Master Thesis

Inequality measurement re-reexamined Stephan Kampelmann November 7, 2007

Master Thesis in Applied Econometrics Faculté des Sciences Economiques et Sociales Université des Sciences et Technologies de Lille 59655 Villeneuve d’Ascq Cedex

Directeur de mémoire Florence Jany-Catrice Responsable de formation Nicolas Vaneecloo i

Contents 1 Introduction and methodology 1.1 Why inequality measurement is still relevant . . . . . . . . . . . . . . . . .

1.2 Discussing the undiscussable: inequality as convention . . . . . . . . . . . .

1.3 Some basic terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 6 13

2 An 2.1 2.2 2.3

internal history of the academic discourse From constant inequality to complex inequalities . . . . . . . . . . . . . . .

Recent developments: generalisation of methods . . . . . . . . . . . . . . .

Closer to ‘truth’ or away from ‘normal communication’ ? . . . . . . . . . .

15 16 59 61

3 Revision of inequality in the IEWB 3.1 Introduction to the Index of Economic Well-Being . . . . . . . . . . . . . .

3.2 Four dimensions, three inequalities . . . . . . . . . . . . . . . . . . . . . .

3.3 Alternative proposals to measure inequality . . . . . . . . . . . . . . . . .

63 63 68 70

4 Empirical application 4.1 Data treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 Results for alternative inequality statistics . . . . . . . . . . . . . . . . . .

4.3 IEWB with modified equality dimension . . . . . . . . . . . . . . . . . . .

83 83 89 92

Compte rendu du mémoire en français

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Chapter 1 Introduction and methodology 1.1

Why inequality measurement is still relevant: the case of the IEWB

Inequalities, and more generally the distribution of resources, are fundamental problems in economics. We believe it is important and helpful to analyse if and to what extent outcomes can be characterised as unequal. We may even agree with R.H. Tawney when he argues that “inequality is perhaps what economics should be all about” (Tawney, 1964).

Economic inequalities impact on a wide range of societal issues so that they can be dealt with in many different ways. Depending on which angle one chooses to shed light on the topic, the analysis tends to be more philosophical (the question of equity and justice), economic (the problems of incentives and resource allocation), or sociological (the function and role of socio-economic inequalities). It is arguably difficult to discuss all these fields simultaneously and we will not attempt to do so.

Instead, the question of inequalities will be approached in this text from a very specific viewpoint: the discussion will take the Index of Economic Well-Being (IEWB), developed

by Osberg & Sharpe (2005), as point of departure. This index was developed to provide an easily communicable heuristic, allowing users with different backgrounds to make judgements about ‘the big picture’ of economic well-being. It compiles statistical information concerning four dimensions that are thought to be relevant: 1) effective consumption; 2) accumulation of productive assets; 3) equality and poverty; and 4) economic security.

We believe that the IEWB is a useful tool for economic analysis, while the last term should be understood as including not only a purely economic discussion, but also the analysis of economic outcomes in political, ethical or sociological terms. This implies that the ‘experts’ of the economic discipline (statisticians, welfare economists etc.) are not the only users of the IEWB. In order to be a useful heuristic for other actors, their conceptions and representations have to be integrated in the IEWB. However, in the context of an earlier application of the IEWB to France (cf. JanyCatrice & Kampelmann, 2007), we noted that the position of economic inequalities within the architecture of the IEWB is not entirely satisfying. Only one aspect of inequalities 1

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appears in the index: in its current state, the IEWB only includes inequality in the distribution of disposable income. However, Osberg and Sharpe consider the various dimensions mentioned above to be relevant to make judgements about the development of economic well-being. If one accepts these dimensions of economic well-being, then, we argue, inequalities should be evaluated according to exactly these aspects, i.e. according to effective consumption, wealth accumulation and economic security. This inconsistency inspired the present text and its basic research questions: are inequalities correctly taken into account in the index of economic well-being? And if not, how can we improve the index without losing its transparency and intuitive appeal?

Inequality being a complex and somewhat blurry concept, these research questions contain more caveats than one might at first expect. For passing from the abstract concept ‘inequality’ to a concrete empirical measure applicable to income or wealth data is not a neutral technical process. It involves many steps and decisions, and the imminent danger is that some of the content of the original concept ‘inequality’ may be lost along the way.

For a start, we cannot even define easily the very concept we are talking about. Hence, the question of how economic inequalities should appear in the framework of the IEWB leads us to a more general question: how should economic inequalities be measured in the first place?

Is this question worth debating? Why don’t we simply rely on the well-known and widely-used measures of inequality such as the Gini concentration coefficient, like Osberg and Sharpe have chosen to? After all, these measures are readily available and are frequently referred to as objective and legitimate references in public debates on inequality.

We argue that at least three points indicate that there is a strong case for discussing the conventionally used inequality measures: 1. There are not one, but many different indices, coefficients or other statistical instruments which supposedly measure inequalities. Since these tools frequently contradict each other, the choice of any specific measure is not neutral and should be based on plausible and legitimate arguments (we will come back to the contradictions between alternative measures at several times in the text). Of course, the choice of a statistic should correspond directly to the purpose at hand (in our case the application of the IEWB). Therefore, we need to verify if available inequality measures correspond to this purpose. In order to do so, we need to understand what distinguishes the different measures, which judgements are embedded in their set-up and which conventions are integrated in their usage.

2. Like in the case of poverty measurement, the empirical analysis of inequalities contains many controversial issues. For example, in the debates on poverty the question of relative versus absolute poverty frequently incites vivid controversies (cf. Ravallion, 2003). Among other things, this shows that sometimes the very basic question of ‘what is poverty?’ does not allow for any obvious answer. As will be seen below, we think that similar controversies are involved in the measurement of inequalities, although they may be less debated and perhaps less obvious. The fact that these

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controversies have not (yet) penetrated the non-expert debate is perhaps partly due to the technical complexities involved in inequality metrics. The discussion on inequality statistics tends to be dominated by ‘experts’, whose background and interest render the conversation somewhat inaccessible to non-experts (we will give examples of this phenomenon later). Given the imperative of transparency and the rather ‘democratic’ purpose of the IEWB, we submit that the controversial issues in inequality measurement should not be hidden behind the statistical technicalities. On the contrary, they should be subject to open argumentation like the one we attempt to provide in this text.

3. We already argued that if we accept that economic well-being has multiple dimensions, it follows that inequalities should be evaluated in these multiple spaces. However, statistics like the Gini concentration coefficient are not directly applicable to multi-dimensional problems. This creates the practical problem of measuring inequalities in different spaces and the subsequent aggregation. Again, a solution to this problem should be coherent with the overall purpose as defined by the framework of the index of economic well-being.

These arguments directly imply why we cannot fully rely on other reviews of inequality measures that are available in the literature. While these accounts are numerous and a precious source for our endeavour, they do not explicitly address many problems that are specific to the IEWB. In fact, many reviews of inequality measures focus on technical issues (e.g. the problem of decomposability) or only make sense within a certain framework (e.g. the utilitarian approach). For our problem, it is of primordial importance to ensure that the measure of inequality corresponds to the representations and conceptions of its users, i.e. those held by policy makers, average citizens or other individuals wishing to make judgements on economic well-being. We therefore have to find a way to take these conceptions into account. In fact, much of the present text is devoted to establishing a coherence between the usage of the concept of inequalities in “normal communication”, on the one hand, and the process of its statistical operationalisation, on the other.

The specific context in which we discuss inequalities — i.e. as one of the dimensions of the IEWB — does not only determine the criteria we have to use to evaluate different inequality measures. Accepting the general IEWB framework also excludes from our analysis some of the questions which have been largely debated in the literature on inequality measurement. Due to the nature of the problem at hand, this text will not deal with the following important, but for us irrelevant questions: 1. We will not elaborate on the more fundamental question which has been much debated since Sen made it the central issue of his book Inequality Re-Examined, namely the question ‘inequality of what?’. In fact, the choice of the dimensions of economic well-being already answers the question in which space inequalities should be evaluated (the alternative spaces that have been proposed in the literature include individual income, household income, capabilities, ‘functionings’, primary goods etc.). Due

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CHAPTER 1. INTRODUCTION AND METHODOLOGY to the nature of the focus of this text, we are exclusively interested in the inequalities as regards the different dimensions of the IEWB. As a consequence, whenever the terms equality or inequality appear in the text, they are implicitly referred to as being evaluated in the space of the IEWB dimensions. This allows to skip extensive branches of the literature on inequality, including the ethical questions raised by Rawls (1971) or Sen (1992), unless these contributions touch on other issues closer to the purpose at hand. The question of ‘equity vs. equality’ is assumed to be subsumed by the philosophical debate excluded from the discussion.

2. We are not directly concerned with issues involved in the measurement of welfare as such. The IEWB is above all a pragmatic and rather descriptive approach to economic well-being. The index is interpreted as a positive, descriptive measure of the state of economic reality. The widely discussed problems in welfare-based inequality analysis such as interpersonal comparability or the dichotomy of ordinal versus cardinal welfare will therefore be ignored.

3. The question of the causes of inequalities will not be discussed in this text. The IEWB is an instrument allowing to evaluate economic outcomes over time and was not explicitly conceived for the analysis of causalities. Consequently, we will focus on the question of the extent of inequalities and not on their origin. In addition, it may be argued that the issues of empirical measurement and conceptual clarity should ideally precede the question of their origin. However, that this is not always the case can be seen in the controversy on poverty measurement where certain definitions of what counts as ‘being poor’ are often determined by the alternative causes of social deprivation: being poor due to an unfavourable socio-economic climate during childhood is not the same as being poor due to consecutive losses in risky stock market speculations. Even if this problem is certainly important in the context of inequality measurement, it will be excluded from the discussion.

These exclusions restrict the scope of the analysis somewhat. On the other hand, the relationship between non-scientific conceptions and empirical measurement may be of relevance for research not directly concerned with inequalities. In a sense, the research questions at hand touch on the foundation of statistics and empirical representations of concepts referring to social objects. The complex interplay between scientific work and the usage of concepts, for instance in policy evaluations, is very visible in the case of empirical measurement. Our analysis is hopefully a good illustration of the underlying difficulties and relationships.

The present text is structured in four chapters. The first chapter starts with an examination of the nature of concepts like inequality. This will be done with loose reference to theories borrowed from the sociology of knowledge, the conventionalist approach in economics and Alain Desrosières’ history of statistics. We will sketch our methodological framework and end the chapter with a terminology of concepts frequently re-appearing in the text.

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The second chapter applies the methodology presented in the first chapter to the academic discourse on inequality measurement in economics. In an attempt to assemble elements of an internal history of inequality statistics, understood as conventions, the scientific contributions in this field judged to be most relevant will be presented: these are the works of Vilfredo Pareto, Max O. Lorenz, Corrado Gini, Hugh Dalton, Henri Theil, Anthony B. Atkinson and Amartya Sen. We will also give an overview on more recent developments in the literature on inequality measurement. This internal history is not to be understood as a synthesis of the theoretical genesis, nor as a comprehensive overview on all existing inequality measures. It is an attempt to illustrate some crucial changes in the common body of knowledge on empirical inequality analysis. The historical perspective will not only allow us to trace the evolution of relevant conventions chronologically, but also will serve to shed light on the process of legitimation of these measures. Since legitimacy and conventions are closely related, it is important to understand why and when certain conventions have evolved.

In chapter three, two alternative ways of measuring inequalities in the framework of the IEWB are proposed. The legitimacy of these measures is based on a mix of acceptable conventions and a remise en question of conventions judged to be less plausible given the purpose of the IEWB. The text argues that it is essential for the given research question to confront the internal history of the academic discourse with the use of inequality measurement in the IEWB. The latter introduces the representations and usages of the term ‘inequality’ by actors external to the academic discourse and the argumentation will be structured around the confrontation between external and internal considerations. For readers not familiar with the IEWB, a brief overview of its methodology is provided in the beginning of the chapter.

The final chapter presents an empirical application of the IEWB to the case of France.

The effect and the sensitivity of the alternative inequality measures proposed in chapter 3 will be tested. This chapter will draw on an earlier application of the IEWB (op. cit.) and use data from the French household survey Budget de Familles.

As to conclude this introduction, some words of a more personal nature may not be entirely misplaced. My interest in inequalities is embedded in a wider personal project and it has been for me an intriguing field of research for some considerable time now. I first had the opportunity to work on the empirical measurement of well-being during an internship at the Centre Lillois d’Etudes et de Recherches Sociologiques et Economiques (Clersé) early 2006 under the inspiring guidance of Florence Jany-Catrice. Further appointments in 2006 and 2007 allowed me to gather some experience in field applications of synthetic socio-economic indicators in the French region Nord-Pas de Calais and the Brazilian state Acre. This Master thesis hopefully represents a junction between my past work and some ideas I would like to develop in a later Ph.D. During the latter I intent to explore the issue of inequalities in a comparative European perspective based on the EU-SILC data, on the one hand, and the theories of socio-economic systems, on the other. In order to build a solid epistemological and conceptual base for my future empirical work, I wanted to dedicate my Master thesis to the more fundamental questions associated with the topic:

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where do today’s standard measures of inequalities come from? On what theories and justifications rests their legitimacy? And, above all, do they correspond to the concept of inequalities as it is used in “normal communication”, in the sense that Sen (1973) employs the term1 ?

1.2

Discussing the undiscussable: the concept ‘inequality’ as a convention

We already mentioned the vast body of research that has developed around the concept of inequalities in political economy, welfare economics, political science, sociology and other related fields. At first sight, adding an additional account to the panoply of contributions — including dozens which employ the same strategy we are about to use of arranging the different inequality measures side by side so as to gauge their similarities and differences — seems to be a futile endeavour. And yet, we submit that some serious issues related to the discussion of inequalities have been missed or not sufficiently treated by most writings.

These issues arise from two interlinked observations, the first being related to the concept of inequalities itself, and the second to its use in the academic discourse: 1. The nature of the concept ‘inequality’ gives rise to serious epistemological questions that cannot be ignored if the discussion is to be useful and scientific. Is there a ‘true’ definition of income inequality2 ? And if this is not the case, how do definitions of the concept emerge?

2. There is an unavoidable and important relationship between the scientific discussion on inequalities and the use of the same concept in ‘normal communication’. Does the usage of the term inequality in the academic discourse correspond to the way in which it is employed outside the scientific arena? Would a discrepancy of meanings be a serious problem or only a minor inconvenience?

Sen (1973, 1997) has noted in the context of his inequality discussion some of the complexities we want to evoke: he frequently refers to the “usage of the term inequalities in normal communication” as some kind of constraint for the scientific discussion. However, the lack of a systematic analysis calls for a more explicit treatment of the two issues 1

Cf. our discussion of Sen’s contribution and his use of “normal communication” as extra-academic constraint in Section 2.1.7 on page 55.

2 As stated above, we excluded the question “inequality of what?” from our research problems. In our text, the term ‘inequality’ refers implicitly to economic inequality ‘in the sense of the IEWB dimensions’.

Since this expression is rather long and some of our results apply also to the discussion of inequality outside the IEWB, we will often use ‘income inequality’ as a generic term to say ‘quantifiable economic inequalities’. Similarly, we will not use quotation marks around the term ‘inequality’ every time it appears in the text, although this would probably make sense in the context of our approach. To make the reading easier, we will drop the quotation marks in most passages. This is similar to Luckmann & Berger’s (1966) strategy of not putting the term ‘reality’ systematically in quotation marks: they note that “this would be stylistically awkward”.

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cited above. We argue that such a ‘head-on’ approach will benefit immensely from an attempt to incorporate results obtained in fields like the history of statistics, the sociology of quantification and the sociology of knowledge. Indeed, since we are dealing with ‘social facts’, it appears quite obvious to turn to the works of disciplines which precisely analyse important aspects of these ‘facts’, such as their genesis, their nature and the procedures involved when they are discussed. While we do not pretend to offer a sociological analysis of the concept ‘inequality’, we merely want to clarify some of its key features and their implications for our research questions.

The nature of the concept ‘inequality’ First, and perhaps most importantly, the concept ‘inequality’ is not a fact. Objects like stones, trees and rivers, however, are facts. In the theory of the American philosopher J.R. Searle (1995), the latter class of objects is referred to as brute facts, which exist independently from human opinions about them. However, these objects are not the only things qualified as facts. Searle holds that other things such as money, a screwdriver or a car share important characteristics with brute facts, since in general any given five euro note is money and any given Renault is a car, independent from one’s particular opinion on the matter. Searle calls this second class institutional facts. The point is that in contrast to brute facts the latter do not exist unless they are constructed by some sort of process involving human interaction. And yet, they are normally considered to be real. It is true that a five euro note is money, even if without humans the same object would cease to be money.

A whole branch of sociology has committed itself to analyse the process during which social reality is constructed. The illuminating contribution of Berger & Luckmann (1966), building on the classic theories of Marx, Nietzsche, Durckheim and Weber, showed that social objects we commonly consider to be real and true are in fact constructed through conversations between humans, who are a priori capable of assigning a completely different meaning to the same object. The tremendous implications of the theory of the construction of social reality can best be understood by contrasting it with the Platonian view according to which ‘ideas’ such as truth, beauty and inequality exist independently from human interference. Putnam (1981) summarised the idea that there exists no single true description of the world by coining the phrase “there is no God’s eye point of view”. From this observation it is not far to see the main argument of what is referred to as the ‘relativist’ position in epistemology, according to which any opinion on how things are is just as true as any other. We immediately feel, however, that there must exist some very strong arguments against the radical relativist viewpoint, otherwise sensible authors such as Noble Prize Winner Amartya Sen would not dedicate much of their time to the question of how we can measure the true extent of inequalities in the world. We have already mentioned Searle’s ‘institutional facts’, and it turns out that it is reasonable to believe that for many social objects we are fairly limited in the way we decide what is real and what is not real.

As Putnam (ibid.) clearly points out, the theory that social reality is constructed does not lead to the relativist attitude of ‘anything goes’ since this construction is constrained by

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several factors: 1) our experience tells us that certain things are true and others are not.

It is impossible to believe that the fact ‘humans can jump out of the window and fly’ is true (at least not after trying); and 2) our view of what we regard as true depends on the complex structure of conceptions that we carry with us at all times. According to Putnam, what makes a phrase or a theory rationally acceptable is its adequation and coherence with internal and mutual beliefs which can be “theoretic” and “experimental”.

While this reduces the space of possible descriptions of reality somewhat, the puzzle that different people may have different conceptions remains, and the problem of relativism is hence not really solved. The classic exit from this impasse put forward by sociologists lies in the processes that construct the social objects we believe to be true. Analogue to a language which is shared by the people that rely on it to communicate, humans tend to form communities in which — given a certain shared context, culture and beliefs — it is again possible to speak of ‘objectivity’, since all speakers use a similar set of conceptions (Prieto, 1975; Putnam, ibid.).

This brief overview on the theory of the social construction of reality aimed to clarify the nature of the concept ‘inequality’. It is obvious that the social sciences — as opposed to the natural sciences — do not deal with what we have called ‘brute facts’, but with aspects of social reality, whose elements are not objective per se but constructed through a process of objectivation. Due to the nature of social reality, the process of ‘objectivation’ is carried out by social groups of some form or another and not by isolated individuals.

The concept of inequality that we are dealing with clearly is one that enables us to conceive social facts. It is, as we have seen, necessarily part of the constructed social reality. However, we generally refer to these concepts as objects when we discuss the state of economic well-being. An example might help to illustrate this point. In a debate on the effects of accelerating GNP growth in an emerging country, say Brazil, an advocate of the market might make the following statement: “The increase of GNP ultimately makes all Brazilians better off, I therefore anticipate that inequalities in this country will decrease.”

A possible counterargument could sound like this: “The economic growth of the Brazilian economy tends to be concentrated in sectors that are not accessible to the rural population or those who live in the favelas around the urban agglomerations. Since these people are excluded from the increasing standard of living, inequalities will increase.” In this example both sides have a different opinion about the effect of growth on inequalities. Nevertheless, they both use the term ‘inequalities’ as if it was referring to an object whose nature itself is not discussed. If the discussion was set in a federal ministry, the two sides might try to prove the correctness of their reasoning by pointing at the evolution of some statistical series reflecting a decrease or an increase of inequalities. This, too, is only possible if both sides treat inequality measures as valid: the statistics have become autonomous from the process of objectivation.

The fact that we tend to treat social constructs as objects is of course vital for any communication. If we considered all social objects simultaneously as constructs, we could not engage in any sensible argument with anyone else. And yet, A. Desrosières (1993) has shown in his history of statistical reason how highly relevant it is to engage in this activity he calls “discuter l’indiscutable”, and we wish to apply some of his lessons to the case of the

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measurement of inequalities. By linking the histories of the State, statistics and economic thought, Desrosières illustrates how some of the ‘objective’ references such as the annually published statistics are in fact the result of conventions.

What is a convention? The term forms the centrepiece of the so-called Economic theory of conventions which holds that conventions are an alternative mechanism of coordination and decision-making in economic situations. In a classic article, Favereau defined conventions as a “dispositif cognitif collectif ” (1989, p. 295) and underlined the functional character of conventions. For our purpose it is not necessary to go beyond the surface of the theory of conventions: the concept of a “dispositif cognitif collectif ” does not require a complex theoretical underpinning and we will use the term ‘convention’ only to underline the general idea of social construction of empirical measures.3 However, we should mention that a convention typically contains an arbitrary element (i.e. the choice between equally valid alternatives), and a more intentional element. Sometimes the latter becomes more visible, for instance when the legitimacy of a particular convention is contested. Gadrey & Jany-Catrice (2007) provided a discussion of conventions in a context close to our problem.

They analyse the different approaches to measurement of economic well-being (among others they discuss the IEWB) and distinguish between two types of conventions. On the one hand, there are statistical conventions resulting from the necessity to make choices between different nomenclatures, data treatment methods, evaluation methods etc. On the other hand, the authors identify conventions that directly relate to wealth and well-being. The latter group of conventions “concernent la représentation globale de ce qui compte et de ce qui devrait être compté au titre de la richesse d’une nation, et de la contribution de diverses activités ou patrimoines” (p. 103). According to Gadrey and Jany-Catrice both types of conventions are interrelated. In an interesting application of the concept of conventions to the current debate on economic well-being, they argue that it is the second, non-technical set of conventions that has lost some of its legitimacy since more and more actors step up to question whether the conventions currently in use actually represent what these actors consider to be wealth, progress or well-being. By doing so, they reverse the process of ‘objectivation’ of the traditional methods (e.g. judgements about well-being based on GNP growth) and attempt to replace them with new and in their view more legitimate ones (e.g. judgements about well-being based on the IEWB). In the context of our problem of analysing inequalities within the Index of Economic Well-Being, the conventions relative to the question of what should count are mostly already answered by the choice of dimensions. We are therefore more concerned with the technical conventions and whether they are legitimate and coherent with respect to the overall representation of well-being in the Index. In other words, in this text we want to analyse how inequalities can be empirically measured given the constraint that the technical conventions should reflect the representations of inequalities held by its users.

