality due to mobility was around 8% in the ﬁrst phase (late 1980s/early 1990s) and remained at similar levels in the second phase, covering the ﬁrst half of the 1990s (Table 7). This index then dropped in the next two phases, reaching 5.4% in the second of them. In these phases, therefore, the discount for mobility was less than during periods of high inﬂation. The fact that this correction has been decreasing reﬂects the aforementioned consolidation of household positions in the income distribution. It can be concluded, therefore, that the increase in inequality since the mid-1990s, documented in several studies based on cross-sectional data, partly underestimated the increase in the concentration of permanent household incomes. Inequality measured by the average of Gini coefﬁcients

Greater Buenos Aires: Gini Coefﬁcients of the Inequality of Household Labor Incomes. High Inﬂation Phase Coefﬁcient

Conﬁdence interval

Stabilization Phase Coefﬁcient

Lower Upper bound bound Gini coefﬁcient of average income Average of cross-sectional Gini coefﬁcients Gini coefﬁcient of riskadjusted average income Coefﬁcient R: Adjustment of inequality for mobility (%)

Conﬁdence interval

Recovery Phase Coefﬁcient

Lower Upper bound bound

Conﬁdence interval

Recession Phase Coefﬁcient

Lower Upper bound bound

Conﬁdence interval Lower bound

Upper bound

0.452

0.432 0.472

0.392

0.375 0.409

0.444

0.428 0.459

0.447

0.43

0.463

0.492

0.476 0.507

0.423

0.405 0.443

0.472

0.468 0.475

0.472

0.462

0.482

0.491

0.469 0.513

0.441

0.423 0.46

0.497

0.477 0.516

0.504

0.487

0.521

8

7.5

5.9

Income Mobility in Argentina

Table 7.

5.4

Source: Authors’ elaboration based on PHS.

313

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rose 11% between the second and the last periods, while the concentration of average incomes grew 14%. The analysis of risk-adjusted income inequality strengthens these results, since it also remained fairly constant when the ﬁrst and last periods are compared, in contrast with a small reduction showed by the inequality of cross-sectional data.

7.3. Convergence or Divergence in Incomes In the previous section, we analyzed the changes in indicators that account for the intensity of income mobility in the sense usually considered in the literature – those changes in real incomes that produce rearrangements in the relative positions of the recipient units or alterations in the distances between them. Even though we concluded that there was a reduction in the intensity of income mobility in Argentina between the late 1980s and the late 1990s, no reference was made on the relationship between this phenomenon and the convergence or divergence of incomes. Some studies on the country, which use either longitudinal data (Fields & Sa´nchez Puerta, 2005; Albornoz & Mene´ndez, 2002) or pseudo-panels (Navarro, 2006), reveal the presence of mobility that leads to income convergence. Therefore, it seems important to explore whether the latter characteristic did occur and, also, the reasons that would allow explaining why the latter phenomenon could have taken place in a period of persistent increases in income inequality measured with static data. As it was already mentioned in Section 3.2, the simultaneous coexistence of convergence without reductions in the income concentration of each period is only possible in the presence of rank mobility and signiﬁcant variations in income levels. Speciﬁcally, a considerable proportion of the increases experienced by those who improve their relative position must be proportionally higher than the reductions of those whose positions worsen. This is precisely the argument that Fields and Sa´nchez Puerta (2005) pointed out when analyzing longitudinal data regarding the incomes of the employed of all the urban centers of Argentina during the period 1996–2002. Longitudinal data on household incomes for Greater Buenos Aires (GBA) provide evidence of the existence of some degree of convergence (Table 8). It has been already pointed out that part of this mobility can be due to measurement or reporting errors, and therefore a methodological alternative

Income Mobility in Argentina

315

Table 8. Variation in Households’ Average Income between Subsequent Observations By Quintiles of the Initial Distribution. Quintiles

%

1 2 3 4 5

63.2 20.2 4.0 4.3 12.3

Source: Author’s elaboration based on PHS.

is to resort to pseudo-panels or ﬁctitious cohorts. Therefore, it seems interesting to carry out exercises that explore the convergence or divergence of per capita household income mobility in the long run with this type of data (see Section 4.2). Long periods – the ones covered by these pseudopanels – are more appropriate for the study of mobility patterns. The frequent use of longitudinal data covering relatively short intervals is due to the lack of surveys speciﬁcally designed to gather this type of data. In order to analyze the pattern of mobility that emerges from both longitudinal data and pseudo-panels, we used model (9) indicated above. With regard to the former, we considered the pool made of all the pairs of observations derived from the PHS’s dynamic panels, which were used in previous sections of this paper. For the case of pseudo-panels, the model is speciﬁed in Eq. (10), which has the logarithm of the cohort’s average income in t as the dependent variable, and the logarithm of income in (t 1) as the independent variable: ln ycðtÞ;t ¼ a þ b ln ycðt1Þ;t1 þ ucðtÞ;t

(10)

Both sets of data suggest some degree of convergence, as it can be seen in Table 9; it seems to be lower when the model is computed with the pseudopanel mobility. However, as was discussed above, the advantages of having pseudo-panels as a source of information with less reporting errors increase as average incomes of more homogeneous units are compared. The available databases bring us one step closer in the search toward homogeneity. Therefore, following other similar studies (Antman & McKenzie, 2005), we divided the sample according to the level of education of the household heads. Hence, two strata were deﬁned for each cohort: one groups those households with low schooling (up to incomplete high school) while the other includes the rest of households.

