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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 Rıó

University of Vigo, Vigo, Spain

Joseph Deutsch

Bar Ilan University, Ramat Gan, Israel

Henar Dıéz

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


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


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

x

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

1

2

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

4


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


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

6


(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


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

8


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


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.

10


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,

12


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


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


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).


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


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).


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


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,


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


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


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


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


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


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.


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

26


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


27

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

28


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 rećiprocite´ geńe´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.


31

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


33

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,


35

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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FRANK COWELL AND UDO EBERT

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,

39

Inequality and Envy

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


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.

41

Inequality and Envy

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)

42


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.

43

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.

44


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

46


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

47

Inequality and Envy

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

49

50

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

52


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


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

54


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


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

56


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


57

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

58


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


59

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

60


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


61

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

62


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

64


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

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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 !

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 Educacioń 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 Educacioń Universidades e Investigacioń 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 Ećonomiques et de Gestion de Tunis, CIRPEÉ, 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 Rıó and RuizCastillo (2000), recently generalized by Del Rıó 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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CORAL DEL RIÓ AND OLGA ALONSO-VILLAR

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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215

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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