The concept of conventions necessarily shifts the focus away from scientific ‘truth’ and underlines strikingly clear the differences between the invariants in natural sciences and 3

For our purpose it is also not relevant that the term ‘convention’ tends to be used with a slightly different meaning in sociology and economics.

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the arbitrary elements of conventions and cognitive constructs in social sciences. However, Desrosières in his analysis goes much further than simply pointing at the arbitrary character of the conventions involved in the process of ‘objectivation’ which transforms the social facts into ‘real’ objects. In his history of the raison statistique, he lays out some of the ingredients that have the power — given specified context, culture and beliefs — to turn certain concepts (such as averages, probabilities, unemployment or national accounts) and their empirical expressions into the references that consequently become indiscutable. Since we are ultimately interested in constructing an inequality dimension in the IEWB that rests on a solid basis, we can use some of Desrosières lessons. We will rely on his approach in two ways, one related to the method he employs, and the other to some of his results.

First, in contrast to most other accounts on inequality measures, a considerable part of this text will be in form of a chronological account. Just as in Desrosières’ work, this approach allows to trace the moments in which conventions have been modified or new ones appeared. Highlighting the turning points in their historical order is arguably not only didactically useful. It also allows to show why and when important conventional choices were made and, ultimately, whether these choices are legitimate and coherent in light of the purpose of the IEWB. However, the account will take a different form as regards at least one crucial point: Desrosières, a trained sociologist and civil servant for the French statistical authorities, could rely on his extensive experience to sketch what he calls an internal and an external history of statistical reason. The former refers to the history of the knowledge itself, together with the instruments, results, theorems and demonstrations.

The latter, linked to the work of the French sociologist Michel Callon, analyses the practical operations involved in the scientific process such as the laboratories, their financing, the scientific careers, and the networks of actors. Being aware of the tremendous knowledge required for an external history of the measurement of inequalities, we restrict ourselves to the hope of providing some insights into its internal history. Some elements of the interplay between internal and external actors in the context of inequality measurement can be found in Desrosières (1993) and Nivière (2005), although their examples refer mostly to the case of poverty statistics. A more complete discussion of external factors on the analysis of inequality can be found in Jenkins & Micklewright (2007). In their summary of recent developments in this field they include an account of the major changes in the policy environment in both industrialised and developing countries. Although more indirectly, Jenkins and Micklewright’s text is also revealing in terms of personal trajectories of some of the involved actors, especially the one of Atkinson. The latter’s activity had an enormous impact not only on the internal history of inequality analysis, but through his numerous collaborations and research supervisions also on the external side (ibid, p.20). While we will not attempt to trace the external history of inequality measurement, we will leave the internal discourse in Chapter 3 and confront it with the requirements of the IEWB, whose users are thought to be economists and external actors.

Second, the analysis will rely on some of Desrosières’ results in a way that was probably not intended by their author. In fact, and perhaps due to his methodological background as sociologist, Desrosières mainly describes the transformation of social facts into ‘real’ objects. By contrast, this text will adopt a more normative point of view and re-interpret

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some of his results for the purpose of argumentation: if, as Desrosières has shown, the legitimacy of statistical measures relies on the coherence between the scientific and non-scientific spheres, a measure of inequality should take non-scientific conceptions of inequalities into account. This leads to the second important issue related to inequality measurement, namely the importance of the relationship between the scientific discourse and the use of the concept by other actors.

The relationship between the scientific discourse on inequalities and ‘normal communication’ When talking about inequalities, one has to clarify the position with respect to a vast body of epistemological questions. We have chosen our camp by accepting the importance of conventions and the idea that the concept inequality is a social construction, whose objectivity relies in fine on processes of inter-subject communication. Not taking this into account leads necessarily to the danger of acting as if the object at hand did not belong to the canon of the social sciences. The second issue which appears to be another corner stone of any sensible discussion on the measurement of inequalities is the unavoidable and important relation between the scientific discourse and the use of the concept in normal communication.

In the literature on economic inequality, Sen (1975) is probably most aware of this issue: “In a trivial sense it is, of course, the case that one can define ‘inequality’ precisely as one likes, and as long as one is explicit and consistent one may think one is above criticism.

But the force of the expression ‘inequality’, and indeed our interest in the concept, derive from the meaning that is associated with the term, and we are not really free to define it purely arbitrarily” (pp. 47-78). An example of this problem frequently appears with approaches that define income inequality not in terms of income, but in terms of another space like welfare or utility. As Sen (1997) has shown, it is possible that the same alteration of an income distribution can yield simultaneously decreasing utility inequality, unchanging income inequality and increasing inequality as evaluated by an Atkinson-type index (cf. our discussion on p. 118). The meaning of the term inequality varies from one approach to the next, and some authors seem to be less concerned with this concept-stretching than others.

We submit that Sen’s qualification “not really free” could be worded much stronger when one scrutinizes the origin of the “meaning” that Sen has in mind. The meaning of the term ‘inequality’ is, as has been seen above, not something that can be proven by an isolated individual without any reference to the common body of social constructs. In order to make sense, and in fact for any concept in economics to make sense, this ‘meaning’ has to relate to the representations and conceptions of the users of the particular measurement instrument, in our case the potential users of the IEWB. When discussing inequality, economists frequently justify the relevance of their work with its usefulness as some sort of policy instrument (evaluation, decision making, advice etc.). Now, both relevance and usefulness of academic inequality measurement are severely limited if the academic meaning deviates from the one held collectively by the users of the policy instruments like public

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administration staff, elected officials and others. Obviously, due to its character as social construct, it is impossible that two actors at any point in time will attach an identical meaning to the term inequality. This is a result obtained by Rogers & Kincaid (1981) in the context of communication theory and based on the imperfect and indefinite character of the language we have to rely on in normal communication. But, according to the same authors, it is possible for the meaning to converge during the conversation via continuous feedback loops. In other words, concepts are co-constructed trough communication.

This is the point where some of Desrosières’ results are re-interpreted. Taking his descriptive account of the history of statistics as an argument, it can be argued that it should matter who participates in the co-construction of the concept and measurement of inequality. A technical monologue held exclusively in scientific language can at best win a ‘pseudo-legitimacy’. To gain full legitimacy, it is necessary to verify whether the ideas embedded in characteristics like “first, second and third order stochastic dominance” or “partial quasi-orderings” effectively correspond to the representations of the users of those inequality statistics based on these technical constructs. The process of ‘objectivation’ should be a co-construction, and not a monologue-like construction carried out exclusively by technical specialists. Desrosières’ descriptions of the link between the scientific representations and other linguistic spaces seem to point in this direction: “Fondant son originalité sur son autonomie par rapport à d’autres langues, religieux, juridique, philosophique ou politique, le langage scientifique a une relation contradictoire avec ces derniers. D’une part, il revendique une objectivité et, par là, une universalité qui, en cas de réussite de cette revendication, fournissant des points d’appui et des référents communs aux débats des autres espaces: c’est l’aspect ‘science incontestable’. Mais cette autorité, qui trouve sa justification dans le processus d’objectivation lui-même et dans ses exigences strictes d’universalité, ne peut s’exercer que pour autant qu’elle participe à l’univers de l’action, de la décision, de la transformation du monde.” (ibid., p. 14) “L’espace de représentativité des descriptions statistiques n’est rendu possible que par un espace de représentations mentales communes, portées par un langage commun, balisé notamment par l’État et par le droit.” (ibid., p. 397) While in our case the State and the law referred to in the last sentence should probably be replaced with ‘the different users of the IEWB’, the general message of these quotations is the following: without interaction and a reasonable degree of semantic coherence, the scientific output on inequality measures can neither serve as common reference for public debate nor have representative value of social facts. If a measure of inequality is not semantically coherent with the representations of its users, one should speak of ‘pseudolegitimacy’ which could result from the absence of an efficient dialogue between technical specialists and users. In contrast to the perhaps more intuitive concept of poverty, the technical complexities involved in inequality metrics may create obstacles for an efficient co-construction of empirical measurement of inequalities. However, in the absence of an

1.3. SOME BASIC TERMINOLOGY

13

efficient co-construction the meaning attached to the term inequality might not converge (in the sense of Rogers & Kincaid mentioned above) — it might even diverge. In the latter case the academic output is stripped of its relevance and, for that matter, also of its scientificness (Wissenschaftlichkeit).

1.3

Some basic terminology

After what has been said in the preceding section, it would be somewhat incoherent to start our discussion with a clear-cut definition of inequality. We argued that the concept is constructed over time and by different actors, and it is precisely this process which will be the object of our analysis in Chapter 2. However, some preliminary remarks about terms related to the concept of inequality may be useful.

Although the literature makes frequent use of analogies and equivalences corresponding to the term ‘inequality’, a distinction between related, but nevertheless distinct concepts should be made. As a matter of fact, inequality has been expressed in terms of concepts like ‘concentration’, ‘diffusion’, ‘dispersion’, ‘entropy’, ‘variation’, ‘range’, and many others.

While inequality is obviously related to these concepts, it has nevertheless an independent semantic content and is thus not identical to concentration nor to dispersion. Our approach of examining inequality as a convention allows to see the difference between “inequality as concentration” and “inequality as difference”, or “inequality as dispersion”. Thinking of inequality as identical to any of these alternative concepts clearly misses important elements of the debate on inequality measurement.

Next, we should be cautious never to confuse the related, but nevertheless distinct notions of inequality and poverty. In public debate, poverty and inequality tend to be used as an almost inseparable pair. This is due to the fact that both are conventionally placed within a wider category of socio-economic problems in which we could also include the theme of social cohesion. However, it should be borne in mind that inequality refers to questions regarding different parts of the distribution of economic assets, while poverty is concerned with the fate of those at the lower end of the distribution. The distinction between poverty and inequality is of course clearer if we stick to a concept of absolute — as opposed to relative — poverty. With absolute poverty, a poor is thought to be poor regardless of the socio-economic position of other individuals. In this case, it is possible to imagine a population with inequality but without poverty, and vice versa. On the other hand, if poverty is defined as something related to the position of others, there can be no poverty without at least some inequality. In recent decades, the measurement of poverty has tended toward a more relative approach and thus made the distinction between inequality and poverty somewhat less clear. However, as we proposed for concepts like concentration, we should not think of the two as being identical.

The last item of terminology refers to the different types of descriptive devices which can be found in the literature. It is useful to divide these alternative ways of representing empirical inequalities into two categories, each of which has special implications. Rosenbluth has divided the descriptive devices into the following two types (1951, p. 935):

14

CHAPTER 1. INTRODUCTION AND METHODOLOGY 1. A table or chart by means of which different parts of a distribution may be compared.

2. An over-all index for comparing different distributions as a whole.

The best-known examples of the first type is the Lorenz curve or the cumulative frequency distribution. The Gini concentration ratio is the most frequently used exponent of the second type of descriptive device. Each type has advantages and differences. According to Rosenbluth, it “can be said of any summary measure, such as an index number, average, or higher moment of the frequency distribution, that there is an infinity of changes in the data to which the measure does not respond” (ibid., p. 936). The use of any onefor-all summary measure therefore implies disregarding certain variations in the income distribution that are judged to be negligible. On the other hand, a chart depicting the entire income distribution is more responsive to almost all changes in the distribution.

However, it is often hard to draw a conclusion on the overall development of inequalities on the basis of a ‘type one’ device. This is why both methods of representing empirical distributions have their merit.

Chapter 2 An internal history of the academic discourse since 1895 The literature on inequality measurement in economics is a vast field. Even the more restricted subject of income and wealth inequality has probably grown beyond the possibility of coherent synthesis. An illustration of the sheer quantity of key readings is the length of the bibliography in Sen’s “On Economic Inequality” which stretches over 31 (!) pages.

Mastering the relevant literature clearly is the work of a lifetime and the present author is aware of his limitations in this respect.

Since it has been argued in the first chapter that a chronological perspective on the internal academic discourse might be useful to highlight the conventions involved in inequality measurement, the tough choice of selection arises. However, this choice is less difficult than it appears at first sight. The approach of analysing inequality as a convention naturally leads to selecting contributions according to their impact on conventions.

And, as can easily be verified, few contributions in the field do not refer explicitly to Pareto distributions, the Lorenz curve and the Gini ratio. These devices have become common knowledge and are arguably among the key descriptive instruments in inequality measurement. We therefore decided to have a closer look at the argumentations put forward by Pareto, Lorenz and Gini before these measures became conventional and apparently legitimate representations of inequalities. A second set of contributions with strong influence on the way inequality is apprehended in economics consists of the welfare-based statistics developed by Dalton, Atkinson and Sen. As a matter of fact, it would be difficult to find an article on the welfare implications of inequality which does not draw on the ideas of at least one of these three authors. Finally, it would be very restrictive to ignore Theil’s impact on the internal discussion on inequality statistics. Not only his own measure has become a frequently used tool, but also the general theme of ‘decomposability’ continues to have a significant impact on the scientific literature. We felt it to be preferable to discuss these seven measures in some depth — without any illusion that we have come at any point close to comprehensive accounts of all relevant aspects — than to include other important authors like Shorrocks, Bourguignon, Anand or Foster.

For each of the seven authors discussed separately in this chapter we have tried to 15

16

CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

acquire as broad an overview as possible on their respective articles and books. While this is a relatively uncomplicated undertaking for Pareto1 , Theil, Atkinson and Sen, it is more difficult to access original texts by Dalton and Gini. The discussion of Dalton’s measure is therefore almost exclusively based on his article from 1920 published by the Royal Economic Society. Some of Gini’s texts are to the present day only available in Italian language and difficult to obtain. It seems that the Internet Age has not yet overcome the remoteness from English-speaking circles of Gini’s “Variabilità e Mutabilità” that Dalton noted back in 1920.

In short, the added value of the internal history of the discourse on economic inequality below is neither completeness nor technical discussion. Its purpose is to emphasise the relationship between theoretical definitions, empirical representations and their impact on conventions. Given the prominent role of the measures we discuss, this approach will allow to develop a critical stance as to the legitimacy of inequality measurement in the context of the IEWB.

2.1 2.1.1

From constant inequality to complex inequalities Pareto’s Law: constant or decreasing inequality?

Vilfredo Pareto (*1848, †1923) is well known as a precursor of the quantitative analysis of income distributions. According to our knowledge, Pareto is the inventor of the first quantitative assessment of inequalities. Yet, the fact that he also proposed an inequality statistic which he derived independently from the famous ‘Pareto Law’ is hardly ever discussed: Pareto’s inequality measure is all but absent from today’s debates on inequality. This may be due to his decision to combine his measure of inequality and his “loi de la répartion de la richesse”. In fact, Pareto combined the two ideas and showed how inequalities could be measured in terms of this law. Consequently, his measure of inequality was discredited as soon as the Law had become subject of controversy. We are primarily interested in the former, i.e. in the way Pareto defined inequalities and how he proposed to measure them empirically, independently from his law. To separate the two issues, it is useful to first briefly discuss Pareto’s Law and afterwards analyse how the measure of inequality fits into this framework.

The discovery of constants in the income distribution Before Pareto used inductive methods to identify general patterns in income distributions, classical economics focused almost exclusively on the question of production. If inequality was treated, it was in terms of categorical differences as in Marx’ focus on class distinctions.

Probably the only quantitative study on the distribution of incomes prior to Pareto’s first 1

We could draw on the compilation in French of the Pareto’s writings on income distribution by Giovanni Busino from 1965.

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

17

article on the topic in 1895 was Otto Ammon’s Die Gesellschaftsordung und ihre natürlichen Grundlagen (Jena, 1895), which Pareto had read.

Pareto repeatedly expressed his preference for political economics as ‘hard science’ (cf. his Cours, published in 1896), and therefore treated the income distribution as a quantitative phenomenon. He was inspired by the French liberal Paul Leroy-Beaulieu, who wrote fourteen years before Pareto’s first article on income distribution and fourteen years after Marx published his Kapital : “L’influence des lois économiques sur la répartition des richesses est un sujet beaucoup moins exploré que l’influence des mêmes lois sur la circulation. [...] Sans doute les volumes sur ce qu’on appelle les questions ouvrières abondent, mais la plupart sont absolument vides, sans rien de précis, de positif et de scientifique.” (quoted in Busino, 1964) The essence of Pareto’s Law is simple. After having observed a strikingly similar distribution pattern in all his data sets, Pareto proposes the following formula supposedly valid at all times and in all places: log N≥y = log A − α log y

(2.1)

Where “y is an amount of [individual] income, N≥y is the number of persons in receipt of that or a higher income, A and α are constants, the former varying with the total number of incomes considered, the latter a constant indeed since it proves to be nearly the same for different countries, about 1.5” (Edgeworth, 1926, pp. 712-713; annotation harmonised with our text).

The parameter α in this formula is referred to as the ‘Pareto-coefficient’ (we will come back to its interpretation in terms of inequalities below). The Pareto-coefficient is the slope of a straight line linking the logarithms of y and N≥y and lies according to Pareto’s empirical evidence around 1.5 in all stable economies. Hans Staehle, writing in 1942, summarised Pareto’s enthusiastic reaction to the apparent constancy of this coefficient as follows: “In 1895, Pareto presented his discovery as ‘a simple empirical law’; in 1896, he spoke of it as a ‘loi naturelle’ ; and in 1897, he made it the main basis for the third book of his Cours” (ibid, p.78). Hence, Pareto decided to raise his statistical observation to the rank of a natural law: the apparent constancy of α could not be the result of chance, and therefore a law must govern the shape of distributions.

A concept of inequalities based on relative poverty Until this point, only Pareto’s discovery of an apparent constancy of the slope parameter α have been presented. This is the centrepiece of Pareto’s discussion of income distribution.

His interpretation of higher and lower values of this parameter has been discussed, nor his view on inequalities. Pareto himself hesitated several years to discuss the matter. In a paper published two years after his first article on income distribution, he still refused to give any definition of the notion ‘decrease of inequalities’: “Il vaut mieux éviter ce terme ambigu” was his crisp statement in 1897. Yet, in the second volume of his Cours, he eventually gives in and asks himself: ‘‘Mais quelle est la vraie signification des termes : moindre inégalités des revenus [...] ?’ ’(ibid, p. 318).

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CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

In light of his postulate of a ‘natural law’, Pareto walks on shaky ground when he discusses changes in inequalities. Still, he makes his diagnosis clear: “Actuellement, dans nos sociétés, il parait bien que c’est ce dernier cas [a decrease in inequalities] qui se vérifie, et un grand nombre d’observations nous font connaitre que le bien-être du peuple s’est, en général, accru dans les pays civilisés” (ibid, p. 323). This is of course somewhat inconsistent with the idea of stable distribution shapes, as will be seen below in more detail.

Pareto’s definition is again influenced by Leroy-Beaulieu’s ideas, who proposed a concept of relative poverty and a somewhat blurry notion of social progress: “Les progrés du bien-être de la classe inférieure de la population sont [...] plus rapides que ceux de la classe moyenne et de la classe élevée. Sans arriver à un nivellement des conditions qui est impossible [...] le mouvement économique actuel conduit à une moindre inégalité entre les fortunes."

And Pareto adds: "La diminution de cette inégalité sera donc définie par le fait que le nombre de pauvres va en diminuant par rapport au nombre des riches. [...] En général, lorsque le nombre des personnes ayant un revenu inférieur à x augmente par rapport au nombre des personnes ayant un revenu supérieur à x, nous dirons que l’inégalité des revenus augmente.” (ibid., p. 320) To see the implications of this definition, Pareto defines an inequality measure in mathematical form. Keeping the notation of equation (2.1) with the total population N , a decrease of inequality occurs when the following expression increases: uy =

N≥y N

(2.2)

To examine inequalities, it is thus necessary to evaluate uy at all levels of income. It is important to notice that the definition of inequality and the statistic in equation (2.2) are derived independently from Pareto’s Law. Whether or not the latter holds empirically does therefore not affect the validity of the former. However, being a true believer in the constancy of the observed regularities, Pareto proceeds to combine the two ideas and showed how his law (as expressed by equation (2.1) above) fits into the definition of inequality and formulates the following property: A higher value for the coefficient α indicates higher inequalities, and vice versa.

Instead of having to apply equation (2.2), this property allows to take a short cut and look directly at the coefficient α to see whether inequalities increased or decreased. To prove this relation between α and uy , Pareto defines h as the minimum income in the data set (which is not to be confused with any kind of legal minimum wage). It follows that N≥h is equal to the total population N and that the measure of inequality (2.2) is confined in the interval [0, 1]. The lowest value is attained at y = k, the maximum income, and the highest value at y = h. Combining equations (2.2) and (2.1), Pareto shows that: N≥y = uy = N≥h

A yα A hα

α h = y

(2.3)

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

19

From equation (2.3) it can be seen that if a distribution is described by Pareto’s Law, then inequalities are lower for all levels of y if the value of α is higher (since it is assumed that y > 1 and α > 0). A change in inequality would thus be an obvious departure from the idea of a fixed value of α. An example will illustrate the relationship between the coefficient α and the measure of inequalities we just derived. Fig. 2.1 below compares three distributions D1 , D2 and D3 that all satisfy Pareto’s Law. They all have the same minimum income (we have chosen h = 1, so that log h = 0), the same maximum income, and the same parameter A. Hence, they differ only with respect to the slope coefficient α. The distribution D1 is characterised by α1 = 1.5, D2 has α2 = 0.7, and D3 α3 = 0.05.