316

Table 9.

Annual Income Mobility (Estimates With Longitudinal Data and Pseudo-Panels). Longitudinal Data All households

Regression coefﬁcient of income of the previous year Lower bound Upper bound P-value Adjusted R2 Number of cases

Cohort All households

Households with heads born between 1940 and 1964

Cohort-strata All households

Households with heads born between 1940 and 1964

0.735 0.740

0.737 0.744

0.732 0.751

0.690 0.718

0.927 0.938

0.927 0.930

0.748

0.740

0.685

0.690

0.919

0.936

0.729 0.766 0.000 0.541 9,269

0.718 0.762 0.000 0.543 6,387

0.568 0.812 0.000 0.534 173

0.503 0.878 0.000 0.471 105

0.867 0.971 0.000 0.860 346

0.867 1.004 0.000 0.858 210

Source: Authors’ elaboration on PHS.

LUIS BECCARIA AND FERNANDO GROISMAN

Correlation coefﬁcient Pearson Spearman

Households with heads born between 1940 and 1964

Pseudo-Panel Data

317

Income Mobility in Argentina

Table 10.

Income Mobility (5 and 15 years interval. Estimates of the regression with pseudo-panel data).

Coefﬁcient of income of the previous period Lower bound Upper bound P-value Adjusted R2 Number of cases

5 Years Before

15 Years Before

0.855 0.719 0.990 0.000 0.61 170

0.956 0.772 1.140 0.000 0.73 70

Source: Authors’ elaboration on PHS.

The model, when estimated for cohort-strata, result in lower levels of mobility than that of the ‘‘pure’’ cohorts; in particular, the coefﬁcient turns out to be very close to one, thus accounting for the existence of a very low degree of convergence among family incomes. If we consider the ﬁve cohortstrata that cover the whole period19 – thus excluding those cohorts that ‘‘enter’’ and ‘‘exit’’ along the period – the estimated coefﬁcient is not statistically different from unity. These results are consistent with those obtained when indicators of correlation between income of successive periods are used: they are also larger for cohort-strata data (Table 9). The model was also estimated considering ycðtnÞðtnÞ , with different values for n, of 5 and 15 years. This procedure aims at assessing the effects of income variations that take place within periods of those durations. The results (Table 10) show that there is no fundamental change in the panorama with respect to interannual mobility. This means that, at least in Argentina during those years, the probability of convergence did not increase with period length. Furthermore, it appears to be an opposite outcome as the coefﬁcient increases as the lengths of periods were extended.

8. CONCLUSIONS The macroeconomic stabilization achieved in the early 1990s reduced the variability of family incomes. Nonetheless, the growth of occupational instability as from the middle of that decade meant that ﬂuctuations in current family incomes persisted into the early 21st century and remained high, although less so than in the high inﬂation phase. It is worth noting here the different impact that inﬂation reduction had on households of the

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different socioeconomic groups. In the case of those headed by lowschooling individuals, the stabilizing effect was fully discounted by occupational variability. When the analysis is made on the basis of individual incomes, the impact of variability is greater because there is no reduction and even increases among employed persons of low-education levels. These patterns of current-income ﬂuctuations are reﬂected in the gap between the trends of average family labor income and family labor income adjusted for risk, which was lower in low-strata households. In conjunction with the (slight) reduction in household incomes instability recorded between the ﬁrst and the last phases, the distances moved by family incomes became increasingly smaller. As a result, the positions of households in the income distribution tended to consolidate, causing growing segmentation between households of different types. The above discussion shows that low-income families not only beneﬁted less from income stabilization, but also faced additional difﬁculties in improving their relative position. The panorama of growing inequality in the income distribution since the early 1990s, as reported by various studies based on current/cross-sectional incomes, is also appropriate for describing what happened to changes in the distribution of more permanent incomes. Inequality in the latter actually increased slightly more than in current incomes because of the decrease in mobility recorded throughout that period. A general conclusion to be drawn from the analysis of this paper is that inequality in the early 2000s was similar to that recorded in the late 1980s. An evaluation of this similarity should take account of the fact that periods of high inﬂation were accompanied by sharply worsening distribution. Even when the comparison is made with the third (growth) phase, rather than the ﬁnal (recessionary) phase, there is no reduction compared to the years of hyperinﬂation. This would appear to support the hypothesis that increasing differentiation in terms of labor instability accentuated the increase in inequality among more permanent incomes. The results attained for the aggregated period of the last 20 years conﬁrm that there was limited global income mobility. In fact, by using ﬁctitious cohort-strata, we tend to corroborate that the initial differences in household incomes remained largely unchanged also throughout longer periods. This indicates that the sharp deterioration in income distribution registered since the late 1980s reﬂected gradual changes in the remunerations of the households’ resources. Hence, these facts seem to support the view that the growing inequality in Argentina was a result of causes associated more to, for example, a persistent trend in the labor market toward the

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exclusion of certain individuals, than to unfortunate events that the mere pass of time – or the effects of the life cycle – can counteract.