We can see from Fig. 2.1 that for any level of y above one, N≥y is highest for D3 , second log y D1

D2

D3

log N≥y Figure 2.1: Pareto curves for different levels of the coefficient α. highest for D2 , and lowest for D1 . Since N is the same for all three distributions, it follows that: uy (D1 ) < uy (D2 ) < uy (D3 )

∀y > 0

This result is of course due to the linear relation between the logarithms of y and N≥y postulated by Pareto’s Law. In general, we can say that for two distributions Du (αu ) and De (αe ) it is true that: uy (Du ) ≤ uy (De )

if

αu ≥ αe

(2.4)

This means that the distribution Du is at least as unequal as De for all levels of y. If distributions can be described by Pareto’s Law, the slope coefficient α is indeed a summary statistic for inequality defined in the sense of uy .

However, this result applies of course only to distributions for which equation (2.1) — Pareto’s Law — presents a reasonably good fit. In other cases, we might not observe a logarithmic distribution or anything close to Pareto’s Law, and then we obviously cannot use the coefficient α to judge whether inequalities increased or decreased. It is this problem that led Dalton (1920) to the conclusion that Pareto’s measure of inequality “evades any judgement” (ibid., p. 354) since it presupposes a unique determination of the distributional shape. However, it is important to see the relationship between three elements involved in Pareto’s argument: he argues that his measure of inequality as expressed in equation (2.2)

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CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

leads to the conclusion that the coefficient α indicates a rise or a fall in inequality, given that the distribution can be described by the equation (2.1). Clearly, this line of reasoning contains a contradiction if one interprets Pareto’s Law as a stable relationship similar to a natural law, which excludes by definition changes in inequalities. However, what Dalton probably missed is that equation (2.2) as a measure of inequalities can be applied even if a distribution displays a different shape, i.e. without insisting on Pareto’s Law. In other words, Pareto’s measure of inequality is not the coefficient α, but the measure uy . Only if Pareto’s Law holds we can use α as a short cut to evaluate uy .

Pareto’s definition and his measure of inequality were derived separately from his alleged law, although Pareto himself presented a summary measure of inequality in terms of this law. Equation (2.4) summarizes this relationship. However, in our analysis of the internal history of the academic discourse on inequality measurement, we are more interested in the alternative definitions and measures of inequalities. Distribution theory (i.e. questions of how shapes of distributions can be described or explained) is less important for our purpose. Therefore, we will now discuss some properties of Pareto’s measure of inequality that are true independent from his Law. Three points are of interest for us: First, the definition of inequality and the empirical instrument of equation (2.2) Pareto proposed cannot be used directly as a summary measure of inequality: this is why he has to take a detour via his Law to obtain a summary measure. Referring to the terminology introduced in Section 1.3, Pareto is the first to base a ‘type two’ measure (his α) on a ‘type one’ measure (uy at different levels of y). The statistic uy gives an impression of the extent of inequalities at different points in the distribution. A change in the shape of the distribution might lead to increasing inequality at some levels of income, and decreasing inequality for others. An unambiguous answer to the question whether inequalities diminished or increased over a certain period can only be given if uy rises or falls for all levels of y.

Second, if one accepts the definition of equation (2.2), inequality is sensitive to changes in the average income: both equal and proportionate additions to incomes result in less inequality. We speak of equal additions to incomes when all incomes are raised by the same amount. A proportionate addition raises all incomes but leaves their relative share in the total income unchanged. A simple example will illustrate this property — in the literature referred to as ‘mean sensitivity’ or ‘mean dependence’ — in the case of Pareto’s measure of inequality. Imagine the income distribution DA = (1, 2, 3, 4, 5, 6) among six individuals.

We now add two money units to each income. The new distribution will be called DA0 , and its values are (3, 4, 5, 6, 7, 8). Next, we transform the distribution DA by multiplying each individual’s income by the factor 4 to arrive at the distribution DA00 = (4, 8, 12, 16, 20, 24).

Obviously, the distribution DA0 corresponds to the case of equal additions to DA , and DA00 to proportionate additions to DA . Fig. 2.2 graphs Pareto’s inequality measure for the three distributions DA , DA0 and DA00 for the relevant range of income levels.

Without being a formal proof, it can be seen immediately from Fig. 2.2 that DA0 and DA00 have equal or higher levels of uy than DA for all incomes. Hence, according to Pareto’s definition of inequalities, (strictly positive) equal or proportionate additions to incomes decrease inequality for at least some income levels. This illustrates that uy is not mean independent.

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

21

uy 1

uy (DA0 )

uy (DA00 )

uy (DA )

y

Figure 2.2: Pareto’s inequality measure for different income distributions.

Although Pareto is not entirely clear on this point2 , the analysis should not be extrapolated to comparisons across different income distributions, but only to variations of the same distribution. In fact, if a distribution has a higher minimum income than the maximum income of another distribution, it is not necessarily more unequal. Imagine, for example, the distributions DA00 with the higher mean represents the richest country in the world, say Luxembourg, and the distributions DA with the lower mean corresponds to a small island in the Caribbean with an economy based on barter. In this case, we could not immediately say which of the two distributions is more equal, even if uy evaluated for Luxembourg would probably be higher for all levels of y than the uy of the Caribbean island.

The notions of ‘the rich’ and ‘the poor’ referred to in Pareto’s definition of inequality make more sense for a given population than for cross-country comparisons. It does not contain the notions of ‘absolute poor’ or ‘absolute rich’ and Pareto clearly stresses the difference between pauperism and inequality in his Cours. The mean independence should therefore be interpreted only for a given population.

Third, Pareto’s definition can be interpreted as making a distinction between concentration and inequality. The author reasons in terms of relative numbers of rich and poor, and not in terms of income shares. Unlike later approaches to inequality, the aggregate income does not enter the picture. Instead of income concentration, the measure is similar to what is referred to as relative poverty in today’s literature on deprivation and social exclusion. The standard formula used today to calculate relative poverty K is: K≡

N<P L N

where N is the total population, and N<P L is the number of people below the poverty 2

In fact, he provides an illustration in his Cours in which two populations with very distinct shapes are compared (ibid., p. 318). His judgement on inequalities in this case leads us to think that Pareto did not apply his measure of inequalities for comparison across different distributions, but only to gauge inequalities of any particular distributions. In the context of the measure presented in the text, Pareto does not speak of higher or lower inequalities, but only whether they increase or decrease.

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CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

line P L. This measure is relative because the poverty line is conventionally defined as a proportion of the median income and therefore reflects poverty relative to the most frequently observed income in the distribution. Pareto’s measure of inequality is equivalent to a relative poverty rate evaluated not only at the poverty line, but at all income levels. In fact, the relative poverty rate is simply a particular point of Pareto’s inequality measure, N L . The intuition for this measure of namely y = P L so that K = 1 − uP L = 1 − ≥P N inequality is therefore based on relative poverty and not on concentration.

Impact on conventions The reason why we are interested in Pareto’s measure is that his work was one of the first to analyse income inequalities with quantitative methods. In many ways he influenced the early research on income distribution and initiated several lines of inquiry. Consequently, much could be said about the impact of his ‘discovery’ of constants in income distribution.

Since we are more interested in the impact of his measure of inequality — and less in his contribution to distribution theory in general — we will try to be as brief as possible as regards the reception of his Law. It is interesting to note that several decades after Pareto first presented the hypothesis of a constant slope many authors still adhered to his propositions. 35 years after Pareto’s original article Davis (1941) writes: “No one, however, has yet exhibited a stable social order, ancient or modern, which has not followed the Pareto pattern at least approximately” (ibid, p. 395).3 Still in the late 1960’s some authors like Aigner and Heins felt it necessary to remark that “our results suggest that the Pareto notion of a fixed α (1.50) should be re-evaluated” (1967, p. 16).

Despite the fact that Pareto’s Law still attracted sporadic support until the second half of the last century, it had soon become subject of critique for various reasons. Edgeworth had criticized that Pareto’s specification “does not fit the phenomena at its lower extremity”. This means that the functional form specified in Pareto’s Law fitted later empirical data only for incomes above a certain threshold and not for the entire distribution. In later applications, the range over which equation (2.1) holds was therefore open to discussion and a serious defect for a ‘natural law’. In 1933, Yntema presented a comparison of several inequality indicators with respect to their performance in empirical applications.

He qualified Pareto’s coefficient as “unstable” (and hence not constant as Pareto had assumed) and “insensitive” (since other measures capture more of the differences between alternative distributions). In 1936, Gini proposed an alternative specification for income distributions with a more sensitive slope coefficient and showed that the range of 1.1 to 1.9 Pareto had found for his coefficient actually presented a considerable difference in terms of concentration.4 3

Davis went further than Pareto in his conclusions. He argued that attempts to move away from the natural values embodied in Pareto’s Law would create inevitably economic and political distortions, and interpreted the French Revolution, the Spanish Civil War or the weakness of the French military in the wake of the German invasion during WWII as deviations from the natural level of α (ibid., p. 435).

4 For a discussion of the reception of Pareto’s Law until the 1940’s, see Bowman (1945).

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

23

The fiercer the stability of his Law was questioned, the more Pareto’s inductive framework became inconvenient for research on inequalities. Contrary to Pareto’s quest for constants, the comparison between the degrees of inequality across distributions and the ensuing welfare implications became soon the main research questions in this field (Pigou (1912, 1920), Dalton (1920)). Since Pareto’s theory was “based upon a supposed law, according to which, if the total income and the number of income-receivers are known, the distribution is uniquely determined” (Dalton, 1920, p. 354), the analysis of differing shapes and their determinants called for other tools, which were soon found in the works of Max O. Lorenz and Corrado Gini (see Sections 2.1.2 and 2.1.3). When we add to this the contradiction between Pareto’s belief in constant distribution patterns and his observation of diminishing inequalities, it becomes clearer why his measure of inequalities uy all but disappeared from the academic discourse. However, we should keep in mind that Pareto derived his definition of inequalities independently from his analysis of distributions and on the grounds of a notion of relative poverty. It had and has therefore a right on its own and dismissing his measure of inequality due to the controversy around his ‘law’ means throwing out the baby with the bath water. This seems to be was has happened, since his intuition to imagine inequalities as a ‘poverty rate’ evaluated at all levels of income was not directly criticized – only the fact that his coefficient α failed to reflect inequalities according to some other definition was reproached.

Even if his inequality measure uy as such is not used any more, Pareto influenced directly or indirectly the state-of-the-art measurement of inequality until the present day.

His heritage includes: 1. The use of quantitative methods to analyse inequality. Pareto wanted the measurement in this field to be ‘scientific’, and he was a pioneer in replacing a purely qualitative analysis (e.g. inequalities defined as socio-economico-political positions of classes within the overall system) with a quantitative measurement based on empirical distributions. This was and is of course not the only practised approach, but it nevertheless appears to be the dominant one in economics until today.

2. By linking his measure of inequality to his Law, Pareto led the way by identifying a summary measure of inequality. He showed that the coefficient α suffices to say whether inequalities decreased or increased. He pioneered the method of defining an ‘index’ or ‘summary measure’ that synthesises available information on inequalities into a single number. This has become standard practice and has not been seriously questioned until Sen’s critique of complete orderings (see Section 2.1.7 on page 55).

These two points are relevant for our problem of measuring inequality in the context of the Index of Economic Well-Being. We will have to employ quantitative methods since the form of an index is hardly possible in qualitative terms. Pareto’s approach was inspired by demands of writers like Leroy-Beaulieu for inequality measurement to be more “precise”, “positive” and “scientific”. Similarly, by choosing the way of quantification, the IEWB wants to contribute ‘hard facts’ to the debate on well-being, otherwise Osberg and Sharpe would have adopted the form of a literary account.

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CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

Since the IEWB works with a measure in index form — and not, for example, a purely graphical expression of well-being — the dimension ‘inequality’ will have to come in form of a summary measure. In this respect Pareto’s example makes aware of the caveats that this procedure may contain. We have seen that a summary measure risks to be decontextualised.

The summary measure α in the way Pareto uses it makes only sense if one accepts his definition of inequality. However, later authors (including Gini, 1915; Dalton, 1920; and Lorenz, 1905) judged this summary measure in light of their (respective) definitions. We argue that an intuitive and easily communicable measure is probably best suited to avoid the risk of misinterpretation and erroneous conclusions inherent in any quantitative statistic that enters the public debate. One has to go through some calculations to show that, under certain circumstances, the coefficient α could be understood as a summary measure. As will be seen below, the combination of Gini coefficient and Lorenz curve was a more efficient solution to this problem.

2.1.2

The Lorenz curve: a new focus on concentration

While Pareto’s measure of inequality we presented in the preceding section occupies a rather marginal place in today’s literature on inequality measurement, the heritage of M. O. Lorenz’ (*1876; †1959) famous article published in 1905 figures rather prominently.

Although the exact details of the computation of the Lorenz curve5 can attain an astonishing degree of complexity in real-world applications, the essence of it is simple: “The method is as follows: Plot along one axis cumulated per cents. of the population from poorest to richest, and along the other the per cent. of the total wealth held by these per cents. of the population” (Lorenz, 1905, p. 217).

If every individual receives an equal amount of income, this method yields a straight line from the origin to the point (1, 1). In all other cases it will be bent in the middle so that an area between this straight line and the empirical curve appears. Lorenz adds that “the rule of interpretation will be, as the bow is bent, concentration increases” (ibid, p.

217).

We will illustrate this method with a hypothetical example. Imagine two distributions of 100 e: D1 (6, 7, 8, 9, 10, 12, 12, 12, 12, 12) and D2 (4, 5, 6, 8, 8, 12, 12, 14, 15, 16). The cumulated percentages corresponding to each of these distributions are given in Table 2.1.

The associated Lorenz curves are illustrated in Fig. 2.3 in which the values between actual incomes have been interpolated. According to the interpretation Lorenz proposes, the distribution D1 is less concentrated than distribution D2 .

Lorenz’ graphical approach has immediate intuitive appeal and many advantages: the curves can be drawn and compared for populations differing in size and total income; the graphical interpretation is not distorted by the use of logarithms; through the proposed 5

The authorship of the graphical method presented in this section is commonly attributed to Lorenz and it is his article from 1905 that has become the standard reference in this context. However, Bowman (1945) indicates that other authors could claim to have invented the Lorenz-type presentations: “The same idea was introduced almost simultaneously by Gini, Chatelain, and Seailles” (ibid., p. 617). This is an irrelevant issue for our problem.

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

Cumulated % of population 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Distribution D1 Income Cumulated % of total income 0 0 6 0.06 7 0.13 8 0.21 9 0.3 10 0.4 12 0.52 12 0.64 12 0.76 12 0.88 12 1

25

Distribution D2 Income Cumulated % of total income 0 0 4 0.04 5 0.15 6 0.15 8 0.23 8 0.31 12 0.43 12 0.55 14 0.69 15 0.84 16 1

Table 2.1: Lorenz table for the distributions D1 and D2 . interpretation of the ‘bent’, the approach can easily be extended into a summary measure (although Lorenz himself does not provide one in his original article). These precious qualities contributed to the immense success of Lorenz’ approach and to the widespread use of Lorenz curves to the present day.

However, it is often ignored that this success is based on more than an eloquent graphical method. The deepest footprint Lorenz leaves on the path of inequality discourse is not the graph itself, but his exclusive focus on concentration that has since become conventional. The title of his article leaves no room for doubt: Lorenz is interested in a ‘method of measuring the concentration of wealth’. From the beginning, he uses the term ‘equality’ as the opposite of concentration, i.e. as synonym of ‘diffusion’. Lorenz is one of the first authors to propose a dichotomy between the extremes ‘equality on the one end, full concentration on the other’. Armed with this definition of inequality he easily dismisses almost all measures that have been discussed before him: the methods proposed by Wolf, Soetbeer, Holmes and Pareto all do not pass his test because they fail to equate concentration and inequality. We saw above that the Pareto defined inequalities in such a way that proportionate increases of all incomes should be registered as a decrease in inequalities.

However, since a proportionate increase leaves the relative shares in the total income of each individual unchanged, i.e. the degree of concentration is constant, Pareto’s measure is in contradiction with Lorenz’ definition of inequalities. Lorenz’ strong language indicates that he has no doubt that his focus on concentration is the only acceptable way to discuss inequalities: the other measures are prone to “error” and “fallacies” instead of being simply based on alternative definitions of inequalities that have to be refuted.

Implicitly, Lorenz assumes that when people speak about inequality they make no interpersonal comparisons in absolute terms like the one included in the following statement:

26

CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE % of income

% of income

1

1 Line of equality

Line of equality D3

D2 D1

D1 1 % of pop.

(a) Without intersection

1 % of pop. (b) With intersection

Figure 2.3: Lorenz curves.

‘The gap between the poorest beggar and the richest capitalist has increased from 100 e to 100000 e. Therefore inequality has increased.’ Since the concentration of the aggregate income might have remained unchanged in this situation, inequality did not necessarily increase according to Lorenz’ definition. Pareto’s definition also included a relative element (the number of incomes below y relative to the number of income above y), but since his measure is evaluated at different levels of income, it also contains an absolute element.

Lorenz, on the other hand, is probably the first to understand inequality as an entirely ‘relative’ concept and presents a measure that corresponds precisely to this idea.

Lorenz also notes the problem of ambiguous decisions in case of intersecting curves.

For concentration is unambiguously higher in one distribution than in another only if its Lorenz curve is always at least equal and at some point more bent than the curve of the competing distribution. This can easily be illustrated with a diagram. We keep the distribution D1 of our previous example and add a new division of the 100 e according to D3 (8, 8, 8, 9, 9, 9, 9, 10, 10, 20). The corresponding Lorenz curves are depicted in Fig. 2.3 above. In this case, it is hard to judge on the comparison of overall concentration. Lorenz argues to interpret this situation as a tendency towards equality in the lower half and a contrary tendency in the upper half.

Impact on conventions As we did for Pareto’s inequality statistic, we will now try to identify some features of Lorenz’ approach that have become part of the common body of conventions in inequality analysis. Two points could be mentioned: 1. Lorenz’ made a clear and unambiguous equivalence between the terms concentration and inequality. He reasoned with reference to aggregate income: not the absolute

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

27

amounts of income of each individual are important, but their share in total income.

Except for some isolated contributions (see e.g. Kolm, 1976; Blackorby & Donaldson, 1980), this has become the viewpoint of the mainstream inequality literature.

2. Much clearer than Pareto, Lorenz evoked the problem of ambiguous comparisons between different income distributions. We have seen that if two Lorenz curves intersect, we cannot make an immediate judgement about which of the two distributions is more equal. The distinction between comparisons allowing for a clear decision and those who necessitate further analytical steps has become an important and recurring theme in the literature ever since. Through the work of Atkinson the former case has become known as Lorenz dominance.

For our purpose of finding a satisfying measure of inequalities for the IEWB framework, these two points are both relevant. As we have argued in Section 1.1, a main criterion that our statistic should satisfy is its adequation with the representations of potential users. Is the idea of inequality as concentration and the embedded ‘relative’ concept of inequalities something that policy makers or other non-experts would intuitively agree with? It seems that most people — not only radical egalitarians — would give at least some importance to absolute differences between individuals incomes. We will discuss this point in more detail later in Chapter 3. Similarly, we will defer the discussion of the second point to a later stage when we will have presented Atkinson’s approach, which is, in a nutshell, a solution to the problem of intersecting Lorenz curves.

2.1.3

The Gini concentration coefficient: the ideal complement to the Lorenz curve

The Italian statistician Corrado Gini (*1884, †1965) approached empirical representations of income distributions in several alternative ways. One of his methods was very similar to Pareto’s approach of estimating logarithmic specifications to describe the income distribution. His specification was slightly different from Pareto’s Law and was preferred by some authors because it was not charged with the postulate of constant parameters as was the case with Pareto’s α. In the 1920’s, and still until the 1940’s, Gini’s slope coefficient δ and Pareto’s α competed against each other for best describing the shape of income distributions (see for example Dalton, 1920; Davis, 1941; Bowman, 1945).6 But Gini also proposed a second summary measure of income concentration. This measure, the classic ‘Gini coefficient of concentration’, possesses the valuable advantage of being independent of any mathematical formula for which the empirical distributions has to display an acceptable fit. In fact, the computation of the Gini coefficient of concentration 6

After Pareto’s death, Gini himself intervened in the debate around ‘Pareto’s α vs Gini’s δ’. In his lecture given at a research conference of the Cowles Commission in 1936 he argued in favour of the superiority of his δ by showing that Pareto’s coefficient was less sensitive than his own estimator. His coefficient δ displayed more variations among different income distributions and cast further doubt on Pareto’s alleged constancies. On this see Bowman (1945).

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CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

involves no regression at all. This is the first reason why the measure was convenient for many empirical applications until today. A second reason is that it constitutes an ideal complement to the Lorenz curve, which had become in the 1940’s, and probably even earlier, “undoubtedly the technique most commonly used to indicate differences in the degree of inequality of different income distributions” (Bowman, 1945, p. 617).

In 1912, Gini presents a variety of indices of variation in a book entitled Variabilità e Mutabilità. Two of the proposed formulas are the absolute mean difference and the relative mean difference. The absolute mean difference (AMD) is defined as the arithmetic average of the differences, taken positively, between all possible pairs of incomes. To obtain the relative mean difference (RMD) one simply has to divide the AMD by the arithmetic average of all incomes.7 In mathematical form, we can write these two measures in the following way: PN PN AMD = AMD RMD = µ

i=1

j=1 |yi N2

− yj |

(2.5)

PN where

µ=

i=1

yi

N

(2.6)

In an article — again in Italian language — published in 1914, Gini explains the relationship between the RMD and the Lorenz curve. This was a “remarkable relation” at the time (Dalton, 1920, p. 354). To link the RMD and the Lorenz curve Gini defines two areas: first, the ‘area of concentration’ which is the area between the line of equality and the Lorenz curve; second, the ‘area of maximum concentration’ is the area that would be circumscribed by the Lorenz curve that results from the extreme case in which one individual receives all income and all others nothing. Gini shows that the ratio of the area of concentration to the area of maximum concentration is equal to half the relative mean difference. This is Gini’s ‘concentration ratio’, also referred to as ‘Gini index’ or ‘Gini coefficient’, and can be written as: G≡

AMD RMD area of concentration = = area of maximum concentration 2µ 2

(2.7)

After Dalton (1920) introduced the Gini concentration ratio to economists outside Italian-speaking circles, this measure has become the standard reference for empirical analysis of income inequality and almost turned into a household name beyond the scientific sphere. The straightforward interpretation in terms of the Lorenz curve and its easy computation8 made it a convenient summary statistic for income inequality.