NOTES 1. The monthly rate ﬂuctuated around 7% during 1990 before rising to 27% in February 1991 as a result of the devaluation and other measures (such as rate hikes). In December, it had already plunged to 1%. 2. The privatization process was implemented rapidly because in addition to supporting the goal of withdrawing the state from productive activity, capital inﬂows were essential to support the external and ﬁscal account balances. 3. It is also a measure that is sensitive to the degree of inequality in the society and, therefore, is unsuitable for comparison between countries. For example, the same 10% increase in incomes could represent a quantile change in one country, whereas in another it could mean staying in the same income bracket. 4. Nominal hourly incomes can vary merely as a result of changes in earnings from a given job, but they can also vary as a result of moving from one job to another. The impact of this effect was not calculated. 5. Estimations made with larger coefﬁcients did not alter the results obtained. 6. The time period over which more permanent incomes are calculated matters because the longer the period, the smaller one would expect the differences between average incomes to be. 7. Fields and Sa´nchez Puerta (2005) and Albornoz and Mene´ndez (2002) use a similar procedure. 8. Sometimes ‘‘quantile’’ regressions are used; see, for example, de Fontenay, Gorgens, and Liu (2002). 9. PHS methodology can be found in www.indec.gov.ar. The survey scheme was changed substantially in 2003. 10. Although this procedure makes it possible to work with a large number of observations, the phenomenon of attrition can introduce sample biases that have not yet been investigated. 11. This is Argentina’s main urban agglomeration, accounting for 30% of the country’s population and 40% of its total urban inhabitants. 12. It is worth to highlight that similar results were obtained when considering other educational groupings. 13. The databases from which longitudinal information are available only from 1987 onward. 14. A similar result was obtained by Gutie´rrez (2004) for the recessionary period 1998–2002. 15. Similar results are conﬁrmed in the analysis by Cruces and Wodon (2003) for 1995–2002. 16. The results of the comparison should not be altered when taking the average of the three coefﬁcients that can be calculated between pairs of consecutive observations. 17. This emerges from a consideration of conﬁdence intervals for the differences in correlations calculated by bootstrapping techniques.

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18. Similar results were obtained with other indicators of inequality. 19. Those whose household heads were born in any of the 5-year periods within 1940–1944 and 1960–1964.

REFERENCES Albornoz, F., & Mene´ndez, M. (2002). Analyzing income mobility and inequality: The case of Argentina during the 1990s. Paris, unpublished. Altimir, O., & Beccaria, L. (2001). El persistente deterioro de la distribucio´n del ingreso en Argentina, Desarrollo econo´mico, vol. 40, No. 160, Buenos Aires, Institute of Economic and Social Development, January–March. Antman, F., & Mckenzie, D. (2005). Earnings mobility and measurement error: A pseudo-panel approach. Mimeo: Stanford University. Arrow, K. (1970). Essays in the theory of risk bearing. Amsterdam: North-Holland. Beccaria, L., Esquivel, V., & Maurizio, R. (2002). Desigualdad y polarizacio´n del ingreso en Argentina, Red pu´blica, No. 2, Buenos Aires. Beccaria, L., & Groisman, F. (2005). Las familias ante los cambios en el mercado de trabajo. In: L. Beccaria & R. Maurizio (Eds), Mercado de trabajo y equidad en Argentina. Buenos Aires: Prometeo. Beccaria, L., & Maurizio, R. (2004). Movilidad ocupacional en el Gran Buenos Aires. El trimestre econo´mico, 71(283). Burgess, S., Gardiner, K., Jenkins, S.P., & Propper, C. (2000). Measuring Income Risk, CASE Paper No. 40. London, Centre for Analysis of Social Exclusion, London School of Economics. Cruces, G., & Wodon, Q. (2003). Risk-adjusted poverty in Argentina: measurement and determinants. Argentina Crisis and Poverty. A Poverty Assessment (2). Washington, D.C.: World Bank. de Fontenay, C., Gorgens, T., & Liu, H. (2002). The role of mobility in offsetting inequality: A nonparametric exploration of CPS. Review of Income and Wealth, 48(3). Fields, G. (2004). Economic and social mobility are really multifaceted. Ithaca: School of Industrial and Labor Relations, Cornell University. Fields, G., & Sa´nchez Puerta, M. L. (2005). Remuneration Mobility in Urban Argentina, document prepared for the World Bank. Galiani, S., & Hopenhayn H. (2000). Duracio´n y riesgo de desempleo en Argentina, Mercado de Trabajo y Relaciones Industriales series, Buenos Aires, Fundacio´n Argentina para el Desarrollo con Equidad (FADE), unpublished. Gottschalk, P., & Danziger, S. (1998). Family income mobility: How much is there and has it changed? In: J. A. Auerbach & R. S. Belows (Eds), The inequality paradox: Growth of income disparity. Washington, D.C.: National Policy Association. Gutie´rrez, F. (2004). Dina´mica salarial y ocupacional. Ana´lisis de panel para Argentina: 1998-2002. Working paper no. 11. La Plata, Universidad Nacional de La Plata. Hills, J. (1998). Does income mobility mean that we do not need to worry about poverty? In: A. Atkinson & J. Hills (Eds), Exclusion, employment and opportunity. London: Centre for Analysis of Social Exclusion, London School of Economics. Hopenhayn, H. (2001). Labor Market Policies and Employment Duration: The Effects of Labor Market Reform in Argentina. Research Net Working Papers no. R.407. Washington, D.C., Inter-American Development Bank, February.