7

Similar to the not completely certain authorship of the Lorenz curve, some writers have argued that it was not Gini, but F. R. Helmert and other German writers in the 1870’s who discovered the relative mean difference. For more on this question see David, 1968 8 The adjective ‘easy’ is of course to be understood relatively. Some authors like Xu (2003) have shown that the Gini concentration ratio can be computed in many different ways, including some rather sophisticated approaches. As a matter of fact, the relative mean difference presented in the text is only one way to calculate the concentration ratio. Other methods include the geometric approach, i.e. direct

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

29

Being a standard reference in inequality measurement, it is important to note that Gini proposed his statistic as a measure of concentration for any quantitative variable. Its use to describe income distributions is only one possible application. Gini was not concerned with welfare considerations and hence his measure of concentration can also be applied to other objects than income distributions, such as wealth, shoes or fleas, which arguably makes the inequality measure appear very objective and neutral. In fact, he reminded Dalton of this universal applicability in 1921 in a reply to Dalton’s article on welfare-based inequality measurement.

Impact on conventions In empirical applications, the use of the Gini concentration ratio itself has become a technical convention. Gini’s alternative statistic, the absolute mean difference, did not have a convenient graphical counterpart like her sister, the relative mean difference, and was therefore seldom employed. As Lorenz-type diagrams were more and more mainstreamed in the period before WWII, the concentration ratio imposed itself as the standard measure. Dalton (1920) showed that — according to his list of principles derived from the utilitarian framework — Gini’s concentration ratio even fared well as a measure of inequality in terms of welfare. Criticism set in much later with Atkinson’s article in 1970, who discussed some of the implicit normative judgements contained in the Gini measure that might be objectionable (see our discussion in Section 2.1.6 on page 45). Still a bit later, Paglin (1975) argued that the Gini concentration ratio (together with the Lorenz curve) mistakenly confuses intra-family and inter -family inequality and sought to remedy this problem by replacing Lorenz’ line of perfect equality with a curve that allows to distinguish between intra- and inter-family income. Even if Atkinson’s and Paglin’s criticisms received much attention in the specialised literature (the former more than the latter), the Gini concentration ratio remains by far the most widely used statistic for inequality measurement. Two of its features that have had a particularly strong impact on conventions can be singled out: 1. Gini’s contribution led to a further focus on summary statistics that synthesise the relevant information on inequality into a single index. Gini’s relative mean difference — as an ideal complement to the Lorenz curve — provided a summary statistic that allowed for a straight-forward graphical interpretation. Other index numbers, like Pareto’s slope coefficient α or Gini’s own regression parameter δ, could also be represented graphically, but had the inconvenience of not being universally applicable: only distributions with a reasonably good fit to a given specification could be ordered with respect to the regression coefficient δ. Gini’s concentration ratio overcame this problem. Being an all-round summary statistic, the concentration ratio thereby fostered the idea that judgements in terms of inequality are possible between all possible pairs of distributions. It was a solution to the problem noted by Lorenz that intersecting concentration curves do not allow for an immediate decision on calculation of the areas in the Lorenz diagram, the matrix approach, the covariance approach...

30

CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE which distribution is more unequal or more concentrated. Gini’s ratio is a way to weigh increases of concentration against the decreases in other parts and allows to rank all thinkable distributions with respect to their concentration without losing its graphical interpretation.

2. The second important impact of Gini’s relative mean difference is that it further consolidated the view that inequality and concentration are essentially the same concepts.

Since the Lorenz curve is a tool for depicting concentration, the Gini ratio is also a measure of concentration. The fact that Gini’s contribution helped blurring the distinction between inequality and concentration is all the more interesting given that in his book Variabilità e Mutabilità he also presented measures for absolute dispersion, namely the absolute mean difference. We should be careful not to overestimate the power of graphical representation. Yet, it seems not completely aberrant to formulate the hypothesis that the reason why it was the relative mean difference and not its absolute sister that has become the standard reference in inequality metrics may be more the result of graphical representations than of conceptual arguments.

Both these points are relevant for our discussion of inequality in the context of the IEWB. As mentioned earlier, the well-being measure we want to implement has the form of an index, and inequality eventually has entered it as a synthetic one-for-all statistic. By measuring inequalities in this way, we follow Gini in the idea that it is possible to capture the essence of an income distribution’s inequality in a single number.

A second conclusion from the success of the couple Gini/Lorenz tells us something about the process of reception and penetration of inequality measures. Perhaps just as important as conceptual purity or mathematical correctness is the communicability and intuitiveness for a given inequality statistic in order to be accepted. It seems that communicability is the key feature for successful penetration, especially in empirical applications that aim at a public of scientific and non-scientific actors. One of the IEWB’s intellectual fathers, Lars Osberg, has noted this point as far back as 1985. We will cite the relevant passage from the paper which also set the intellectual base for what had later become the IEWB. Osberg’s observations also contains a possible explanation for the continued success of the Gini measure despite several proposals to amend it: “[...]the Theil index is the only appropriate measure to use to disaggregate economic inequality, but this measure is not used all that often in empirical work, largely because it is extremely hard to communicate in anything other than algebraic terminology. On the other hand, the continued appeal of measures of inequality based on the Gini index is no doubt due largely to their easy graphical interpretation. The importance of easy interpretation is illustrated by the fate of two proposed amendments to the Gini index. The Donaldson-Weymark (1980) proposals are technically correct, but they are complex and have received little attention. The “Paglin-Gini” (Paglin, 1975) is a technically incorrect method of inequality decomposition, but it can be presented easily in graphical form and

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

31

soon became rather popular (e.g. Armstrong et al., 1977). The moral of the story appears to be that information will not be used in public debate, whether the debate of the general public or the debate of technical specialists, unless it is easily communicable” (1985, p. 84).

Since we want the IEWB to continue its increasing penetration of public and expert debates, the ‘Gini/Lorenz’ combination is at the same time a reminder of the limits of technical sophistication and a good example of efficient communicability. This issue is one of the recurring themes in this text and we will discuss it in more detail in Chapter 3. However, despite all its convenience, we have to analyse whether we agree on the assumption embedded in the Gini index that concentration is essentially the same concept as inequality. For if we disagree with this equivalence, we should not use Gini’s statistic since it is nothing more and nothing less than a measure of concentration. Gini himself proposed and discussed alternative measures of variability that place more importance on absolute income differentials: the absolute mean difference is the best example of a statistic that reflects income differentials. Again, the implications of the difference between concentration and inequality — a second recurring theme of this text — will be further examined in Chapter 3.

2.1.4

Dalton’s measure of ‘distributional badness’

The influential paper by Dalton (*1887, †1967) on inequality measurement from 1920 is frequently referred to as a pioneering contribution because it inspired an approach continued by welfare theorists such as Atkinson (1970), Kolm (1976), Sen (1973) and many others. In contrast to earlier attempts in measuring inequality as such, Dalton proposed not to describe economic inequality as an object but instead to directly evaluate its impact on welfare.

The rupture introduced by this new approach is visible in Gini’s reply to Dalton from 1921 in which the Italian reminds that the traditional methods of inequality measurement also allowed for applications “to all other quantitative characteristics” (Gini, 1921, p. 124).

By contrast, Dalton’s approach can not be disentangled from the cognitive framework of welfare economics. While the particular welfare function Dalton used was subsequently criticized (see the discussion of the Atkinson’s measure in Section 2.1.6), the shift of the evaluation space persisted and inspired a new catalogue of research questions.

The motivation for Dalton to evaluate inequality in the space of welfare lies in a purposeoriented argument: “For the economist is primarily interested, not in the distribution of income as such, but in the effects of the distribution of income upon the distribution and total amount of economic welfare which may be derived from income” (Dalton, 1920, p. 348). Having identified the purpose of inequality measurement correspondingly, one still has to introduce an important relationship in order to understand what rendered the transition to the new evaluation space so natural and smooth. After all, if economic welfare is regarded as the only relevant object for the economist, the analysis should focus on the relation between income and welfare, without any particular interest in inequality.

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CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

However, Dalton is perhaps the first to formalize the view that, ceteris paribus, a clear link exists between unequal incomes and economic welfare, namely a strictly negative relation. By defining inequality in terms of welfare, Dalton makes use of this negative relationship. He equates a particular arithmetic characteristic encountered in his expression of economic welfare with a concept that exists under the same name in normal communication. Dalton’s argument starts with expressing welfare in algebraic language. This leads to the conclusion that “it is evident that economic welfare will be a maximum when all incomes are equal” (ibid, p. 349). Inequality is thus a phenomenon that coincidentally happens to maximise economic welfare. However, from an obvious and profane observation like ‘welfare is maximised when there is equality’ it still takes a bold move to a definition of inequality in terms of welfare. We thus note the absence of any semantic safeguards contained in Sen’s observation we cited earlier stating that economists are not really free to define the term inequality arbitrarily. Hence, from now on (and this holds for all welfarist measures), we have to bear in mind that we are not dealing with a bottom-up definition based on the concept of inequality and normal communication, but with a top-down definition originating from particular welfare definitions or other more sophisticated approaches.

The welfarist measures of inequality are genuinely concerned with the functional form that links incomes to economic (Dalton) or social (Atkinson) welfare, and their use of the term ‘income inequality’ can only be understood within this particular terminology. In Dalton’s case, inequality (in a normative sense) is defined as the welfare effect of inequality (in a descriptive sense).

We now turn to the specific form Dalton gives to economic welfare and, by extension, to what he calls inequality. He applies the utilitarian framework of representing economic welfare as the simple sum-total of individual welfare. The negative relation between inequality and welfare inherent in the utilitarian system stems from two key assumptions (in addition to the one that individual welfares are additive): 1) all members of the group have the same welfare function w(y) (welfare is symmetric); 2) this function has the property of decreasing marginal welfare to income. Maximising economic welfare in this framework leads to the “very special coincidence” (Sen, 1973, p. 16) that associates an efficiency loss in terms of economic welfare with unequally distributed incomes. Inequality is hence not an independent concern, but enters the analysis only because welfare is not maximised when incomes are unequally distributed. The inefficiency inherent in unequally distributed incomes is illustrated in Fig. 2.4 which represents total economic welfare in a two-person society. The curves are indifference curves of group welfare, which simply means that total economic welfare is the same on all points of the same curve. The closer the curves are to the upper right corner of the diagram, the higher is total welfare. The line Y Y 0 is the line of all possible divisions of the total income Y = Y 0 between the two individuals. We can see immediately that the point A is a rather unequal distribution since person 1 receives a considerably higher share of Y than person 2. The crucial point is that by moving along the line Y Y 0 from point A towards the line of equi-distribution EE 0 , the iso-welfare curves indicate higher and higher levels of total welfare. Maximum welfare, given Y , is attained at point B, where both individuals receive the same share and the indifference curve is tangent to the line Y Y 0 . The further the actual allocation of the income Y is away from

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

33

point B, the greater the extent of welfare inefficiency. income indiv. 2 Y E0 µ B

I2 A

I1

45˚ E

Y0

income indiv. 1

Figure 2.4: Economic welfare in the simple utilitarian framework.

The intuitive reasoning of this two-person example can easily be extended to the case of N individuals, given that the assumptions of an additive total welfare function (additivity), identical individual welfare functions (symmetry) and decreasing marginal welfare (concavity) are retained. In this framework, the criterion used to evaluate ‘distributional badness’ of income distributions is the inefficiency that it creates in not generating the biggest sum-total of welfare. Dalton formalises this idea and proposes the following measure of inequality: nw(µ) maximal total welfare = Pn D≡ actual total welfare i=1 w(yi )

n

where

1X µ= yi n i=1

(2.8)

It can be seen from Fig. 2.4 that D takes its minimum value of 1 when all incomes are equal. In this case, actual total welfare is already at a maximum for a given amount of total income. If all income shares are not equal, D increases the more total income is concentrated. This is Dalton’s “measure of inequality” in its abstract form. However, for an empirical application one needs to spell out the individual welfare function w(y) (the aggregate simply being the sum-total of these individual functions) to compute numerical values for actual and potential welfare.

In order to specify the function w(y), further hypotheses on the relation between income and economic welfare on the individual-level are necessary. Dalton argues that the following hypotheses are plausible: a) the higher the level of y, the lower the effect on w(y) should be for a given proportional increase; b) total individual welfare should tend to a finite limit; and c) welfare for incomes below a certain threshold should be negative.

This leads Dalton to the following specification of the relation between individual income and individual welfare:

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CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

dw =

dy y2

All three additional requirements plus decreasing marginal welfare are satisfied by this specification as can be easily verified. We thus obtain an infinite class of individual welfare functions: 1 (2.9) w(y) = c − y Combining equations (2.8) and (2.9), we obtain an expression for the inequality measure that depends on the parameter c: nw(µ) Dc = P n i=1 c −

nc − nµ = P 1 nc − ni=1

yi

1 yi

(2.10)

The denominator of this expression can be simplified by applying the definition of the harmonic mean yh to the incomes yi : n yh = Pn

1 i=1 yi

Using this in equation (2.10), we can write: Dc =

c− c−

1 µ 1 yh

Dalton therefore proposes this form of D to evaluate inequality as he defines the term.

However, we note the uncomfortable presence of a free parameter c which has to be specified for an empirical application to an observed income distribution. In other words, by placing the problem of inequality measurement in the framework of economic welfare, Dalton creates the necessity to agree on an additional convention: what value should be assigned to c, the reciprocal of the minimum income that yields positive individual welfare? Without convening on a way of imaging the cut-off point of positive welfare, an empirical evaluation of inequalities becomes impossible in the Dalton approach.

Apart from this problem, for Dalton the preceding expression of D provides a satisfying degree of acceptability given his definition of inequality in terms of welfare (and given his definition of welfare). However, we have to bear in mind that the paper was published in September 1920, and that in this context “the corresponding calculations for the geometric and harmonic means are very laborious, when the number of individual incomes is large, and the corresponding approximations, especially for the harmonic mean, are practically impossible, where the statistics show more than a small degree of imperfection.” (ibid, p. 351). This practical impossibility — paired with the problematic choice for the parameter c — forces Dalton to consider an alternative method for inequality measurement. This second option consists in assessing other “plausible measures of inequalities” with the help

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

35

of criteria derived from Dalton’s utilitarian framework of welfare assessment.

These criteria are presented in the form of “principles” and reflect certain consequences of the utilitarian assumptions. Dalton’s strategy consists in testing whether other practically more appealing measures satisfy these principles. By sticking to a particular definition of inequality and evaluating the conformity of available measures according to a test derived from this definition, we meet here another important piece of Dalton’s legacy for inequality measurement. This indirect approach of testing the acceptability of measures in light of their conformity to a list of features has, in fact, become a standard method. It reflects the difficulty to transpose a definition of inequality to the space of empirical evaluations — even if the definition is relatively clear-cut and almost unambiguous as in Dalton’s case.

Dalton’s list of principles — each based on the idea of a hypothetical variation to income distributions — includes four items. It is important to note that these principles are not derived from ‘intuition’, or any other argument except the particular form of the welfare function. Each of these principles can be proven mathematically in terms of Dalton’s assumptions as regards the form of this function. The four principles are: 1. The principle of transfers; 2. The principle of proportionate additions to incomes; 3. The principle of equal additions to incomes; 4. The principle of proportionate additions to persons.

The first principle on this list has entered the literature as the Pigou-Dalton principle of transfers since Dalton reformulates a rationale proposed by Pigou.9 In a nutshell, it holds that any transfer from a richer to a poorer person — provided that this transfer does not alter the relative position of the two persons involved in it — will diminish economic inequality and should therefore lead to a strict decrease of a plausible statistical measure.

The second principle holds that if all incomes were multiplied by a scalar, then the statistical measure should reflect a corresponding decrease in inequality. This result is a direct consequence from the assumption of diminishing returns to income in the welfare function.

From this assumption it follows intuitively that if incomes were multiplied by a scalar, then an individual with a higher income will gain less welfare from this operation than one with a lower income. While the recipient of the higher income will still have a higher level of welfare than his poorer neighbour, the difference in welfare levels will diminish from a proportional addition to all incomes and hence inequality in terms of welfare decreases.

Third, if an equal sum is added (subtracted) to (from) all incomes, inequality diminishes (augments). This is the principle of equal additions to incomes and the reasoning is the same as for the preceding principle. It follows from the principles of equal and proportional additions to income that Dalton’s measure of inequality is — like Pareto’s definition — not mean independent: increases through equal or proportional additions lead to a reduction in inequalities.

9

See Pigou (1912). The transfer principle is formally discussed by Pigou on p. 44.

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CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

The fourth principle of proportionate additions to persons holds that if the numbers of persons at each level of incomes is inflated by their proportion in the total population, inequality remains unchanged. This means that if a given population and its income distribution are merely duplicated, inequality stays constant. As an example, imagine two countries which have the same mean income and the same inequality. According to Dalton’s measure, inequality stays the same whether we calculate the statistic D for each of the two countries separately or one D for both countries combined. As for the three preceding ones, this principle is directly derived from the definition of inequality and the particular assumptions in Dalton’s welfare arithmetic.

Armed with these principles, Dalton can evaluate different statistics as to their coherence with his utilitarian welfare function. The measures Dalton discusses are Gini’s relative and absolute mean differences; the relative and absolute standard deviation; an inter-quartile measure proposed by Bowley (1901)10 ; and the relative and absolute mean deviation. All relative measures in this list are obtained by dividing their absolute sister by the average income.

It is Gini’s concentration ratio (the relative mean difference) and the relative standard deviation that fare best in the comparison. They are both sensitive to transfers from rich to poor at all levels of income (Principle 1). And they both indicate ‘correctly’ diminishing inequalities in case of equal additions to incomes (Principle 3). They also ‘correctly’ remain unchanged in case of equal additions to persons (Principle 4). The Gini concentration ratio and the relative standard deviation fail to reflect diminishing inequalities in case of proportional additions to incomes if all incomes are multiplied by a scalar λ (Principle 2).

Since such a multiplication does not modify the concentration of income, the relative mean difference remains unchanged if incomes are inflated by λ. The relative standard deviation also remains unchanged: the absolute standard deviation increases by λ, but so does the mean. Since the relative standard deviation is the ratio of the two, the proportional increase of all incomes has no effect. However, all the other inequality measures Dalton reviews also contradict Principle 2. The top place in this ranking surely helped to build up the reputation of Gini’s relative mean difference as a practical and theoretically acceptable measure of economic inequality.

Impact on conventions In an immediate reply to Dalton’s article, Gini (1921) can be seen to “admire the simplicity and ease of the method which he suggests for measuring the inequality of economic welfare, on the hypothesis that the economic welfare of different persons is additive” (ibid., p. 124), and then goes on to cite a list of Italian writers or articles in Italian journals related to the topic and which — due to the remoteness from the English-speaking audience — have not yet received the attention they deserved in the eyes of Gini.

For our questions, two features of the Daltonian heritage are important: 1. Dalton’s main contribution was probably to successfully shift the purpose of inequality 10

The formula of Bowley’s quartile measure is B =

Q3 −Q1 Q3 +Q1 .

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

37

measurement: it should not describe income inequality as such or attempt to answer questions like “are inequalities in this country on the rise?”. Dalton’s followers implicitly detached this descriptive element and took the normative judging firmly into their own hands: according to the welfare approach, describing inequality could be bypassed by directly evaluating the income distribution in terms of welfare. On the grounds of this precursor, a bulk of the literature on economic inequality actually measures welfare instead of describing inequality as such.

2. Dalton’s method to use a list of principle-based tests has become conventional in the literature (see among others Theil, 1967; Atkinson, 1970; Sen, 1972; and Kolm, 1976, who uses it as an axiomatic). We already mentioned that the suitability of such a strategy may be attributed to the difficulty of transforming the concept inequality into an empirical measure. Interestingly, not only the method of testing a list of features, but also some of the items themselves found their way into the body of measurement conventions. This holds notably for the Pigou-Dalton principle of transfers.

This is a remarkable phenomenon since the principle of transfers used by Pigou and Dalton is not derived from intuitive views on inequality but from their particular definition of welfare. Dalton derives it within the utilitarian framework, with its special assumptions, and it makes as such only sense if one sticks to his definition of inequality in terms of welfare. Atkinson (1970) and Kolm (1976), who do not explicitly use a framework of additive individual utility functions, stick to the principle of transfers by referring to Dalton’s proposition; Theil (1967) uses it as an argument for a measure not even based on welfare. The Pigou-Dalton-Theil-Atkinson-Kolm-Sen principle of transfer, through its use in various approaches to inequality, appears to be a good example in the inequality literature of what Favereau has called a “dispositif cognitif collectif ”. In other words, it has become a convention.

Given our problem of inequality in a context of an economic well-being index, it seems that the research programme Dalton initiated, with its emphasis on welfare-effects of inequality — as opposed to describing inequalities —, makes us aware of the limits of welfare measurement. We submit that Dalton’s definition in terms of economic welfare led to a clear divergence in meanings of the term ‘economic inequality’ between the Daltonian approach and non-scientific communication. Inequality is often thought of as evoking different normative judgements, and the IEWB can only assist in these judgements if a) either preferences are accurately communicated or b) the normative judgements are left as much as possible open so that the users can form their own opinion on the evolution of well-being. By imposing a welfare criterion instead of attempting to describe as neutrally as possible the evolution of inequalities, we risk to integrate a set of normative views either unknowingly or without being able to communicate it efficiently to the potential users.

For our problem of empirical measurement of the concept of inequality that should rely on common language, we should be aware that Dalton’s approach might be a starting point for a separation between the internal (i.e. within the economic science) debate and other communicational spheres.

38

CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

As for the second conventional feature mentioned above, it seems that the blurriness of the concept of inequality calls for an approach based on an indirect list of desirable features. Such a method has the advantage of imposing some transparency since the different “principles”, “properties” or “axioms” have to be clearly and explicitly stated.

Potential differences in meanings or conceptions can therefore be detected more easily once such a list is spelt out and open to debate. In the context of the IEWB, we nevertheless have to be aware that an over-specified list might divert the focus on less important technicalities.

An example of a mathematically elegant, but hardly communicable (and consequently less used) list is the axiomatic developed by Kolm (1976). Furthermore, Dalton’s list-based test contains an additional choice for the investigator. The approximate character of such a test allows to define acceptability either rather loosely (say, conformity to one principle suffices), or extremely accurately (with a very long list of principles candidate measures have to pass). Dalton identifies and tests four principles, and a list of roughly this length seems to have become a convention ever since.