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Jarvis, S., & Jenkins, S. (1998). How much income mobility is there in Britain? Economic Journal, 108, 428–443. Navarro, A. I. (2006). Estimating Income Mobility in Argentina with pseudo-panel data, unpublished. OECD (Organisation for Economic Co-operation and Development). (1996). Employment Outlook (pp. 59–108). Paris. Shorrocks, A. F. (1978). Income inequality and income mobility. Journal of Economic Theory, 19(2), 376–393.

RISK LEVEL AND INEQUALITY PREFERENCE Liema Davidovitz ABSTRACT Purpose: The purpose of this paper is to investigate whether inequality aversion is inﬂuenced by the risk level. Recently empirical evidence points to deviations from selﬁsh behavior of Homo economicus. Thus, people are not motivated solely by their own monetary payoffs, but are also concerned about issues of equality and fairness. This paper distinguishes between inequality aversion and risk aversion and discusses whether the level of risk affects these motivations. Design: In an experimental framework the attitude toward inequality is separated from the attitude toward risk. A risky environment is generated by a set of lotteries. The subjects had to determine the method for payment, equally (CG) or nonequally (IG), for three lotteries with different levels of risk. The inequality preferences are measured by the level of the selected probability for CG. Findings: The main ﬁnding of this paper is that preferences for inequality are inﬂuenced by level of risk. We found that aversion to inequality was stronger when the level of risk was higher. In the low and medium risk lotteries participants preferred the individual gamble – the nonegalitarian method. Only in the high-risk lottery the participants preferred the common gamble that assured them equal payments. Inequality and Opportunity: Papers from the Second ECINEQ Society Meeting Research on Economic Inequality, Volume 16, 323–338 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1016/S1049-2585(08)16014-8

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LIEMA DAVIDOVITZ

Originality/value: The paper distinguishes between inequality aversion and risk aversion and subjects are allowed to trade one off against the other. Thus, it contributes to the understanding of the interrelationship between income inequality and risk.

1. INTRODUCTION Multiple streams of research have explored the risk-taking construct and compared it to inequality aversion. This paper adds to this line of research by questioning whether the degree of risk affects the degree of inequality an individual will choose. There is ample empirical evidence of deviations from selﬁsh behavior by Homo economicus. Namely, people are not motivated solely by concern for their own monetary beneﬁt, but also care about equality and fairness (Fehr & Fischbacher, 2002). The beneﬁts received from private activity are generally expected to be allocated to the person who takes on the risk of the enterprise. However, previous work has found that higher equality motivates individuals to take on even more risks and challenges (Davidovitz & Kroll, 2004; Davidovich, Heilbrunn, & Polovin, 2006). This positive relationship can be attained if the risk involved in the efforts is shared more evenly among the inequality-averse participants. Using an experimental design based on the portfolio game, Davidovitz and Kroll (2004) found that inequality aversion may reduce and even reverse the negative impact of an egalitarian economy on risky effort motivation. Jewell and Molina (2004) investigated the salary distribution among major league baseball (MLB) players in the USA. They found that the distribution of salaries within MLB teams has a signiﬁcantly negative effect on team success as measured by a team’s winning percentage. They claimed that if this trend continues, MLB may be forced to explore ways to equalize salaries within teams. Carlsson, Daruvala, and Johansson-Stenman (2005) ask whether people are inequality averse or just risk averse. They measure individuals’ preferences for risk and inequality by the choices the participants make between imagined societies and lotteries. They claim that ‘‘individuals may also have a willingness to pay for living in a more equal society per se’’ (pp. 375) and therefore, one must estimate risk aversion and inequality aversion separately.