2.1.5

Theil’s analogy and decomposability

The Dutch econometrician Henri Theil (*1924, †2000) applied methods of information theory to the problem of inequality measurement and thereby gave new impetus to the analysis of alternative statistics for income distributions. The origin of the methodological apparatus that Theil (1967) transposed to inequality questions lies in the analysis of information by Shannon (1948), who in turn applied an analogy between the notions of information content and physical entropy in thermo-dynamical statistics. It appears that de Jongh (1952) was the first to exploit the analogies between information concepts and the partitioning problems typical for economics, but Theil is credited for transforming these correspondences into concrete statistical tools and discussing their properties. These analogies between the problems in thermo-dynamics and other research fields are of course responsible for the persistence of the term “entropy”, which is stripped of any sense in both information theory and economics.11 While it is futile to discuss thermo-dynamics in our context, it may be useful to present some basic elements of information theory in order to foster our understanding of Theil’s inequality statistic and its relevance for our purpose. We begin by presenting the two basic concepts ‘information content’ and ‘expected information content’ used in Theil’s programme.

According to information theory, messages differ with respect to their information content. Broadly speaking, information content captures how useful a message is, in the sense that the message changes our knowledge about certain aspects of reality. If we are already familiar with the content of the message or if we anticipated the information it contains, the information content of the message is rather small. Now, assume that we ignore whether a certain event in the past has happened or not. Assume further that the message in question contains the information that this event has occurred. Then it is intuitively clear that the 11

“Entropia” appears to be based on the Greek en- (in) + trope (a turning).

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

39

information content of this message depends on the probability of the event to happen.

If it was absolutely certain that the event occurred (i.e. the probability of the event’s occurrence equals unity), the information content of the message is nil. On any Sunday, the message ‘yesterday was a Saturday’ contains therefore only very limited information. On the other hand, a message stating that an event occurred which had an infinitesimal small chance of occurring (a message like ‘the NASA discovered alien living forms on the moon’) has a very high information content.

It is therefore intuitive to postulate a negative relation between, on the one hand, the information content, denoted h(x), of this type of message and, on the other hand, the probability x of the event . While infinitely many functional forms for h(x) could satisfy a negative relation, information theory uses one particular function, namely the logarithm of the inverse of the probability x: 1 (2.11) x Since this expression will reappear below in Theil’s inequality measure, it is important to note that the choice of this functional form is by no means arbitrary. As Theil shows, the logarithmic definition of information content is the only form that corresponds to a set of five “natural axioms” (ibid, p. 6) defining the properties of h(). For us, it is not the exact content of these axioms — the reader with an interest in information theory is referred to the first chapter of Theil (1967) — that is relevant. What is more important is the fact that the relationship in equation (2.11) is not arbitrary in the sense that it is the only functional form of h() that corresponds to the axiomatic of information theory.12 From equation (2.11) we conclude that the information content of a message telling us that two independent events E1 and E2 occurred simultaneously is additive. Denoting the respective probabilities of E1 and E2 with x1 and x2 , we can see this additivity when we remember that the probability of both independent events occurring equals x1 x2 . Then the additive information content becomes: h(x) = log

h(x1 x2 ) = h(x1 ) + h(x2 ) If the information content is defined as in equation (2.11), additivity can be written as follows: 1 1 1 = log log + h(x1 ) + h(x2 ) h(x1 x2 ) = log x1 x2 x1 x2 We can now move on to the next concept on which Theil’s inequality measure is built: the expected information content. Suppose that we have a complete system of N independent events E1 , . . . , EN , of which exactly one event will occur. The probabilities of these events are: xi , i = (1, ..., N )

with

N X

xi = 1

and

xi ≥ 0

i=1 12

It does not, however, follow from these axioms which base we should take for the logarithm.

(2.12)

40

CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

We now define a special kind of message: after one of these n events occurred, a definite and reliable message will be received stating which Ei actually happened. However, it is possible to form an opinion on the expected information content of this message before it is received. It is clear that this depends again on the probabilities with which the events occur: if there is certainty that one specific event occurred (‘yesterday was a Saturday’), the expected information content is zero. This reasoning is formalised when the expected information content is defined as the sum of all possible h(xi ), weighted for the probabilities xi . Before the message comes in and tells us which of the n events occurred, its expected information content, denoted H, is therefore: H(x) =

N X i=1

xi h(xi ) =

N X

xi log

i=1

1 xi

(2.13)

where x on the left stands for the vector of the n probabilities. The lowest possible value for the expected information content H(x) is zero. This corresponds to the case when one probability is unity and all others zero.13 The maximum value for the expected information content can be calculated by maximising H(x) with respect to x, given the constraint of equation (2.12). The result of this maximization is that the message has the highest expected information content when all events have the same chance of occurring, i.e. all N events have probability 1/N . The value of equation (2.13) in this case is log N , so that we have: 0 ≤ H(x) ≤ log N

(2.14)

The crucial step in deriving Theil’s measure of inequality14 is to see the formal similarity between probabilities and income shares: they are both non-negative, and they both add up to one. It is thus technically possible to calculate a value for equation (2.13) by substituting the vector of probabilities with a vector of income shares. The income shares are of course derived from a distribution of income among the N income receiving units i. By applying this analogy, the minimum of (2.13) is interpreted as the value that corresponds to complete inequality, and the maximum as complete equality. Theil defined his measure of inequality as the difference between (2.13) and its maximum level log N . Replacing the vector of probabilities x by the vector of income shares s, the statistic is thus defined as: T ≡ log N − H(s) = log N −

N X i=1

13

si log

1 si

(2.15)

Note that the product x ∗ log(1/x) is in general not defined for x = 0. However, here it is defined to be zero.

14 In most of the literature on information theory, the notions ‘expected information’ and ‘entropy’ are used interchangeably. Hence, we could say ‘Theil’s entropy measure’ or ‘Theil’s expected information measure’ and refer to the same thing. However, since entropy refers etymologically to a distinct concept, which is furthermore still used with a different meaning and a different calculation method in other sciences, we should try to avoid the term where possible.

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

41

The measure T — Theil’s measure of inequality — varies between zero (complete equality) and log N (complete inequality).

Decomposition of T The measure T allows identifying two additive components of total inequality: a) the inequality within population groups and b) the inequality between these groups. How many and which groups are identified does not alter total inequality and may be adapted to the investigator’s interest. Since this decomposability is arguably the key added value of Theil’s measure, we will briefly illustrate the decomposition mechanism. The decomposition will also help to shed light on a notion Theil called “aggregation consistency” (ibid, p. 95) in the context of inequality measurement.

We divide the population N in k groups G1 , . . . , Gk , each group containing Ng individuals, so that k X

Ng = N.

(2.16)

g=1

It is straightforward to rewrite equation (2.13) in order to make the k groups visible.

We simply lump some of the summands together according to the criteria which allocates the income receiving units i into one of the k groups. Since H(s) is a sum of N elements, we can rewrite it as a sum of k elements with each forming a sum of Ng elements. Hence, (2.13) becomes:   k X X 1  (2.17) T = log N − si log  si g=1 i∈G g

We define Sg as the share of group g in total income, so that Sg =

X

si

g = 1, . . . , k.

i∈Gg

We can multiply the sum over the income receivers i and the denominator of the logarithms in equation (2.17) by (Sg /Sg ). This yields: T = log N −

k X g=1

Sg X 1 si log Sg i=1 si (Sg /Sg ) = log N −

k X

" Sg

g=1

= log N −

k X g=1

!

Sg

1 log Sg si /Sg

X si i=1

"

g=1

X si i=1

"

= log N −

k X

1 log Sg si /Sg

# X si 1 1 Sg log + log Sg si /Sg Sg i=1

1 + Sg log Sg

# −

k X g=1

#

Sg log

1 Sg

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CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

Next, we add and subtract the sum log N −

k X

Sg log Ng −

g=1

k X

" Sg

g=1

Pk

g=1

Sg log Ng from the preceding equation.

X si i=1

1 log Sg si /Sg

# +

k X

Sg log Ng −

g=1

k X

Sg log

g=1

1 Sg

" # k k X X X si 1 1 = log N + Sg + Sg log Sg − Sg log − log Ng Ng g=1 Sg (si /Sg ) g=1 g=1 i=1 k X

By rearranging the terms we obtain a decomposable expression for T : T = log N −

X

si

i=1

in which: B = log N −

k X

Sg log

g=1

W =

k X g=1

" Sg log Ng −

1 Sg /Ng

X si i=1

1 log Sg si /Sg

1 =B+W si

(2.18)

(Between-group inequality) # (Within-group inequality)

It is easy to see that B has the form of equation (2.15) — the basic inequality measure —, the difference being that the income shares of the individuals i are replaced by the ratio (Sg /Ng ). This ratio captures the differences of per capita income between the k groups.

B is therefore interpreted as between-group inequality. Note that if all group income shares Sg are exactly equal to the share of the different groups in the total population then Sg = Ng /N , for all g = 1, . . . , k. In this case B = 0 and each group’s weight in total income is equal to the group’s weight in the total population. In other words, there is no between-group inequality.

Theil proposes to interpret W as total within-group inequality. The difference in square brackets in the expression for W also has the form of (2.15), only that here the income shares of the individual i in the total income are replaced with the respective shares in the group income. In general, the higher the concentration of income among the members of a group, the higher the expression in square brackets will be, and vice versa. This expression can therefore be interpreted as within-group inequality of group g. Total within-group inequality is then given by W , which is the sum of the k within-group inequalities, weighted for the income share Sg of each group.15 We have seen that Theil’s inequality measure T is derived from an analogy between probabilities in the framework of information theory and income shares in the analysis of income distributions: since income shares are formally similar in that they are non-negative and sum up to one, the concept of ‘expected information’ could be transformed into an 15

Some authors noted an analogy between the decomposition of the Theil measure of inequality and the well-known decomposition in the Analysis of Variation (ANOVA). In the latter, the total variation is decomposed in explained and unexplained variation. On this point see Anand (1983) and Sen (1997).

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43

inequality measure. But does T really measure inequality? Beyond the formal similarities, income shares are not really probabilities and some prudence calls for further tests. To overcome this problem, Theil sets up a list of features to check whether T is indeed a valid inequality measure. The method to verify the acceptability of inequality statistics is thus very similar to Dalton’s list of principles (see p. 37). However, the different items of this list, i.e. the content of the definition, is derived completely differently by Theil compared to Dalton. Theil, without explicit references to the utilitarian, or even a social welfare approach, derives the items of this list partly from conventional usage, partly from his own additional arguments. This can be seen by discussing briefly the desirable features that he identifies and advocates for an inequality statistic to be acceptable:16 1. The measure should be at its minimum value when the distribution is characterised by complete equality, defined as the situation when everybody receives the same share of total income (ibid., p. 91). This is an obvious requirement.

2. The measure should take its maximum value when the distribution is characterised by complete inequality, defined as the situation when one person receives all income and all others nothing (ibid., p. 91). This requirement is more interesting (and less obvious) than the preceding one. We note that by the time Theil was writing, Lorenz’ argument of defining complete inequality as complete concentration had become a convention that did not need any further discussion. Theil mentions it without justification, perhaps assuming it to be a dispositif cognitif collectif. The proof that Theil treats concentration and inequality as equivalents is that later in his book, in a chapter on industries and allocation problems, he employs essentially the same statistic T to measure concentration. For him, “concentration and inequality are essentially the same concepts, this index [the Herfindahl index of industrial concentration] may in principle also be used as a measure of income inequality” (ibid., p. 128).

3. The measure should indicate decreasing inequality if income is transferred from a richer to a poorer person up to the point where the two incomes are equal (ibid., p. 93). This, of course, is the Pigou-Dalton principle of transfers (see p. 37). It is interesting to see that for Theil this requirement is simply an “obvious test”, while Dalton and Pigou still had to go through substantial mathematical proofs. Even more interesting is perhaps that Theil does not even refer to a utilitarian, or even welfare-based evaluation criterion. Again, he seems to rely on the dispositif cognitif collectif, where ‘collective’ is probably restricted to the academic community.

4. The maximum extent of inequality in a situation of complete inequality should increase with total population size N (ibid., p. 92). In fact, Theil does not specify a fixed maximum level an inequality measure may take; only the distribution of income 16

Although we present these features here in list form, Theil spreads them throughout his text. Some of them are presented parallel to the derivation of T , others in his discussion of the “traditional” inequality statistics. This, however, is not an important difference for our problem. To find the passages in which these features are mentioned in Theil (1967), we have indicated the corresponding page numbers.

44

CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE that corresponds to complete inequality is defined. As we have seen above, the highest possible value of T (which is log N ) depends on the population size N . For Theil, the bigger the population, the higher is the potential inequality. The intuition behind this theoretically unlimited maximum extent of inequality is that adding people to a population means also adding potential recipients of an income share equal to zero.

According to Theil, a N -person economy in which one person has all income contains less inequality than a N + k-person economy (k being a positive integer), in which one individual owns everything.

5. A proportional change of all incomes (holding income shares constant) does not alter inequalities. This is a corollary of item 2 on this list and therefore a feature inherited from Lorenz (cf. our discussion p. 26).

6. The measure should easily be decomposable in within-group and between-group inequalities (ibid., p. 123). This means the overall inequality measure should be independent from the different groups in which we might divide the population and that overall inequality is a sum of the different group-inequalities. The decomposability of T is and was its key added value over alternative measures and arguably the most genuine contribution of Theil to inequality measurement. Both ‘decomposability’ and ‘subgroup consistency’ were regarded by later authors as axioms for the acceptability of inequality measures (on this see Sen, 1997, p. 149).

As one might have expected, the Theil measure passes all of these tests and is therefore according to this list an acceptable measure of inequality.

Impact on conventions Theil raised the bar that inequality measures have to pass in order to be acceptable. He added to the other conventional requirements the feature of decomposability. The main rationale to add this property is of practical, and not necessarily of conceptual nature.

Being able to decompose an inequality measure for skin colour, gender or region is in many empirical applications a convenient feature. Consequently, many authors have included decomposability as an axiom to test whether a certain inequality measure is acceptable or not. In our discussion of the recent developments of the academic discourse (Section 2.2 on p. 59), we will see that decomposability engendered a whole new branch of problems in inequality literature.

For our problem related to the IEWB we must, however, decide on at least two issues before we accept some of the conventions embedded in Theil’s approach. First: is decomposability useful for our purposes? Second: is the Theil measure an acceptable statistic for inequality?17 17

Sen (1973) went a step further. While not criticising the usefulness of decomposability, he questioned its plausibility. He noted that “if there is even a modest amount of interdependence between groups in society, an exact separation into between-group and within-group terms may not be attainable. A residual term [...] or some other modification to additivity may be needed to account for overflow or undercounting

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

45

Concerning the first question, our answer will depend on the way in which we choose to aggregate the different inequalities in the three different dimensions (consumption, wealth and economic security). We will come back to this point in Chapter 3 in more detail when we discuss different alternatives to inequality measurement in the IEWB framework.

Even if we decided that decomposability is a desirable feature for our purpose, we would still have to verify whether we think Theil’s measure corresponds to the usage of ‘inequality’ in normal communication. After all, inequality and concentration may often be correlated, but the concepts are not identical. Item 4 on the above list of Theil’s features also calls for some prudence. We think it is not obvious that a completely concentrated two-person economy is necessarily less unequal than a completely concentrated three-person economy or even a completely concentrated thousand-person economy. After all, when one person receives all income and the rest of the people nothing, it is true that all individuals except one are completely equal with respect to their income. The more individuals we add to a completely concentrated distribution, the more people are completely equal. While poverty, justice or other considerations might clearly indicate a profound malaise in a completely concentrated distribution, it seems odd to define complete inequality as a situation in which everybody except one is completely equal.

2.1.6

The Atkinson index: refining the analytical apparatus

Anthony B. Atkinson’s article “On the Measurement of Inequality”, published in 1970, updated many of Dalton’s ideas we presented in Section 2.1.4. In fact, to people without some background in welfare economics, the difference between the respective measures of Atkinson and Dalton may not be obvious. Both define inequality in terms of welfare and measure them in terms of incomes, i.e. they both start by defining a welfare function and then derive an inequality statistic from this specification. Both use the additive framework in which individuals enter the relation between income and welfare symmetrically. Finally, both authors use a benchmark measure to gauge the ‘distributional badness’ of the actual income distribution. And yet, despite these similarities, Atkinson — whose professional career seems to be inextricably intertwined with almost all important research on inequality since 1970, as Jenkins & Micklewright (2007) recently noted — is rightly credited for a considerable improvement of the analytical apparatus of inequality measurement.

Atkinson argues that “any measure of inequality involves judgements about social welfare” (1970, p. 257). In order to make sure that these normative judgements correspond to accepted values, he proposes to spell them out explicitly via the welfare function. Atkinson’s specification of a ‘social welfare function’ includes like Dalton’s ‘economic welfare function’ the assumptions of additivity and symmetry. Both are classic properties of the utilitarian framework: the former simply states that group welfare can be expressed as the inherent in the problem.” (ibid, p.156). The problem of interdependence is raised by a ‘separatist’ element of Theil’s measure, which means that if population and average incomes rest the same and inequality rises in any subgroup without changing the within-inequality of all other groups, then total inequality must necessarily increase. Hence, if inequalities are interdependent between individuals of different groups, then decomposability and subgroup consistency might not be plausible.

46

CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

unweighed sum-total of individual welfare; the latter implies that permutations of individuals leave total welfare unchanged as all individuals are supposed to be equal with respect to their welfare function (i.e. the welfare function adopts an impartial point of view in that all individuals are treated equally). This group welfare function is defined as: W ≡

N X

U (yi )

with

0 ≤ yi ≤ k

and

i = 1, ..., N

(2.19)

i=1

where the constraint on individual incomes excludes negative incomes and those above the maximum income k. The form of the function U (y) is characterised by:18 dU (y) >0 dy

and

d2 U ≤0 dy 2

(2.20)

While this welfare specification in itself is nothing new, Atkinson’s innovation consist in linking it to the Lorenz curve — like Gini related his concentration ratio to the Lorenzian framework (cf. Section 2.1.3). In what has become known as the Atkinson Theorem, he shows that income distributions can be ranked unambiguously according to the welfare function in equation (2.19) if, and only if, the Lorenz curves do not intersect. In other words, non-intersecting Lorenz curves allow for a complete ordering of distributions in terms of welfare. In this case, any specification of the function U (y) satisfying (2.20) will lead to an identical ordering, so that further assumptions on the relation between income and welfare are not necessary.19 The Atkinson Theorem therefore specifies the minimum amount of information about the welfare function necessary to make unambiguous decisions when comparing inequality across different distributions. At the same time, it shows that a more precise specification of the function U (y) is necessary to rank distributions whose Lorenz curves do intersect. In a certain sense, the Theorem is an ex post explanation of why it is so difficult to compare inequalities when two concentration curves are not consistently higher or lower to each other. While Lorenz provided this explanation in terms of his definition of concentration (“higher concentration in one part, lower concentration in another”), Atkinson derived the explanation from a welfare argument: if the Lorenz curves intersect, we need further knowledge on W and U to be able to rank them in terms of total welfare.

18

Sen (1973) pointed out that strict concavity might be a more reasonable assumption that Atkinson’s 2 weak concavity. In fact, if ddyU2 = 0, which Atkinson does not exclude, the maximisation of total welfare is unconcerned with inequality (ibid., pp. 38-39). In this case, the distributions (0,10) and (5,5) would yield the same value for the Atkinson measure of inequality. This is, however, an extreme case which we can neglect without losing generality.

19 The proof of this Theorem can be found in Atkinson (1970), pp. 245-248, and earlier in Kolm (1969).

Like Theil, Atkinson draws strongly on an analogy between probabilities in the theory of choice under uncertainty and income shares. Due to these analogies with probabilities, the theorem of Lorenz-dominance gave rise to the term ‘stochastic dominance’ (cf. Jenkins & Micklewright, 2007, p. 13). Since no probabilities are involved in the case of inequality measurement, this is — like the term ‘entropy’ — arguably an unfortunate and confusing name, which is nevertheless frequently employed.

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

47

Gini’s concentration ratio and Dalton’s measure of inequality are of course two answers to the same question: how can we rank intersecting Lorenz curves? But in contrast to Gini and his relative mean difference, Atkinson argues that we should directly specify W and U in order to obtain clear orderings. This approach, goes the argument, ensures that the normative elements in inequality comparisons are explicitly spelt out and — at least in theory, we may add — open to debate. And Dalton’s measure allowed for infinitely many specifications due to the free parameter c as we have noted earlier (cf. p. 34). While the free parameter in Dalton’s measure of inequality does not alter the order of distributions, different values of c make the levels of his statistic completely arbitrary, which is according to Atkinson a very inconvenient feature. If, for example, inequality in France according to Dalton’s measure is, say, 1.30, this value would have no meaning. It only gives us an idea about the extent of inequality in France if we compare it to a second numerical value calculated at the same level of the parameter c. In this respect, the Gini coefficient is more convenient since a value for France of 0.3, for instance, would contain some information independent from the Gini coefficient of other countries. Both the Gini coefficient and the Dalton measure therefore have disadvantages which Atkinson attempts to avoid.

How then does he narrow down the possible specifications of the function U (y) to one that allows to make judgements in case of intersecting Lorenz curves? Before discussing Atkinson’s specification, we need to introduce his notion of “equally distributed equivalent income”. This can perhaps be explained best with the example of a hypothetical two-person economy. Fig. 2.5 below shows the welfare indifference curves that correspond to the total welfare for different divisions of total income Y between the two individuals. Again, the closer the curves are to the upper-right corner, the higher the total welfare. If the initial distribution is the (unequal) point A, the level of welfare generated by this distribution is the one that corresponds to the welfare indifference curve Iw . Now, it is clear that the same level of welfare could be generated with a lower amount of total income than Y due to the distributional inefficiency of point A. As can be seen from this figure, the lowest amount of total income that could still generate the same level of total welfare is 2 × yede .