Risk Level and Inequality Preference

325

These fairness motives are a component in the utility function of an individual. A loss of utility caused by income differences between individuals can be measured as inequality aversion (Fehr & Schmidt, 1999; Boole & Ockenfels, 2000). Risk aversion and inequality aversion both play a role in the process of decision making when choosing risky ventures, if individuals have the ability to trade one off against the other. In order to understand the complexity of the decision-making process, we have to distinguish between inequality aversion and risk aversion. In most previous research, measures of inequality aversion were based on the same tools as used when assessing a probability distribution with monetary payoffs in terms of risk. The question, therefore, arises whether the choice of an egalitarian distribution is motivated by aversion to inequality or aversion to risk. According to Kroll and Davidovitz’s (2003) approach, the attitude toward inequality should be analyzed as a response to a change in inequality among individuals, while maintaining the moments of income distribution constant. Such a change is simulated when an individual moves from a ‘‘common gamble’’ (CG) to an ‘‘individual gamble’’ (IG) environment. In a CG environment, the results apply to all the individuals in the reference group. In contrast, in an IG milieu, each individual faces a separate gamble and may have a different outcome. The risk in both situations is identical, while only the CG represents a state of total equality. The purpose of this paper is to investigate whether inequality aversion is inﬂuenced by the risk level. We hypothesize that preferences toward inequality are dependent on the level of risk and that aversion to inequality will be stronger when the level of risk is higher. The next section presents a basic deﬁnition of CG and IG. Section 3 describes the laboratory experimental tests. Section 4 presents the results of the experiment. The last section presents conclusions and points for further analysis.

2. BASIC DEFINITIONS Assume an economy with n participants. Let x be a random variable of income with distribution function f(x). Let us deﬁne an ordered statistics of n sample variants out of f(X) by (X1,X2,yXn). By the deﬁnition of the ordered statistics for xiAX, ’i, it must be that x1rx2yrxn.

326

LIEMA DAVIDOVITZ

Assume, also, that the participants in the economy randomly sample their income out of f(X). Let us deﬁne two alternative types of gamble. Deﬁnition 1. ‘‘Common Gamble’’ (CG) All participants sample the same xiAX out of f(X) by one mutual gamble. Deﬁnition 2. ‘‘Individual Gamble’’ (IG) Each participant independently draws an income x out of f(X). In both cases of income gambles, the participants, who are under a veil of ignorance, face the same distribution of income f(X). An IG occurs when, for each individual, a separate gamble is executed; some individuals may have high results and others may have low results. This cannot happen in a ‘‘common gamble’’ where all will have the same results. Thus, the ‘‘common gamble’’ situation presents a less unequal state, yet the risk under the two situations may be the same. Deﬁnition 3. Composite Gamble (COMG) This is a two-stage gamble, in which the ﬁrst stage determines, with probabilities q and 1 q, whether CG or IG will be employed in the second stage. The distribution of the individual income f(X) is independent of the result of the ﬁrst stage of COMG since the distribution of income X under CG is identical to that under IG. Deﬁnition 4. ‘‘More Inequality’’ The lower the probability (q) of CG in COMG (other things being equal), the higher the ex ante inequality among the participants is. Deﬁnition 5. ‘‘More Inequality Aversion’’ If an individual can vote anonymously for q, then the higher his selected q is, the more inequality averse s/he is.

3. THE EXPERIMENT The study’s primary objective is to examine the inequality preferences of participants facing uncertain outcomes. A risky environment is generated through a set of lotteries. The inequality preferences are measured by the level of the selected probability for CG.

Risk Level and Inequality Preference

327

3.1. Subjects A total of 124 volunteer students, all undergraduates from the Department of Economics at the Ruppin Academic Center, participated in the experiment. All the students had taken ﬁnance courses and were familiar with risk investment theories. The instructions were given and the ﬁrst part of the experiment was conducted at the end of a lecture. The students had the option of not participating in the experiment and leaving the classroom at this point. For the second part of the experiment and to collect their payment, the students had to come to the computer laboratory when they had free time. The total time for the two parts of the experiment was about 25 min, 10–15 min for the ﬁrst part and 10 min for the second part. The average payment was 20NIS (about US $5), which is the national hourly student wage. We chose to use only volunteers in order to reveal the respondents’ true preferences. ‘‘When subjects are recruited to an independent location and paid for their appearance, they behave in a less extreme manner than subjects who participate as part of an entire class recruited for an experiment conducted during class period’’ (Eckel & Grossman, 2000). 3.2. Experiment We called for volunteers to participate in an experiment in which they could earn some money. We told them that the volunteers would be arranged in groups of ﬁve and asked to carry out some task for which they would be paid 20NIS (about US $5). We further told them that they could increase or decrease this amount by means of a bet. We emphasized that by betting they could earn more money and even double their 20NIS, but they could also lose the entire amount and end up with nothing. The size of the new reward/ ﬁne would be determined by throwing a dice. The subjects had to choose between two methods for throwing the dice: In Method 1 (CG), the experimenter throws the dice once on behalf of the entire group and all the subjects in the group receive the same reward. In Method 2 (IG), the experimenter throws the dice separately for each student, thereby determining the individual reward/loss. The explanation given to the subjects was: ‘‘You can choose CG, IG or a mixed method. The mixed method is a two-stage gamble, in which the ﬁrst stage determines with probabilities q and 1 q whether CG or IG will be employed in the second stage; you can determine the probability (q).