In fact, if each of the two individuals receives exactly yede as in the distribution C, total welfare would be unchanged compared to point A. In this case, yede is called the equally distributed equivalent income of the distribution A. On Fig. 2.5 we see that the average income µ of the distribution A is higher than the equally distributed equivalent income that corresponds to this distribution. On the other hand, if the distribution was the completely equal distribution of point C, the average income and the equally distributed equivalent income would coincide. This means that the average can never be smaller than the equally distributed equivalent income, which is a result from the concavity assumption in equation (2.20). Given this relationship, Atkinson defines his measure of inequality as a function of the ratio between average and equally distributed equivalent income: yede (2.21) A≡1− µ If, and only if, the average and the equally distributed equivalent income are identical, the index A equals zero. In this case, the distribution is completely equal. Complete inequality

48

CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE income indiv. 2 Y

µ yede

B C Iw

A 45˚

Y0

0

income indiv. 1

Figure 2.5: Illustration of Atkinson’s ‘equally distributed equivalent income’. corresponds to A = 1. For intermediate values it holds that the further the average income µ is from yede , the more unequal is the distribution and the higher the value of the index A. With the notion of an equally distributed equivalent income comes “an intuitive appeal”: “If A = 0.3, for example, it allows us to say that if incomes were equally distributed, then we should need only 70 per cent of the present national income to achieve the same level of social welfare [...]” (ibid, p. 250; notation harmonised with the text).

So far, the measure A only replaces Dalton’s “welfare if the current income was equally distributed” with Atkinson’s notion of “the equally distributed income that would generate an equivalent level of welfare”. The former defines the benchmark against which inequality is measured as the highest level of welfare attainable given the actual total income. The latter uses the lowest level of total income that yields a given level of welfare as a benchmark. These benchmarks are very similar, and they do not give an answer to the crucial issue: how do we compute the exact values of maximum attainable welfare or equally distributed equivalent income? This leads us to the specific form that Atkinson assumes for the function U (y).

The function U (y) enters the inequality measure A via the equally distributed equivalent income yede . The latter is defined as the income which, if equally distributed, would yield the same level of welfare than the actual distribution. Given the definition of total welfare in equation (2.19), this relation can be written as:

N U (yede ) =

N X

U (yi )

i=1

N 1 X U (yi ) U (yede ) = N i=1

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES Inverting the function U (y) gives us an expression for yede as a function of y: !

N 1 X â&#x2C6;&#x2019;1 U (yi ) yede (y) = U N i=1

49

(2.22)

So that the inequality index becomes: A=1â&#x2C6;&#x2019;

U yede (y) =1â&#x2C6;&#x2019; Âľ

â&#x2C6;&#x2019;1

P N 1 N

i=1 U (yi )

Âľ

(2.23)

If U (y) is specified and invertible, we can directly calculate the value of the inequality index. Besides the constraints on U (y) given in (2.20), Atkinson argues to opt for a functional form of U (y) which would make A insensitive to proportional shifts of the income distribution. In other words, if all incomes are multiplied by the same scalar Îť, yede (y) should also increase by the factor Îť. Since the mean of all incomes in this case would as well be multiplied by the same factor, A would remain unchanged. This is precisely the property of mean independence from proportional additions advocated by Lorenz (cf. p. 25) and later by Theil (cf. p. 44). We have argued that this restriction is by no means innocuous given the intuitive feeling shared by many people that if absolute income differentials increase, inequality can rarely be constant. What is Atkinsonâ&#x20AC;&#x2122;s rationale to impose this restriction? His answer to this question is unambiguous and he does not need extensive argumentation to justify the assumption of mean independence: â&#x20AC;&#x153;Now we have seen that nearly all the conventional measures are defined relative to the mean of the distribution, so that they are invariant with respect to proportional shifts. If we want the equally distributed measure to have this property, then [...]â&#x20AC;? (ibid, p. 257). The rationale for specifying U (y) so that A is insensitive to proportional shifts is thus entirely the product of conventions. Once this convention employed by Lorenz, Gini, Theil and now Atkinson is accepted, the possible specifications of U (y) are surprisingly limited. Atkinson argues that this narrows U (y) down to the Arrow-Pratt class of functions with constant relative risk-aversion. In the context of inequality measurement, this is analogous to requiring â&#x20AC;&#x153;constant relative inequality-aversionâ&#x20AC;?, which implies that U (y) has the form:20 y1â&#x2C6;&#x2019; if 6= 1 and â&#x2030;Ľ 0 1â&#x2C6;&#x2019; (2.24) U (y) = log y if = 1 To illustrate that this specification of U (y) yields an inequality index which is insensitive to proportional shifts, we can insert equation (2.24) into A (we stick to the case of 6= 1). To do so, we first have to insert equation (2.24) into yede as we derived it in (2.22). Inverting U (y) yields: 1 U â&#x2C6;&#x2019;1 (y) = [y(1 â&#x2C6;&#x2019; )] 1â&#x2C6;&#x2019; 20

In technical terms, other specifications of U (y) that render equation (2.22) homothetic would also have the desired property of making A mean independent. An example of such a function is U (y) = y , (0 < â&#x2030;¤ 1) used by Sen (1997, p. 128). However, the Arrow-Pratt class of functions with constant degree of relative risk aversion allows Atkinson to use the analogy between â&#x20AC;&#x2DC;constant relative risk-aversionâ&#x20AC;&#x2122; and constant relative inequality-aversionâ&#x20AC;&#x2122; we will explain below.

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CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

The equally distributed equivalent income can therefore be written as: !

N 1 X U (yi ) = N i=1

yede (y) = U â&#x2C6;&#x2019;1

N 1 â&#x2C6;&#x2019; X yi1â&#x2C6;&#x2019; N i=1 1 â&#x2C6;&#x2019;

1 ! 1â&#x2C6;&#x2019;

=

N 1 X 1â&#x2C6;&#x2019; y N i=1 i

1 ! 1â&#x2C6;&#x2019;

And the index A becomes:

A=1â&#x2C6;&#x2019;

yede (y) =1â&#x2C6;&#x2019; Âľ

P N 1

1â&#x2C6;&#x2019; i=1 yi

N

1 1â&#x2C6;&#x2019;

,

Âľ

6= 1

(2.25)

An equiproportional shift means multiplying all income by the same factor. If we multiply all yi by the factor Îť, the preceding equation becomes: P N 1 1â&#x2C6;&#x2019;

N

1 1â&#x2C6;&#x2019;

Îť1â&#x2C6;&#x2019; N

PN

1â&#x2C6;&#x2019; i=1 yi

P N 1 =1â&#x2C6;&#x2019;

Ν¾

=1â&#x2C6;&#x2019;

1â&#x2C6;&#x2019; i=1 (Îťyi )

1 1â&#x2C6;&#x2019;

P N 1 =1â&#x2C6;&#x2019;

N

i=1

N

yi1â&#x2C6;&#x2019;

Âľ

1 1â&#x2C6;&#x2019;

Ν¾ P N 1

=1â&#x2C6;&#x2019;

Ν¾

N

1â&#x2C6;&#x2019; 1â&#x2C6;&#x2019; yi i=1 Îť

1â&#x2C6;&#x2019; i=1 yi

1 1â&#x2C6;&#x2019;

1â&#x2C6;&#x2019;

Îť 1â&#x2C6;&#x2019;

Ν¾ 1 1â&#x2C6;&#x2019;

(2.26)

Since (2.25) and (2.26) are identical, the multiplication by Îť has no effect on the value of A. This is the proof that inequality as measured by the Atkinson index is insensitive to proportional shifts. A similar calculation leads to the specification of A in the case that = 1. We will skip the intermediate steps and give directly the corresponding expression for A. The index A for all positive values of is:  1 P  ( N1 yi1â&#x2C6;&#x2019; ) 1â&#x2C6;&#x2019; if 6= 1 and â&#x2030;Ľ 0 1â&#x2C6;&#x2019; Âľ (2.27) A= 1 P exp ( N log yi )  1â&#x2C6;&#x2019; if = 1 Âľ The advantage of Atkinsonâ&#x20AC;&#x2122;s approach is that it narrows down with relatively few assumptions all the possible specifications of A to the class defined by equation (2.27).

Yet, as was the case with Daltonâ&#x20AC;&#x2122;s specification, again a free parameter appears in the index. In order to apply A empirically, the parameter has to be specified. To make the choice of less arbitrary, Atkinson proposes the following interpretation: â&#x20AC;&#x153;In this case, the question is narrowed to one of choosing , which is clearly a measure of the degree of inequality-aversion â&#x20AC;&#x201D; or the relative sensitivity to transfers at different income levels. As rises, we attach more weight to transfers at the lower end of the distribution and less weight to transfers at the top.

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

51

The limiting case at one extreme is â&#x2020;&#x2019; â&#x2C6;&#x17E; giving the function mini {yi } which only takes account of transfers to the very lowest income group (and is therefore not strictly concave); at the other extreme we have = 0 giving the linear utility function which ranks distributions solely according to total incomeâ&#x20AC;? (1970, p. 257).

This interpretation of the parameter is based on the analogy between risk aversion and inequality aversion that Atkinson introduces. In risk theory, the in the Arrow-Pratt function (2.24) is the â&#x20AC;&#x2DC;degree of risk-aversionâ&#x20AC;&#x2122; and indicates someoneâ&#x20AC;&#x2122;s preference for certain â&#x20AC;&#x201D; as opposed to uncertain â&#x20AC;&#x201D; outcomes. The degree of risk aversion determines how much a certainty equivalent is preferred to a risky gamble. An example of such a choice would be an option between â&#x20AC;&#x2DC;getting 100 e for sureâ&#x20AC;&#x2122; and â&#x20AC;&#x2DC;getting 200 e or 0 e with 50% chance eachâ&#x20AC;&#x2122;. If the degree of risk-aversion is zero, i.e. = 0, these two alternatives yield the same level of utility. For positive levels of the certainty equivalent yields higher levels of utility than the risky gamble. The analogy between the â&#x20AC;&#x2DC;certainty equivalentâ&#x20AC;&#x2122; in risk theory and the â&#x20AC;&#x2DC;equally distributed equivalent incomeâ&#x20AC;&#x2122; in Atkinsonâ&#x20AC;&#x2122;s inequality measure transforms a preference for certainty into a preference for equality. Consequently, the parameter becomes the degree of this preference for equality or, in other words, the degree of inequality-aversion.

The relationship between the parameter and the level of A can be illustrated graphically.

In Fig. 2.6 below we have drawn welfare indifference curves for different values of . As can be seen, the indifference curve for = 0 is simply a straight line. For values above zero the indifference curves are convex, and the degree of convexity increases with the value we assign to . We have drawn several indifference curves in such a way that they cross the point A, which represents an unequal distribution of income between two individuals. As in our example above, this distribution A has the average income Âľ (the average income equals the distance between 0 and point B). In Fig. 2.6 we see that the higher the degree of inequality-aversion , the greater the distance between point B and the intersection of the indifference curves with the 45Ë&#x161;-line. Since it is this very intersection which indicates for each value of the equally distributed equivalent income, we see that the ratio between y ede and Âľ will be greater the higher the value of .

The Atkinson measure is therefore a welfare-based statistic of inequality derived from explicit and transparent assumptions (e.g. insensitivity to proportional increases) that provides a less arbitrary specification of the remaining free parameter (to be understood as â&#x20AC;&#x2DC;degree of inequality-aversionâ&#x20AC;&#x2122;). Atkinson argues that the possibility to allow for different values of is a key advantage of A over the Gini concentration ratio, since the latter also contains some preference for equality, but the extent of inequality-aversion in the Gini measure is not obvious and implicit in its formula. To illustrate these implicit preferences, Atkinson evaluates the degree of inequality-aversion of the Gini ratio by calculating its sensitivity to transfers from rich to poor. The remarkable result is that for the Gini ratio this â&#x20AC;&#x153;suggests that for typical distributions more weight would be attached to transfers in the centre of the distribution than at the tails [...]. It is not clear that such a weighting would necessarily accord with social valuesâ&#x20AC;? (ibid, p. 256). In a nutshell, Atkinson challenges the

52

CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE income indiv. 2

=1 = 0.75 = 0.5 = 0.25 =0

Âľ

B

A

45Ë&#x161; 0

income indiv. 1

Figure 2.6: Welfare indifference curves for different levels of inequality-aversion. dominant Gini ratio in two steps: first, he presented a welfare-based interpretation of the Lorenz curve with his Theorem; then, he shows that the â&#x20AC;&#x2DC;inequality-aversionâ&#x20AC;&#x2122; implicit in the Gini ratio is higher in the middle of the distribution than at higher or lower incomes.

The clear welfare-interpretation of his A and its constant degree of inequality-aversion at all income levels brought the debate on inequality measurement out of the shadow of Giniâ&#x20AC;&#x2122;s dominance. In many ways, Atkinson gave the start signal for a renewed interest in empirical inequality measures and inspired many researchers to develop and refine new welfare-based inequality statistics.

Impact on conventions Due to the prominent place of Atkinsonâ&#x20AC;&#x2122;s contribution in the recent literature on inequality analysis, it is particularly important to point out the impact of his approach on the body of conventions. Interestingly, it seems that an important consequence of Atkinsonâ&#x20AC;&#x2122;s measure was to consolidate ideas that had already been proposed by earlier authors. This is particularly obvious in his first important article on inequality from 1970 in which he twice refers explicitly to â&#x20AC;&#x153;conventionalâ&#x20AC;? methods to justify his approach. We think that five points are relevant for our discussion: 1. The Atkinson measure continues the tradition to use summary statistics to synthesise information on inequality into a single number. The reason to do so seems to be

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

53

purely conventional: â&#x20AC;&#x153;The conventional approach in nearly all empirical work is to adopt some summary statistic of inequality such as [...]â&#x20AC;? (ibid., p. 244).

2. Again with explicit reference to conventions, Atkinson explains why he assumes that proportional increases of incomes should leave inequality unchanged : â&#x20AC;&#x153;Now we have seen that nearly all the conventional measures are defined relative to the mean [...]â&#x20AC;? (ibid., p. 257). With respect to this key assumption, he appropriates the legitimacy that comes with conventional usage and positions his index A in line with Lorenz, Gini, Theil and others. This is a very relevant point since the assumption of mean independence allows Atkinson to narrow down the possible functional forms of his measure to the specific expression of A with a single free parameter we exposed above. Without mean independence, it would be considerably less obvious how to obtain numerical values for A (cf. Kolm, 1976).

3. By placing his approach in the Daltonian welfare framework, Atkinson could present his measure as a continuity to an older â&#x20AC;&#x201D; almost classic â&#x20AC;&#x201D; contribution in inequality measurement. Atkinson consolidated a convention that inequality should be defined and evaluated in terms of welfare: â&#x20AC;&#x153;it seems more reasonable to approach the question directly by considering the social welfare function that we would like to employ rather than indirectly through the summary statistical measuresâ&#x20AC;? (ibid., p. 257).

Through this stance, Atkinson directed the academic discourse further away from direct descriptive approaches to inequality measurement and toward the more indirect method of evaluating â&#x20AC;&#x2DC;distributional badnessâ&#x20AC;&#x2122;.

4. All of the three preceding points illustrate that it is safe to say that Atkinson prolonged several pre-existing conventions in inequality measurement. However, he also added a new requirement to the list of desirable features that summary statistics of inequality should have. He argued that a measureâ&#x20AC;&#x2122;s sensitivity to transfers from rich to poor should vary according to the place in the income distribution: transfers at the lower end should decrease inequality more than transfers between two individuals that are already rich. The situation of a millionaire who passes a sum d to another slightly less well-off millionaire should decrease inequality by less than if d is transferred at subsistence level. The rationale for this additional requirement was Atkinsonâ&#x20AC;&#x2122;s intuition that constant relative transfer sensitivity â&#x20AC;&#x153;is unlikely to command wide supportâ&#x20AC;? (ibid., p. 58). This was an important extension of Daltonâ&#x20AC;&#x2122;s simpler â&#x20AC;&#x2DC;principle of transferâ&#x20AC;&#x2122; and has since become a conventional requirement for inequality measure to be acceptable (cf. Kolm, 1976).

5. Related to the two preceding points on this list is the observation that Atkinsonâ&#x20AC;&#x2122;s approach led to a complexification of the analytical tools of inequality measurement.

This complexification can easily be felt if one attempts to lay out the conceptual foundation of the index A to a non-specialist: not only is it necessary to explain the framework of utilitarian welfare maximisation. For complete comprehension,

54

CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE also the notions â&#x20AC;&#x2DC;equally distributed equivalent incomeâ&#x20AC;&#x2122; and â&#x20AC;&#x2DC;constant relative riskaversionâ&#x20AC;&#x2122; have to be exposed in order to make the core features of A clear to the nonspecialist. Probably the explanation would have to involve a detour via the theory of choice under uncertainty that we have chosen above. Complexification is obviously not objectionable per se, but it reveals that Atkinson prioritizes theoretical over practical issues as the following quote from his article illustrates: â&#x20AC;&#x153;Much of the early literature was in fact concerned with the problem of choosing between the different summary measures, and such properties were discussed as ease of computation, ease of interpretation, the range of variation, and whether they required information about the entire distribution. However, [...] the central issue clearly concerns the underlying assumption about the form of the social welfare function that is implicit in the choice of a particular summary measureâ&#x20AC;? (ibid., p. 253). We cannot state more eloquently that in Atkinsonâ&#x20AC;&#x2122;s list of priorities â&#x20AC;&#x2DC;ease of interpretationâ&#x20AC;&#x2122; clearly ranks behind other considerations of more theoretical nature.

All these points touch core issues of our problem of measuring inequality within the IEWB. We already mentioned repeatedly our intuition that insensitivity to proportional increases of all incomes (or â&#x20AC;&#x2DC;constant relative risk-aversionâ&#x20AC;&#x2122; as Atkinson calls it) might not be an innocuous assumption. We have argued that Atkinson relies without explicit argumentation on several conventional methods and we will have to analyse in Chapter 3 whether we are as willing as he was to adopt these conventions at our turn.

Next, Atkinsonâ&#x20AC;&#x2122;s impact brings up an important arbitrage between theoretical completeness and internal coherence, on the one hand, and â&#x20AC;&#x2DC;ease of interpretationâ&#x20AC;&#x2122; and communicability, on the other. Of course, this arbitrage is not a necessary evil of scientific work since internal coherence and external communicability are not per se opposed. But the nature of the progress in analytical methods that resulted from Atkinsonâ&#x20AC;&#x2122;s contribution â&#x20AC;&#x201D; and especially the cross-fertilization between risk theory and inequality analysis â&#x20AC;&#x201D; seems to indicate that this arbitrage might be an important obstacle for a transparent debate on inequality measures. In a way, Atkinsonâ&#x20AC;&#x2122;s approach contains a profound dilemma: he wants to make normative values more explicit and transparent because â&#x20AC;&#x153;this approach allows us to reject at once those that attract no supportersâ&#x20AC;? (ibid., p. 257); but this very strategy complexifies his measure to such an extent that potential supporters might not be able to step up and defend their normative values. The parameter of risk-aversion might be a case in point since few people could actively participate in a debate on the degree of concavity of the social welfare function. Hence, more and more actors might be excluded from a debate increasingly dominated by technical specialists. As with any dilemma, we might have to chose the smaller evil; in our discussion of inequality measures for the purpose of being a heuristic tool in public debate, we might argue that communicability is more important than technical completeness.

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

2.1.7

55

Sen’s conceptual tour de force

The influential book On Economic Inequality synthesised almost all important issues concerning the empirical measurement of inequality and proposed an important methodological critique. Jenkins and Micklewright refer to it as “Amartya Sen’s conceptual tour de force” (2007, p. 1). Many of Sen’s ideas found their way into mainstream debates. The UNDP Human Development Index, first published in 1990, is a prominent example of Sen’s influence since it contains elements of Sen’s approach. Perhaps the deepest impact of Sen’s critique was the tabula rasa question “Inequality of what?”, which was further developed in Sen (1992). However, from the outset of the present text we excluded this very question from our analysis. Since we have chosen to operate within a predefined framework which already answers the question of what should be counted — the Index of Economic Well-Being — Sen’s more general investigation around the question “Inequality of what?” is arguably less relevant for our current purpose. Nevertheless, On Economic Inequality, combined with the revised and extended edition Sen published with J. E. Forster in 1997, contains an array of conceptual interrogations that touch directly on our preoccupations.

We will therefore present only some of the features of Sen’s approach on the measurement of inequality, and remind ourselves that this necessarily amputates many important results.

We incite the reader to compare or complement our account with the complete texts of Sen (1973, 1992, 1997).

For us, perhaps the most decisive characteristic of Sen’s approach lies in his intermediate position between descriptive or “objective” inequality measurement, on the one hand, and normative or welfare-based assessments, on the other. Sen argues that the concept inequality has a dual character, blending a descriptive element (‘a cake eaten by two individuals is divided into two equal parts’) and ethical judgements (‘this division is good since it yields equal welfare’). When discussing issues like income inequality, the argument goes, the objective and the ethical elements are intertwined: “[...] in some complex problems [...], it becomes very difficult to speak of inequality in a purely objective way, and the measurement of the inequality level could be intractable without bringing in some ethical concepts” (Sen, 1973, p. 3). Sen’s intermediate position between descriptive and normative measurement leads to a dual constraint on statistical representations. Inequality statistics do not only have to take ‘ethical concepts’ and people’s values into account, they also have to correspond to what is thought to be an ‘objective’ way to describe the factual state of economic inequality. This is a more restrictive requirement than a purely normative approach based on the evaluation of total welfare, which does not explicitly require objective description. In fact, a welfare approach on inequality does not necessarily correspond to an ‘objective’ description of reality: a perfectly equal fifty-fifty division of a cake may not correspond to equal welfare: it suffices to chose different individual welfare functions. In this case, equality in the normative sense would correspond to inequality in the descriptive sense. In other words, equality of utility from cake consumption does not necessarily mean equality of cake slices. Sen argued that inequality measures must integrate both of these elements if they are to be relevant. A statistic that defines equality contrary to the ‘objective’ or descriptive notion of equality is according to Sen not usable: “In one way

56

CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

or another, usable measures of inequality must combine factual features with normative ones” (ibid., p. 3). The dual constraint that Sen poses on equality metrics introduces an element to the academic discussion that hitherto had been ignored by most writers: the new element is “normal communication”, which we have already borrowed for our purposes in the introduction. Sen repeatedly refers to the importance to match academic and normal communication, i.e. the non-expert language used outside technical models on welfare inequality. Consequently, his text is coloured with allusions to a ‘common sense idea of inequality’ of which we will cite only some examples: “our conception of inequality” (p. 3); “the sense in which the word is used in normal communication” (p. 39); “the meaning associated with the term” (p. 47); “in terms of definitions corresponding closely to the normal usage of the term inequality” (p. 62); “in normal communication both the normative and the positive aspect can be observed in the use of the concept of inequality” (p. 63); “such a measure seems also to be reasonably close to the non-technical concept of inequality as employed in normal communication” (p. 72); “our standard descriptive understanding of inequality may conflict sharply with the ‘normative measurement’ of inequality” (p. 119).

These quotations illustrate that Sen believes in a relatively clearly defined ‘normal meaning’ of the concept inequality, and makes frequent use of this ‘normal meaning’ as an argument in favour or against features of inequality statistics. In fact, he advocates that the statistics should be “reasonably close” to the conception of inequality in normal communication.

We will argue below in Chapter 3 that this issue may be less obvious than Sen’s usage of the term ‘normal communication’ suggests. It may be worthwhile to analyse if what Sen refers to as ‘normal’, ‘standard’, or ‘non-technical’ corresponds indeed to widely-accepted conceptions of inequality or if it is limited to the internal academic discourse.

Besides this insistence on a descriptive and common sense approach, Sen’s method of evaluating inequality rests thoroughly welfarist. Like Dalton and Atkinson, Sen defines inequality in terms of a welfare function, so that much of the discussion is centered around the question which functional specification of welfare is most convenient and reflects best normative values. In this context, Sen calls for a radical departure from the utilitarian framework of maximising the simple sum-total of identical individual utilities. In our discussion of Dalton’s measure, we already indicated that a “very special coincidence” leads to the fact that equality is associated with maximum welfare (see Section 2.1.4 on p. 32). Since inequality is not a genuine concern but rather a coincidental byproduct of the simple utilitarian framework, Sen proposes to reject it altogether and replace it with a relation between individual incomes and total welfare that directly and explicitly integrates the preference for equality.

A second mise en question is Sen’s stance toward economic inequality as a relative concept, i.e. the idea that inequality measures should remain unchanged if all incomes are multiplied by the same number. This point is somewhat blurred in On Economic Inequality since the argument is structured around the term ‘mean independence’, without distinguishing between proportional and equal additions to income like Dalton had proposed in his list of principles (see p. 35). However, it is quite clear that Sen recognizes the difficulties involved in the assumption of mean independence: “We are caught in a bit of a dilemma here. Making inequality measures independent of the mean income seems

2.1. FROM CONSTANT INEQUALITY TO COMPLEX INEQUALITIES

57

objectionable, but no alternative general assumption about the relationship of the mean income to these measures seems to be acceptable at all” (ibid., p. 71). However, in the expanded edition of his book in 1997, Sen seems to have chosen his camp with respect to mean independence from proportional increases: “What happens if welfare is not homothetic? We lose the property of mean independence in the normative inequality measure, and this can introduce an ‘absolutist’ element in what is standardly thought of as being a relative concept (that of inequality)” (Sen, 1997, p. 128).21 Finally, another relevant element of Sen’s theoretical observations is that completeness may not be a reasonable characteristic of inequality measures. This means that a statistic that generates an unambiguous and complete ranking of all possible income distributions may not accurately reflect the concept of inequality: “[...] inequality as a notion does not have any innate property of ‘completeness’ ” (ibid., p. 47), and measures which generate complete rankings of all distributions display more precision than the notion of inequality itself. If inequality is a ‘fuzzy’ or ‘incomplete’ concept, we may often be faced with comparisons not allowing for a clear judgement — perhaps not even the clear judgement that two distributions are equally unequal. This is an important point since all summary measures we have discussed in this text share the feature of generating complete orderings of distributions in terms of their extent of inequality. If the concept inequality is ‘incomplete’, these measures contain some artificial precision and fill in the evaluative void with arbitrariness.

By the same token, it follows that if two summary measures, say the Gini and the Theil statistics, contradict each other in their assessment of the same distributions, this is not necessarily in discord with the concept of inequality. It could merely be a signal that the underlying comparison does not allow for a clear decision given the fuzzy character of the inequality notion itself. The hypothesis of incompleteness of inequality gives rise to Sen’s meta-measure which he calls “intersection quasi-orderings” (ibid., pp. 72-74). In a nutshell, this meta-measure tries to turn the cacophony of partially contradicting inequality statistics into a polyphony in which only the harmonic and corroborative elements obtain a voice. The more the different measures included conflict among each other, the more the meta-ordering is incomplete and indicates the existence of arbitrary decisions if each of the measures was to be relied upon in isolation. In technical terms, one first defines a set of a priori plausible inequality orderings C j , for j = 1, ..., k. The intersection of these k complete orderings is denoted Q and can be written as: yQx

if and only if

∀j = 1, ..., k : yC j x

(2.28)

In words, the distribution y is ordered higher than the distribution x if this ordering holds for all C j . If, for instance, we define an intersection quasi-ordering by limiting the plausible measures to the Gini concentration ratio and the interdecile ratio D9 /D1 , the ordering Q would only allow us to rank distributions if the ordering C G of the Gini measures does not contradict the ordering C ID of the interdecile ratio. While the intersection quasi-ordering cannot entirely eliminate the arbitrariness — the choice of the measures to be included in 21

Sen’s remark cited above refers to the Atkinson measure in which homotheticity of the equally distributed equivalent income function implies mean independence; cf. the footnote on page 49.

58

CHAPTER 2. AN INTERNAL HISTORY OF THE ACADEMIC DISCOURSE

Q is obviously a partially arbitrary one — it has the merit of indicating the situations in which simple summary statistics may be misleading. However, the ordering Q is in some ways simply a sophisticated version of a much older incomplete ordering which we discussed earlier on. We have seen that the Lorenz curve only generates complete orderings if the concentration curves do not intersect. Intersecting curves may consequently be interpreted as a sign for ‘blanks’ in our decision-making ability. In light of this similarity between Q and Lorenz comparisons and the remaining arbitrariness of the choice of the orderings C j , it is unclear whether Q adds much practical value over the more intuitive Lorenz curve.

However, if one disagrees with the concept of inequality embedded in the latter, then Sen’s intersection quasi-orderings are indeed an attractive alternative.

Impact on conventions We already noted that some of the main results of Sen’s treatment of inequality are derived from his interrogations around the question “inequality of what?” which led to the focus on individual capabilities instead of incomes. These results are beyond the reach of our research questions and have therefore a negligible impact on our discussion. Nevertheless, Sen’ contribution to the measurement of inequality remains highly relevant for us. We want to stress two points: 1. Sen introduced the notion of normal communication and the common sense definition of inequality to the academic discourse. He insisted on the requirement that the scientific measurement of inequality should be “reasonably close” to normal conceptions and thereby imposed a constraint that had hitherto been neglected or only laxly taken into account. However, while Sen refers to ‘normal communication’ as if it is something obvious and easily definable, it seems to be less clear how to interpret what normal usage constitutes and how the coherence between scientific and ‘normal’ language can be verified. Kolm (1976) picked up this point and quotes from personal conversations with friends, Sen himself quotes what he “has heard” in non-technical discussions (ibid., p. 70).

2. The notion of ‘incompleteness’ is an important hypothesis. It calls for prudence not to fully rely on any single summary measure of inequality, no matter how precise or convenient this statistic may be. For if the concept itself is too fuzzy to allow for clear-cut decisions, an empirical measure derived from this very concept can hardly attain a higher degree of accuracy than the concept itself. However, even if the hypothesis of incompleteness is widely discussed in the theoretic literature, the conventional approach in empirical measurement still illustrates that many authors chose precision in preference to partial orderings.

Since communicability is one of the key concerns for a heuristic tool like the IEWB, Sen’s frequent allusions to normal communication are close to our research question. However, the insight that economists ‘are not really free to define inequality arbitrarily’ is for us only a necessary first step. We also need to analyse more precisely how to decide when

2.2. RECENT DEVELOPMENTS: GENERALISATION OF METHODS

59

a scientific definition of inequality is “reasonably close” to normal communication. And, most importantly, we need a more systematic way to ‘read’ and interpret the normal usage of inequality if our measure of inequality within the IEWB is to be useful for public debate.

We have argued that Sen uses allusions to normal communication merely as a constraint to the rather sophisticated usage of the term inequality in the welfare evaluations he discusses or develops. Considering common sense usage as something obvious and given, he does not see the necessity of an efficient dialogue between the two spheres. Implicitly, for Sen it is rather the economist who casually observes normal conversations who creates the link between academic and non-academic discourse. We submit that this vision ignores the problem of missing feedback loops due to the technical complexity of the analytical tools involved in the state-of-the-art in inequality measurement. Again, we observe that inequality statistics are mostly constructed by technical specialists — who assign a central place to notions like ‘third order stochastic dominance’ or ‘quasi-concavity’ in their highly developed cognitive apparatus — rather than being co-constructed between experts and users. In light of these observations, we may argue that Sen’s intersection quasi-ordering Q is an analytically attractive innovation, but it is less clear whether it enhances the transparency of inequality measurement as a heuristic for public debate.

2.2

Recent developments: generalisation of conventional methods

Sen’s “conceptual tour de force” will be the last contribution we discuss in detail. To conclude from this that the internal history of empirical inequality measurement has come to an end after the publication of the extended edition of On Economic Inequality in 1997, or even after the first edition in 1973, would be wrong. In the past four decades, inequality measurement underwent considerable mainstreaming effects, with many new and important research fields entering the literature. However, we argue that the most important parts of the standardly used conceptual framework — i.e. the basis of conventional methods — in the field of income and wealth inequality has probably been shaped in the period between Pareto’s La courbe de la répartition de la richesse and Sen’s On Economic Inequality. More recent contributions have, by and large, focused on extensions, refinements or improvements of existing approaches. This does of course not mean that these contributions are less useful or less innovative. But for our question of measuring inequality in the framework of the IEWB, it was more important to analyse the foundations of today’s state-of-theart measurement than the current form of the methods. The impressive development of analytical methods in the last decades was only possible because they could refer to a more or less coherent body of conventions, and it was the genesis of this body of conventions that we have tried to sketch in the present chapter.

Before we continue, we should stress that what we said in the preceding paragraph holds only for our research problem of measuring inequalities within the IEWB framework, and not for inequality analysis in general. In fact, in the wider field of inequality analysis

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important conceptual changes occurred so that the conventional methods have been altered significantly in many ways. The increasing use of multi-dimensional inequality measures is perhaps the most influential development and modified our perception of inequality in general. Different influences — many of which emanating from very diverse actors including civil society, governmental agencies and researchers — have led to an expansion of the uni-dimensional monetary view on economic inequality and toward a perspective in which differences in other spheres like social status, education, working conditions, access to health services or other utilities etc. are also taken into account. In France, the Réseau d’Alerte sur les Inégalités regularly voices concerns regarding the predominance of monetary measures and presented an alternative inequality measurement called the Baromètre des inégalités et de la pauvreté, or BIP40. To a certain extent the Human Development Index developed by the UNDP is a result of similar interrogations. Lars Osberg and Andrew Sharpe’s Index of Economic Well-Being, which we used as point of departure for the present text is another example of multidimensional approaches. Our research questions are thus a result from this shift toward an extension of the evaluation space and the use of multi-dimensional measurement.

However, within the more restricted field of income and wealth measurement, a process of consolidation and improvement of conventional approaches seems to have marked the recent decades. This can be seen in the extremely useful overview on inequality analysis by Jenkins & Micklewright (2007). While these authors underline the importance of the above mentioned tendency toward multidimensional approaches to inequality, most of the recent contributions to the field of income and wealth inequality they discuss have been of more technical than conceptual. In terms of their conceptual frameworks, recent contributions are all more or less based on the ideas developed by Lorenz, Gini, Theil, Atkinson and Sen. 1. The Lorenz curve and Atkinson’s Theorem have been extended into generalised Lorenz dominance and stochastic dominance by Shorrocks (1983), Foster and Shorrocks (1987) and others.

2. Atkinson’s measure has been generalised and gave birth to the “Atkinson familiy” (Jenkins & Micklewright, 2007, p. 13). The derivation of classes of parametric summary indices with explicit normative characteristics that Atkinson proposed was systematized and extended.

3. Theil’s decomposable index has inspired the generalized entropy class of inequality measures, developed by Bourguignon (1979) and others.

4. Other developments, such as the systematic treatment of sampling errors and the derivation of confidence intervals for inequality measures proposed by Beach and Davidson (1983), were also directly based on Lorenz and generalized Lorenz curves.

5. Paglin’s (1975) critique not to confuse intra-family and inter -famliy inequality was essentially an extension of the Gini concentration ratio and the Lorenz approach.

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This list suggests that the recent developments in empirical inequality measurement of wealth and income have focused on the improvement of analytical methods. This progress is a consequence of the fact that older contributions, above all the Lorenz-Gini concentration measure and the Dalton-Atkinson welfare approach, have become accepted and legitimate conventions. Due to the generalisations and extensions that occurred in recent decades, it is safe to say that the exercise of inequality measurement today is a much more complex and technical undertaking than it has ever been before.

2.3

Closer to the ‘truth’ or further away from ‘normal communication’ ?

In this chapter we have presented the scientific contributions that share the property of having influenced significantly the way in which inequality is traditionally measured in economics. We have tried to assess their impact by naming explicitly the conventions that appear to be most relevant for our purpose of measuring inequality in the framework of the IEWB. The nature of the IEWB implies necessarily to leave the internal debate within the scientific community and bring in more external considerations such as ‘who are the users of inequality measures?’ and ‘how do these users think about inequalities?’.

For if the IEWB in general, and its inequality dimension in specific, is to be useful and legitimate, the output of the internal debate on inequality measurement has to correspond to its external usage. From the elements we have presented in this chapter we can already identify some general problems that arise from the confrontation between internal scientific discourse and external usage.

The arbitrage between communicability and analytical completeness There is an important arbitrage between the purity and completeness of the scientific treatment of economic inequality, on the one hand, and the communicability and transparency of the resulting statistics, on the other. Due to the analytical progress over the past hundred years this arbitrage has become more and more uncomfortable and it is difficult to verify the coherence of the scientific discourse with normal communication. The shift from purely descriptive inequality analysis to welfare-based representations — initiated by Dalton and continued by Atkinson, Sen and others — has largely contributed to this complexification.

In order to communicate welfare measures, the underlying welfare framework has to be explained in more or less detail. This is particularly obvious in the case of parametric indices such as the Atkinson measure since the notion of inequality-aversion it contains can hardly be understood without some insights into the theory of welfare functions, including the degree of concavity of indifference curves at different levels of welfare. The Atkinson measure is also a good example of the paradox involved in the process of improving inequality measurement in economics. As a matter of fact, the rationale of the systematic derivation of parametric welfare functions is to produce inequality statistics that correspond closely to normative values of society. In a way, the objective of improvements in the analytical

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apparatus has been to come closer to the ‘true’ concept of inequality given societal representations. However, we note that this very objective has led to the exclusion of more and more actors from the debate on inequality and the construction of inequality indicators.

For example, it is hard to verify whether the idea that welfare indifference curves display a constant degree of convexity and are all radial copies of each other actually corresponds to inequality in the common-sense usage of the term. Osberg phrased the general problem faced by economists very eloquently: “Public debate might well be improved if we could consider explicitly some of the aspects of economic well-being [...], but public debate will not be assisted by an incomprehensible deluge of esoteric statistics” (1985, p. 73). To make it clear: we do not cast any doubt on the sincerity or even the accuracy of the scientific results we have discussed in this chapter. But economics arguably stands to gain by raising awareness of the arbitrage we mentioned above.

The advantage of a descriptive approach In light of this arbitrage, the distinction between descriptive and normative measures of inequality becomes more relevant again. The explicit inclusion of normative values almost mechanically increases complexity and decreases communicability. The more we want to emphasise the interaction between internal and external spheres, the stronger will be the case for employing purely descriptive measures — even if these include some implicit value judgements. These implicit values could be the kind of sacrifice we have to make by moving away from analytical completeness toward external communicability.

Is inequality a relative or an absolute concept?

Given the difficulty to verify whether the complex scientific ‘truth’ corresponds to normal communication, we are ipso facto in an uncomfortable position when we want to analyse if current academic conventions are acceptable for the purpose of the IEWB. However, we intuitively feel that some of the more fundamental conventions require further attention.

The convention contained in the Lorenz curve, the Gini concentration, the Theil measure and the Atkinson index of assuming that only the ratios of incomes, and not absolute differences, are ‘inequalities’ seems questionable. As a matter of fact, the emphasis on absolute income differences is nothing new. Kolm (1976) and Blackorby & Donaldson (1980) have already stressed the plausibility of this alternative point of view. Interestingly, these contributions are extremely technical — Kolm employs a mathematically elegant approach with a rigorous axiomatic — and did not significantly alter the standard body of conventions (which is also why we excluded them from a detailed analysis in this chapter).

We will present further arguments in favour of an absolute element in inequality statistics in the next chapter.

Chapter 3 Inequality measurement within the IEWB framework 3.1

A brief introduction to Osberg and Sharpe’s Index of Economic Well-Being

The Index of Economic Well-Being was above all conceived as an instrument for public debate. In his first article on the topic from 1985, Lars Osberg explicitly mentions the Royal Commission on Economic Prospects, a Canadian body charged with the assessment of economic policies of the government, as an institution that is concerned with evaluations of overall economic well-being and therefore a potential user of the IEWB. In another example, Osberg derives the necessity of a synthetic indicator from the need to evaluate the performance of politicians (he mentions Ronald Reagan, who asked his electorate in 1980 the question ‘Are you better off today than you were four years ago?’; ibid., p. 49).

An explicit objective of the IEWB is to present an alternative to Gross Domestic Product (GDP) as measure of economic welfare, mainly because “national income accounting measures may sometimes not agree with popular perceptions of trends in economic wellbeing” (Osberg & Sharpe, 2005, p. 311-312). As a matter of fact, the correspondance between normal communication and statistical representations appears repeatedly in the contributions of the two Canadian authors. In fact, the IEWB is presented as a measure closer to “popular perceptions”.

Furthermore, Osberg and Sharpe stress that the IEWB is not a single objective number: “It is more accurate, in our view, to think of each individual in society as making a subjective evaluation of objective data in coming to a personal conclusion about society’s well-being” (ibid., p. 313). The authors conceive their statistical tool to be useful for both public administration staff and common people. They argue that the IEWB can help individuals to make informed choices: “Citizens are interested in evaluating the well-being of their country, partly because all adults are occasionally called upon, in a democracy, to exercise choices 63

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CHAPTER 3. REVISION OF INEQUALITY IN THE IEWB (e.g. in voting) on issues that affect the collectivity (and some individuals, such as civil servants, have to make such decisions on a daily basis). [...] Hence, although self-interest may play some role in each individual’s evaluation of societal outcomes, citizens have a number of reasons to ask questions of the form: ‘Is my country better off ’?” (ibid., p. 313).

The authors clearly point out that “the purpose of index construction should be to assist individuals — e.g. as voters in elections and as bureaucrats in policy making — in thinking systematically about national outcomes and public policy” (ibid., p. 314). The user of the IEWB should therefore not be thought of as a technical specialist using well-being statistics as material for sophisticated scientific analysis, but rather as a non-expert (the “citizen”) looking for an informed vision on overall well-being.

Genesis of the IEWB The founding-stone of the house that was to become today’s index of economic well-being was already laid by Lars Osberg in 1985. This article contained a concrete framework of variables divided into four categories, although no empirical application was included at the time. The variables retained on a preliminary list were selected in terms of data availability and contextual fit. In the last twenty years, this framework underwent only minor modifications and is hence to a large extent identical with the presentation in the next section.

Despite the concreteness and applicability of Osberg’s measurement tool, which includes no formal model but a weighted index of components assumed to represent societal wellbeing, no implementation was published for 13 years. Osberg & Sharpe (1998) eventually presented the first index of economic well-being for Canada, covering the period from 1971 to 1997. Two years later, Osberg & Sharpe (2001) tabled an index for the United States. Osberg & Sharpe (2002) extended the framework to allow for a comparison of seven countries: Australia, Canada, Germany, Norway, Sweden, UK, and the USA for the period from 1981 to 1996. An updated and slightly modified version for this set of countries can be found in Osberg & Sharpe (2005). This last version recommends the use of the IEWB as substitute of per capita GDP in the UNDP’s Human Development Index (HDI).

To this end, the authors slightly modify their scaling technique and apply a logarithmic conversion to all sub-indicators. In addition to these societal IEWB, Osberg and Sharpe proposed applications of the general four dimensional framework to the labour market for North America (Osberg & Sharpe, 2001) and 16 OECD countries (Osberg & Sharpe, 2003).

The IEWB has received international attention in debates on social well-being (cf. OECD, 2002).

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Overview of the IEWB methodology The IEWB consists of four main dimensions which in turn are set up by a varying number of variables.1 In all, the full fledged indicator as in the seven-country version requires 38 different variables . The four dimensions, as stated in Osberg & Sharpe (2005), are: “(1) Effective per capita consumption flows — which differ from the consumption of marketed goods and services included in GDP by including the value of government services and adjusting effective per capita consumption flows to account for household production, changing household economies of scale, leisure and life expectancy.

(2) Net national accumulation of stocks of productive resources — which adds net changes in the value of natural resources stocks, environmental costs, net change in level of foreign indebtedness, net accumulation of human capital and R&D investment to the net investment in tangible capital and housing stocks now measured in GDP. (3) Income distribution — the intensity of poverty (incidence and depth) and the inequality of income.

(4) Economic security — from the financial implications of job loss, illness, family break-up and from poverty in old age.”

The different components of the IEWB often blend very distinct concepts. For instance, the index includes such different variables as average household size and the level of net foreign indebtedness. Few people would argue that these variables are direcetly comparable. Nevertheless, Osberg and Sharpe put forward the proposition that economic decision making and policy evaluation necessarily boils down to “ ‘adding it all up’-across domains that are conceptually dissimilar”. This procedure of “adding it all up” does not mean that we leave the grounds of an economic, material analysis of well-being. In Table 3.1, as well taken from Osberg & Sharpe (2005), each of the four IEWB dimensions is linked to a time period and a conceptual view.