328

LIEMA DAVIDOVITZ

If you write q ¼ 100%, then we use Method 1, the common gamble. In this case the same results, either high or low, apply to all individuals in your reference team. If you write q ¼ 0%, then we use Method 2, the individual gamble. In this case each individual faces a separate gamble and her/his results may differ from those of her/his team members. If, for example, you write q ¼ 70%, then we’ll toss a false coin with probability of 70% for CG and 30% probability for IG. The result determines whether we continue with CG or IG.’’

The subjects had to determine the way the dice would be tossed for three lotteries with different levels of risk (equal expectancy and different standard deviation). We called the levels: high risk, medium risk, or low risk. For some subjects the lotteries were presented in a decreasing order of risk and for others the reverse. We used the Random Lottery Incentive approach. This approach suggests that during the experiment participants are asked to perform some tasks. At the end of the experiment one of the tasks is selected randomly and the subject plays the lottery and is paid accordingly (Starmer & Sudgen, 1991). This approach is rather attractive because it is relatively low cost and allows the collection of a greater amount of data.

3.3. The Method The experiment comprised three phases. 3.3.1. Phase I: The Task The task was to determine the price for different lotteries in which one can gain 10,000NIS or nothing with different levels of chance. We had a number of motives for creating this hypothetical gamble as a task. First, we wanted to give participants the feeling that they were entitled to a reward (compensation for their labor). Secondly, this task serves the purpose of providing a stake that can be lost later and ﬁnally we hoped to determine whether the participants were risk averters. After the participants completed the ﬁrst task they were told that they were now entitled to a reward on which we were asking them to bet (see Appendix A).

329

Risk Level and Inequality Preference

3.3.2. Phase II: The Bet The subjects were faced with three lotteries. For each lottery they had to decide the method for payment: equally (CG) or nonequally (IG). The participants were notiﬁed in advance that only one of the lotteries would apply to them and that this lottery would be determined randomly (see Appendix B). 3.3.3. Phase III: The Payment Payment was made using a four-stage random lottery. Stage 1 – we selected a student from each group of ﬁve subjects whose answer would determine the group’s payment. Stage 2 – we selected the lottery (high risk, medium risk, or low risk). Stage 3 – The selected student’s sheet (Stage 1) is examined and the selected lottery is played (Stage 2). Stage 4 – in this stage the extra reward/ﬁne is determined. If the answer for Stage 3 was CG – we throw the dice once for the entire group. If the answer was IG, then we throw the dice for each student and pay each accordingly. Diagrams 1 and 2 illustrate the experiment.

The Experiment

The Task Decide the price of the following lotteries

You are entitled to 20NIS

The Bet Decide the method of payment for 3 lotteries

The Payment Or

Only one will apply to you

End

Diagram 1.

Flow chart of the experiment per participant.

330

LIEMA DAVIDOVITZ

Payment

Select a student from the group: 1,2,3,4,5

Select a lottery: A,B,C

Bet according to the above answer: Probability q for CG____

q CG

1-q IG

CG was selected Throw the dice once for all the group

Diagram 2.

IG was selected Throw the dice for each student separately

Flow chart of the experiment for the experimenter.

4. RESULTS The subjects determined the prices of the different lotteries in which one can gain 10,000NIS or nothing with different levels of chance. From Table 1 we can see that almost all the subjects were risk averters. The price (certain equivalent) was lower than the expected value of the lottery in most cases (93.55). Only 6% of the students showed a risk-prone attitude and gave a price that was above the expected value. As can be seen from the results, there is a monotonicity preserving, CE increase with the expected value of the lotteries. 4.1. The Bet Each subject was presented with three lotteries for which she had to select the method for throwing the dice by determining the probability for CG.

331

Risk Level and Inequality Preference

Table 1. Lottery

1

The First Phase Lotteries. 2

3

4

5

6

7

8

Expected value (EV) 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 Mean price (CE) 315 482 745 1,101 1,454 1,903 2,306 2,920 Price Standard deviation 523 803 1,128 1,446 1,735 2,131 2,383 2,738 Minimum price 0 0 0 0 20 20 20 30 Maximum price 2,000 3,000 4,100 5,200 6,000 7,400 8,500 9,600 CEoEV 93.5% 93.5% 93.5% 93.5% 95.2% 93.5% 93.5% 93.5% CE ¼ EV 6.5% 6.5% 4.8% 4.8% 4.8% 4.8% 4.8% 4.8% CEWEV – – 1.6% 1.6% – 1.6% 1.6% 1.6%

Table 2.

The Selected Probability of CG for Each Lottery.

Lottery

Mean

Standard Deviation

N

Lot A: low risk Lot B: medium risk Lot C: high risk

0.3067 0.4932 0.6589

0.3299 0.2384 0.3364

124 124 124

In order to test the difference between the selected probabilities in the three lotteries (low risk (A), medium risk (B), and high risk (C)), we used a GLM repeated measures data test. To rule out framing effects, 64 of the students answered the questionnaire going from the low-risk lottery (Lot A) to the high-risk lottery (Lot C) while 60 of the students answered the questionnaire in reverse o rder – namely, going from the high-risk lottery (Lot C) to the low-risk lottery (Lot A). We did not ﬁnd any signiﬁcant effect for the order (F(1.244,151.779) ¼ .030, po.907). Consequently, we treated all the participants as one group. Table 2 shows the selected probability of CG (q) for each lottery. We can see that the selected probability of CG (q) increased with the level of risk. In the low and medium risk lotteries participants preferred the IG – the nonegalitarian method (qo0.5). Only in the high-risk lottery did participants vote for qW0.5 – namely, they preferred the CG that assured them equal payments. By looking at the descriptive statistics of the answers we can see that in the low-risk lottery (Lot A), 77.4% of the students selected qr0.5, in the medium risk (Lot B) 60.5% selected qr5, and only 35.5% of the students selected qr0.5 in the high-risk lottery (Lot C).

332

LIEMA DAVIDOVITZ

The risk effect of the selected probability of CG was signiﬁcant (F(1.244,153.017) ¼ 40.43, po.000). The results are also shown in Fig. 1. Table 3 shows the results of the Bomferroni test, which compares every pair of lotteries. As mentioned above, lottery C (Lot C) is riskier than lottery B (Lot B), which is riskier than lottery A (Lot A). The negative results show that the selected level of equality (CG) is higher when risk is higher and all the

proportion of CG

1

0.5

0 low risk

medium risk

high risk

level of risk

Fig. 1. The selected probability of CG for each lottery. The ﬁgure shows the selected probability of CG (q) for each lottery. Notice that the selected probability of CG (q) increased with the level of risk. Only in the high-risk lottery the participants preferred the common gamble (CG) (qW0.5).

Table 3. Pair

Lot A–Lot B Lot A–Lot C Lot B–Lot C

The Selected Probability of CG-Paired Sample Tests. Mean Difference

0.187 0.352 0.166

Standard Error

0.03 0.049 0.027

Signiﬁcancea

0.000 0.000 0.000

The mean difference is signiﬁcant at the 0.05 level. a

Adjustment for multiple comparision: Bonferromi.

95% Conﬁdence Interval for Differencea Lower bound

Upper bound

0.259 0.470 0.232

0.115 0.234 0.099

Risk Level and Inequality Preference

333

differences in the mean selected probability of CG between the lotteries are signiﬁcant.

5. CONCLUSION The main ﬁnding in this study is that inequality preferences are inﬂuenced by the level of risk. We should not refer to the attitude toward inequality as a ‘‘global property.’’ Individuals are not either risk averters or risk seekers. We should refer to inequality as an economic ‘‘good.’’ Consequently, even inequality averters will agree that increasing inequality is not necessarily bad (Welch, 1999). Accordingly, in our approach, we gave participants the possibility of deciding the level of inequality they preferred in each situation. We found that aversion to inequality was stronger when the level of risk was higher. Carlsson et al. (2005) found higher parameter values of inequality aversion for women. This ﬁnding may be explained by the fact that women tend to be more risk averse. Nonetheless, it is not straightforward to generalize the ﬁndings from a ﬁve-person setting to a social setting. Further investigation and study are needed. A comparison of differential-oriented kibbutzim and collective-oriented kibbutzim in Israel in 2004 showed a difference of distribution in the operational risk level between the two organizational cultures: the more collectiveoriented cultures show a higher degree of risk taking. The proportion of risky entrepreneurship was higher in the egalitarian organizations (Davidovich et al., 2006). We found that inequality preference is inﬂuenced by level of risk. As Beckman, Formby, Smith, & Zheng (2004) concluded behavior toward risk and inequality may be more complex than expected. An additional explanation to inequality aversion under risk is that due to loss aversion (Prospect Theory) people are hurt more by others gaining more than them than they enjoy it that others gain less than them. The more vivid the difference between the group peers, the more we expect inequality aversion. Future research will have to test the inﬂuence of the level of risk on inequality aversion. As Holt (1986) pointed out, the level of rewards affects motivation. The expected reward in this experiment was 32.5NIS (one and a half times the hourly wage for students), which in our view indicates that motivation was maintained. This task in phase one involves only gains, we did not check the attitude toward risk over losses, and therefore cannot conclude whether the subjects are risk averters.

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REFERENCES Beckman, S. R., Formby, J. P., Smith, J. W., & Zheng, B. (2004). Risk, inequality aversion and biases born of social position. Further experimental tests of the leaky bucket. Research on Economic Inequality, 12, 73–99. Boole, E. G., & Ockenfels, A. (2000). ERC: Equity, reciprocity and competition. American Economic Review, 90, 166–193. Carlsson, F., Daruvala, D., & Johansson-Stenman, O. (2005). Are people inequality-averse or just risk-averse. Economica, 72, 375–396. Davidovich, L., Heilbrunn, S., & Polovin, A. (2006). Inequality and entrepreneurial risk-taking of organizations. The ICFI Journal of Entrepreneurship Development, III(4), 52–66. Davidovitz, L., & Kroll, Y. (2004). On the attitude towards inequality. Research on Economic Inequality, 11, 137–148. Eckel, C. C., & Grossman, P. J. (2000). Volunteers and pseudo-volunteers: The effect of recruitment method in dictator experiments. Experimental Economics, 3, 107–120. Fehr, E., & Fischbacher, U. (2002). Why social preferences matter – the impact of non-selﬁsh motives on competition, cooperation and incentives. Economic Journal, 112, c1–c33. Fehr, E., & Schmidt, K. M. (1999). A theory of fairness, competition, and cooperation. The Quarterly Journal of Economics, 114(3), 817–868. Holt, C. A. (1986). Preference reversal and the independence axiom. American Economic Review, 76, 508–515. Jewell, R. T., & Molina, D. J. (2004). The effect of salary distribution on production: An analysis of major league baseball. Economic Inquiry, 42(3), 469–482. Kroll, Y., & Davidovitz, L. (2003). Inequality aversion versus risk aversion. Economica, 70, 19–29. Starmer, C., & Sudgen, R. (1991). Does the random-lottery incentive system elicit true preferences? an experimental investigation. The American Economic Review, 81(4), 971–978. Welch, F. (1999). In defense of inequality. The American Economic Review, 1–17.

APPENDIX A. FIRST TASK (Translated from Hebrew) This assignment is for research purposes only. You have the right not to participate. We intend to award you 20NIS for your participation. Lottery A Probability

Return

0.8 0.2

0 10,000

The maximum price you are willing to pay for this lottery is ______________

Risk Level and Inequality Preference

335

Lottery B Probability

Return

0.7 0.3

0 10,000

The maximum price you are willing to pay for this lottery is ______________ Lottery C Probability

Return

0.6 0.4

0 10,000

The maximum price you are willing to pay for this lottery is ______________ Lottery D Probability

Return

0.5 0.5

0 10,000

The maximum price you are willing to pay for this lottery is ______________ Lottery E Probability

Return

0.4 0.6

0 10,000

The maximum price you are willing to pay for this lottery is ______________ Lottery F Probability

Return

0.3 0.7

0 10,000

The maximum price you are willing to pay for this lottery is ______________

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LIEMA DAVIDOVITZ

Lottery G Probability

Return

0.2 0.8

0 10,000

The maximum price you are willing to pay for this lottery is ______________ Lottery H Probability

Return

0.1 0.9

0 10,000

The maximum price you are willing to pay for this lottery is ______________ Thank you for participating in this research. You are entitled to a reward of 20NIS for your participation, but you can increase or decrease this amount by means of a bet.

APPENDIX B (Translated from Hebrew) In this questionnaire you are the representative of a group of ﬁve students. You will be asked to choose a method of a bet. The reward is not ﬁxed; the size of the new reward/ﬁne is to be determined by throwing dice. The dice has six sides, each side presents a 1/6 chance of a reward/ﬁne.

WILL THE GAMBLE BE INDIVIDUAL OR FOR THE ENTIRE GROUP? There are two methods for throwing the dice: In Method 1 (CG) – Common-group Gamble – the experimenter throws the dice once on behalf of the entire group and all the subjects in the group receive the same reward. In Method 2 (IG), Individual Gamble – the experimenter throws the dice separately for each student, thereby determining the individual reward/loss.

337

Risk Level and Inequality Preference

You have to determine your willingness (q) to assume a common-group gamble for the entire group.

Determination of method of gamble

q Probability for Common Gamble

1-q Probability for Individual Gamble

If you write q ¼ 100%, then we use Method 1, the CG. In this case the same results, either high or low, apply to all individuals in your reference team. If you write q ¼ 70%, the probability of identical results for the entire group is 70%. We will toss a false coin with probability of 70% for CG and 30% probability for IG. The result determines whether we continue with CG or IG. If you write q ¼ 40%, the probability for identical results for the entire group is 40%. We will toss a coin with probability of 40% for CG and 60% for IG. The result determines whether we continue with CG or IG. If you write q ¼ 0%, then we use Method 2, the IG. In this case each individual faces a separate gamble and her/his results may differ from those of her/his team members. After you answer this questionnaire one of the lotteries will be selected randomly. Lottery A (low risk) Return NIS Probability

275 1/6

225 1/6

175 1/6

75 1/6

25 1/6

25 1/6

I would like with probability __________ that the gamble will be common for the entire group. Lottery B (medium risk) Return NIS Probability

450 1/6

350 1/6

250 1/6

0 1/6

100 1/6

200 1/6

338

LIEMA DAVIDOVITZ

I would like with probability __________ that the gamble will be common for the entire group. Lottery C (high risk) Return NIS Probability

650 1/6

450 1/6

300 1/6

50 1/6

200 1/6

400 1/6

I would like with probability __________ for the gamble to be common for the entire group.

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