Concept ‘Typical citizen’ or ‘representative agent’ Heterogeneous citizens

Time period Present Future Average flow of current in- Aggregate accumulation of come productive stocks Distribution — income in- Insecurity of future income equality and poverty

Table 3.1: Concepts in the IEWB — Source: Osberg & Sharpe (2005).

There is no homogeneous method of compiling the variables inside each dimension. In some cases multiplicative indices are used, in others weighted averages or simply sums of 1

The description of the IEWB methodology in this section is necessarily superficial. For more detailed explications the interested reader is referred to Osberg & Sharpe, 2005.

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the underlying variables. In addition, each variable has its own definition. In most cases the conventional nature of this definition does not require a detailed description (e.g. the variable personal consumption per capita simply uses the definition specified by the national accountancy). However, other variables were introduced by Osberg and Sharpe and their interpretation requires more information (e.g. the computation of the value of leisure involves information such as average wages, average numbers worked, national hours of unemployment etc.). In order to provide a transparent view on the overall indicator and to offer a glimpse at its complexity, the following figures illustrate how the four sub-indicators are created. The graphs are only horizontally exhaustive, in the sense that more vertical levels could be included to depict in more detail the construction of all variables.

Personal consumption p.c.

*

Index of household size

+

Value of leisure

+

Govern. spending (goods & services)

*

Index of life expectancy

=

Real total consumption p.c.

Figure 3.1: Component 1 — consumption flows.

The first IEWB component is a measure of effective per capita consumption, whereas adjustments are made to take into account household economies of scale, the value of leisure, individual consumption of government spending and life expectancy.

The second component — real stocks of wealth per capita — includes a monetarised variable measuring the negative externality of economic activity on the environment. Osberg & Sharpe (2005) proposed to estimate the social cost of greenhouse gas emissions in wealth by multiplying total emission with a fixed cost per ton.

Component 3 (equality and poverty) is constructed exclusively with three variables and is itself a synthetic indicator of monetary equality and poverty. The product of poverty rate and poverty gap ratio is referred to as ‘poverty intensity’. It should be noted that this measure could be replaced by the modified Sen-Shorrocks-Thon index as discussed by Osberg & Xu (2000), thus integrating besides the poverty rate and the (average) poverty gap ratio also the inequality of poverty gaps .

Component 4 (economic security) includes as well certain poverty rates and gaps (Fig. 3.4). However, it does not serve the purpose of measuring the poverty itself, but rather the risk to economic security borne by economic agents. The IEWB proposes to

3.1. INTRODUCTION TO THE INDEX OF ECONOMIC WELL-BEING Real capital stock p.c.

67

+

Real R&D stock p.c.

+

Human capital stock p.c.

+

Real net foreign debt p.c.

+

Real social cost of CO2 p.c.

=

Stocks of wealth

Figure 3.2: Component 2 — stocks of wealth.

0.75*

Poverty intensity

+

Gini 0.25* coefficient of incomes =

Equality and poverty

Figure 3.3: Component 3 — equality and poverty.

identify only some relevant economic risks and trace their evolution, instead of measuring and compiling all conceivable risks. It should be noted that this notion of security is — analogous to the monetary poverty rate — not the ‘true’ security that agents enjoy.

It is merely a proxy of the material risks that arise from uncertain factors like illness, unemployment or divorce.

Once the evolution of the four basic components is known, a weighting scheme allows aggregating them into a single number: the index of economic well-being. The weights attached to each dimension are ex natura rei arbitrary. Osberg and Sharpe have argued that a transparent aggregation is essential so that each user can modify the weights according to her preferences and to help consensus-building on societal preferences. The two weighting schemes currently in discussion are 1) equal weights for all sub-components, and 2) a more prominent impact of consumption, namely 0.4 × consumption flows + 0.1 × wealth stocks + 0.25 × equality + 0.25 × security.

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CHAPTER 3. REVISION OF INEQUALITY IN THE IEWB (Risk from unemployment) *(P1 /P )

+

(Risk to financial security from illness) *(P2 /P )

+

(Risk from single parenthood poverty) *(P3 /P )

+

(Risk from poverty in old age)*(P4 /P )

=

Economic Security

Figure 3.4: Component 4 â&#x20AC;&#x201D; economic security. Legend: P1 = Population aged 15-64; P2 = all persons; P3 = married woman with children; P4 = population aged 45-64. P = P1 + P2 + P3 + P4 .

3.2

Four dimensions, three inequalities

In the preceding section we have presented the IEWB in its original form proposed by Osberg and Sharpe. We emphasised our opinion that the Index is a useful tool for public policy analysis, which motivated our application of the IEWB to French data in an earlier communication (op. cit.). Nevertheless, we noted an inconsistency in its internal structure that arises from the way in which the IEWB accounts for economic inequality. As a matter of fact, the current version of the Index only measures income inequality, thereby neglecting the distribution of wealth and economic risk. However, the choice of the four dimensions we presented in the preceding section is based on the assumption that not only income, but also wealth and economic risk have a significant impact on economic well-being.

Consequently, not only the level of consumption, wealth and economic risk should enter the IEWB, but also the inequalities in exactly these dimensions. In other words, if we believe that economic well-being is composed of multiple dimensions, economic inequality should also consist of these multiple elements. We argue that most people think of inequalities of wealth or economic risk as being important. Measuring inequalities in the three IEWB dimensions would render the overall Index more consistent and provides a more accurate vision on economic well-being.

Fig. 3.5 illustrates our proposal to analyse three aspects of economic inequality. The three IEWB components effective consumption per capita, accumulated stocks of wealth per capita and economic security give rise to three spaces of inequalities: the distribution of per capita consumption; the distribution of accumulated wealth per capita; and inequality of exposure to economic risks. The modified third IEWB component that results from our proposal is presented in Fig. 3.6.

On a conceptual level, we think that this amendment is consistent with the idea of economic well-being contained in the IEWB and relatively easy to communicate. However, it is considerably more difficult to measure economic inequality in various dimensions simultaneously, and hence to operationalise the logic of Fig. 3.5. Translating our conceptual modification into a quantitative measure is a difficult endeavour given the incomplete

3.2. FOUR DIMENSIONS, THREE INEQUALITIES

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Real total consumption

Stocks of wealth

Economic security

Distribution of p.c. consumption

Distribution of accumulated wealth p.c.

Inequalities of risk exposure

Economic equality Figure 3.5: Proposal for inequality measurement in the IEWB.

0.75*

Poverty intensity

+

0.25*

Economic equality

=

Equality and poverty

Figure 3.6: Modified Component 3 â&#x20AC;&#x201D; equality and poverty.

nature of the data. To be entirely consistent with the definitions presented in the preceding section, a measure of the inequality of effective consumption per capita would have to take into account the differences in household sizes, leisure consumption, consumption of goods and services provided by the government and, finally, the differences in life expectancy among the households. The second IEWB dimension, accumulated stocks of wealth, is even more problematic: next to household differences in per capita capital stocks, we would have to account for differences in the level of human capital stock between households. And it is unclear how to measure the distribution of R&D capital and the incidence of the national foreign debt or of the environmental liabilities on particular households. Also in the third dimension, economic security, inequality between households is problematic.

In fact, some risks are by definition not borne by all households since they are restricted to a particular socio-demographic profile (e.g. old age and single mother poverty). This renders comparisons of economic security, for instance between a single mother household and a household of retirees, very delicate. We will discuss these problems in more detail in Section 4.1.

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Alternative proposals to measure inequality

In the preceding section we treated the question which inequalities we would like to measure: our empirical measurement should reflect inequalities in the IEWB dimensions effective consumption, accumulated wealth and economic risk. We now need to operationalise the inequality measurement.

In Chapter 2 the most important inequality statistics have been discussed. We have paid particular attention to point out in how far these measures modified the often implicit conventions in today’s analysis of economic inequality by scientific experts. To summarise the most relevant points, it is safe to say that the internal, i.e. academic, discourse in economics can be characterised by a large consensus on several conventional methods: a) if theoretical correctness and simplicity conflict with each other, it seems that technical completeness is frequently given priority over ease of interpretation and communicability (cf. Atkinson, 1970, p. 253); b) quantitative analysis is undoubtedly the dominant approach to economic inequality; c) the acceptability of alternative inequality statistics is conventionally tested indirectly with the help of a list of desirable features; d) despite Sen’s critique on the arbitrary element in complete orderings, most authors continue to employ summary statistics to compare the degree of inequality of different distributions; e) concentration and inequality are widely regarded as “essentially the same concept” (Theil, 1964, p. 128).

We have argued in Section 3.1 that the IEWB user should not be thought of as a technical expert, but rather as the average citizen looking for comprehensive information on the ‘big picture’ of economic well-being. The radius of actors thus extended beyond academic circles and we therefore have to confront the scientific representations with a more external viewpoint. Clearly, inequality measurement in the IEWB should be as close as possible to what Sen referred to as “normal communication”, and what Osberg and Sharpe call “popular perceptions”. At the same time, we have to be aware that these notions will not allow us to identify a precise definition of inequality. This is due to the fact that there is not one ‘typical citizen’, but a heterogeneous mass of different perceptions, representations and values as regards inequality. In other words, it is impossible to prove the correctness of any definition if our only criteria is the blurry notion of normal communication. Nevertheless, we argue that the legitimacy of the IEWB depends on the degree in which its inequality measure is co-constructed and takes into account internal and external considerations. We therefore examine to what extent the conventions listed above are legitimate in light of the specific purpose and the potential users of the IEWB. First, it is obvious that the arbitrage between theoretical completeness and communicability has to be re-considered. Not only the final result, but also details of the computation of inequality measures have to be transparent and easily communicable. Otherwise the important feedback from external actors on these statistics is very difficult and measures risk to be constructed by experts rather than being co-constructed.

Second, a quantitative approach to the measurement of inequality remains useful. If individuals seek to compare different aspects of well-being, eventually the inherent diversity has to be reduced in order to evaluate the overall development. The index approach proposed by Osberg and Sharpe is a transparent and useful way to aggregate the compo-

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nents of economic well-being and preserves the possibility for individuals to attach different weights to each of these components.

Third, the convention to spell our different features of inequality measures and test their acceptability can be an efficient way to stimulate feedback from external actors as it renders underlying assumptions and characteristics more explicit. The fuzzy character of the concept inequality contributes to the attractiveness of such an indirect approach to the construction of inequality measures. However, special attention has to be paid to render the list of desirable features accessible and avoid unnecessary technical complexity.

Fourth, the completeness of summary measures is both an advantage and a problem for inequality measurement in the framework of the IEWB. Although Sen’s intersection orderings probably reduce effectively the extent of arbitrary information, at the same time they may increase the amount of redundant or unnecessary information. The advantage of an intersection ordering is that it allows to identify situations in which comparisons between distributions are difficult. But to construct a particular intersection ordering, an agreement has to be achieved as to which summary measures should be included in the ordering. And eventually, all included statistics have to be computed and their results compared. This inflates the complexity of inequality measurement, as users are confronted with a panoply of different measures based on a variety of concepts. By contrast, the use of a single approach diminishes considerably the amount of unnecessary information contained in the well-being index. With respect to what we said about the arbitrage between theoretical completeness and communicability, we argue in favour of a single approach which nevertheless should allow for different normative opinions.

Fifth, it is doubtful whether inequality is always thought of as concentration and as being independent to proportional increases. This point merits to be discussed in some detail.

We are here confronted with two different normative views on the concept inequality: on the one hand, we have relative inequality, which is insensitive to proportional increases of all incomes, and absolute inequality on the other. Of course, both views are a priori valid and it seems that both are able to generate support from many people. However, as we have seen in our discussion, mainstream academic literature in economics tends to take relative inequality for granted, which is why we focus here on the arguments indicating that popular perceptions may at least in part regard inequality as containing an absolute element.

Arguably the most frequently used expression in normal communication to describe inequality is the “gap between rich and poor”. We think that few people would seriously argue that inequality is not at all related to the gap between rich and poor. While the terms ‘poor’ and ‘rich’ are both problematic and difficult to operationalise in statistical terms, the point is that a gap does not exclude a difference in absolute terms and therefore does not systematically refer to ratios rather than differences of monetary amounts. It is not aberrant to believe that most people are unconscious whether they refer to an absolute or a relative concept when they speak of the gap between rich and poor. Nevertheless, we should note that this frequently used expression for inequality can hardly be interpreted as a clear indicator for the relative point of view.

In line with the idea of inequality as gap, it can often be observed in public debate on

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economic inequality that people tend to oppose the fate of the lower classes with the number of the proverbial ‘millionaires’. Sometimes even governmental reports have to include the number of millionaires in their assessment of economic inequality. This can be seen in the two reports on poverty and wealth published by the German Ministry for Labour and Social Affairs (BMAS) in 2001 and 2005. Both reports contain warnings to interpret the number of millionaires as a sign for increasing wealth or inequality. Nevertheless, in a chapter under the plain title “Millionäre”, the interested citizen finds information about the evolution of the number of inflation-corrected millionaires in Germany. The reason is simple: “In the general discussion the notion of the millionaire is often used as a synonym for property wealth” (BMAS, p. 47) and, combined with the number of non-millionaires, part of the common perception of inequality. Hence, although one million is an absolute amount money, many people tend to think that an information on the number of millionaires is useful to evaluate economic inequality. When in the process of economic growth the number of millionaires increases, while at the same time the low-skilled worker only notes a mere plus of 20 eon his payroll, he may very well be ignorant enough to contradict mainstream economics and interpret this development as an increase in inequality — although income ratios might have remained the same.

It should be noted that the notion of ‘absolute’ inequality is not completely absent from the academic debate. A precursor in this field is Kolm (1976), who cites from personal conversations to come to the conclusion that it is “no less legitimate to attach the inequality between two incomes to their difference than to their ratio” (ibid., p. 419). Unsurprisingly, Kolm brings in non-scientific actors to illustrate his point: “In May 1968 in France, radical students triggered a student upheaval which induced a workers’ general strike. All this was ended by the Grenelle agreements which decreed a 13 % increase in all payrolls. Thus, labourers earning 80 pounds a month received 10 pounds more, whereas executives who already earned 800 pounds a month received 100 pounds more. The Radicals felt bitter and cheated; in their view, this widely increased incomes inequality. But this would have left unchanged an inequality index Ir [the Atkinson index] computed according to the above formula. [...] In other countries (I have been quoted examples from England and The Netherlands), trade unions are more clever and often insist on equal absolute, rather than relative, increases in remuneration, so as to avoid the above effect. And I have found many people who feel that it is an equal absolute increase in all incomes which does not augment inequality, whereas an equiproportional increase makes income distribution less equal or more unequal — and these were people of moderate views.” (ibid., p. 419) Kolm translates this reasoning into a class of absolute measures of inequality and discusses some of its properties. However, with some exceptions like the extension of Blackorby & Donaldson (1980), the absolute statistic introduced by Kolm has not significantly altered the academic focus on relative measures of inequality. This may be due to the fact that his text is extremely technical, without easy graphical interpretation and rather inaccessible

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73

to most lay readers. Nevertheless, Kolm should be credited for pointing out that the dichotomy of absolute versus relative measures contains a political dimension. As a matter of fact, the letter ‘r’ with which Kolm indexes the Atkinson measure in the above quote stands for “right”. Kolm’s own measure based on inequality of differences (instead of ratios) is indexed with “l” for “left”. While this division into political camps should not be taken too literally, Kolm uses it to interpret the many scientific contributions in economics in favour of relative measures: they “tend to support Abba Lerner’s contention that economic science tends to shift its servants to the right” (ibid., p. 420).

Thirty years after Kolm’s introduction of ‘absolute’ inequality, the proponents of this view still tend to be associated with a ‘left-wing’ political stance. Mostly individuals who are critical of market capitalism and economic globalisation tend to emphasise growing differences in absolute income levels between inhabitants of the same country or between different economies. Martin Ravallion, in an examination into the reasons why globalisation often tends to provoke diametrically opposed opinions, points out that relative inequality is not the only defensible concept. Ravallion underlines that ‘anti-globalisation’ protesters do not necessarily get the numbers wrong when they criticize free trade and the faith in economic growth. Perhaps they simply do not have relative, but absolute differences in mind: “Perceptions on the ground that ‘inequality is rising’ appear often to be referring to this concept of inequality” (2003, p. 742).

In one of the few experiments on the question carried out by Amiel & Cowell (1999) with students from Israel and the UK, the results show that 40 % of the participants thought about inequality in absolute terms. However, we have reason to believe that Behavioural Economics can hardy be expected to solve the problem of agreeing on a relative or absolute measure of inequality. If we follow Kolm and Ravallion and interpret the alternative views on inequality as reflecting political attitudes, democratic elections are probably better suited to answer this question than scientific experiments.

Together, the five points we have discussed in this section indicate what type of inequality measure we should employ for the IEWB. We have expressed our opinion that communicability is one of the key features in our context and should be preferred over more theoretically complete instruments such as intersection orderings. Furthermore, we want to keep the index form of the IEWB and therefore look for a quantitative summary measure. The opposition between absolute and relative measures of inequality cannot be decided upon since it may reflect political or ethical opinions which tend to co-exist in most democratic societies. The citizen should be able to form an opinion about economic wellbeing according to her opinions and we should not impose either of the two alternatives.

This is consistent with Osberg and Sharpe’s stance to leave the weights of the dimensions open to discuss so as to match the user’s values. In the next section we will illustrate why we think that the following two alternative measures for the three-dimensional inequality in the IEWB satisfy these considerations: 1. An easily communicable and intuitive measure of inequality is a three-dimensional mean difference. When we think of each dimension of inequality as a dimension of a space, each household can be treated as a point in this space. The inequality between

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CHAPTER 3. REVISION OF INEQUALITY IN THE IEWB two households is then the gap — or distance — between their respective points and the total inequality is the average gap between all household. This is an absolute measure of inequality.

2. All distances of first measure can be divided by the mean of the different dimensions.

This yields a three-dimensional relative mean difference. While this sacrifices some of the intuitive appeal of distances between points in a space, it has the advantage of being very similar to the Gini ratio. It is a relative measure of inequality.

These two measures are far less sophisticated than the state-of-the-art statistics derived from welfare functions and thus subject to their criticism. However, they are also less complicated than welfare-based measures and therefore better suited for the IEWB. They should be thought of as a compromise in the arbitrage between theoretical purity and communicability: both are based on the same concept of inequality as gap, but nevertheless allow for two alternative value judgements.

3.3.1

Measuring differences: a geometric approach

The two measures we presented in the preceding section have the advantage of allowing for a relatively easy graphical representation. Since the dimensions of inequality we want to analyse form an Euclidean space, the absolute differences between households can simply be calculated as Euclidean distances. The Euclidean distance between two points Pi = (pi,1 , pi,2 , . . . , pi,n ) and Pj = (pj,1 , pj,2 , . . . , pj,n ), in a Euclidean n-space, is defined as: v u n q uX Pi Pj = (pi,1 − pj,1 )2 + · · · + (pi,n − pj,n )2 = t (pi,d − pj,d )2 d=1

In a one dimensional space, which we will denote d, the Euclidean distance between two points is equal to the absolute difference between their coordinates. If we write the two points as Pi = (pi,d ) and Pj = (pj,d ), their Euclidean distance is equal to: q d Pi Pj = (pi,d − pj,d )2 = |pi,d − pj,d | It is easy to see that Gini’s absolute differences we presented in Section 2.1.3 can be expressed as one-dimensional Euclidean distances. We recall the formula for the absolute mean difference (AMD): PN PN i=1 j=1 |yi − yj | AMD = N2 Gini’s AMD is thus the average Euclidean distance between all possible pairs of the N points Pi , i = 1, · · · , N in a one-dimensional Euclidean space. Interpreting the incomes as one-dimensional vectors, the AMD can thus be written as: PN PN p (pi,d − pj,d )2 i=1 j=1 AMDd = (3.1) N2

3.3. ALTERNATIVE PROPOSALS TO MEASURE INEQUALITY

75

To obtain an expression for the relative mean difference in Euclidean space, we replace the average income µ by the average length λ of the income vectors: PN q 2 pi,d i=1 AMDd where λ= RMDd = λ N The advantage of using the Euclidean space is the possibility to increase the number of dimensions for which we want to evaluate inequalities. If we want to measure inequality in n dimensions, the absolute and relative mean differences become: PN PN pPn 2 i=1 j=1 d=1 (pi,d − pj,d ) AMD = (3.2) N2 PN qPn 2 i=1 d=1 pi,d AMD RMD = where λ= λ N For up to three dimensions the inequalities can be easily represented graphically. We will illustrate this with an example of three households and a three-dimensional space (these dimensions could represent different dimensions for which we want to measure inequality).

Each household is characterised by one value for each of the three dimensions. These three values are interpreted as a vector in the three-dimensional space (we can think of them as a point in this space). In our example, we assign the respective values of P1 = (0.5, 0.5, 0.5), P2 = (0.5, 0.5, 0) and P3 = (0, 1, 1) to the three households. Their distances are illustrated in Fig. 3.7. We can also compute the AMD for the three points in our example. From the dimension 2

P2

P2 P3

P1 P2

P3 P1 P3

P1

dimension 1

dimension 3 Figure 3.7: Illustration of three-dimensional inequality as geometric distances. formula 3.2 we see that there are nine possible pairs between the three points: AMD =

P1 P1 + P1 P 2 + P1 P 3 + P2 P1 + P2 P2 + P 2 P3 + P 3 P1 + P3 P2 + P3 P 3 32

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CHAPTER 3. REVISION OF INEQUALITY IN THE IEWB

The numerical values for these distances are: p √ √ P1 P2 = P2 P1 = 02 + 02 + 0.52 = 1/2; P1 P3 = P3 P1 = 0.52 + 0.52 + 0.52 = 3/4 p √ P2 P3 = P3 P2 = 0.52 + 0.52 + 12 = 1 1/2; P1 P1 = P2 P2 = P3 P3 = 0 The AMD is therefore: AMD =

2(1/2 +

p p 3/4 + 1 1/2) = 0.576 9

This means that, in our example, the AMD for the three households is 0.576. We can also compute a multi-dimensional RMD for the three households. For this we need the average length λ of the household vectors: √ √ √ 0.52 + 0.52 + 0.52 + 0.52 + 0.52 + 02 + 02 + 12 + 12 = 0.996 λ= 3 And hence: RMD =

AMD 0.576 = = 0.578 λ 0.996

From uni-dimensional mean distances...

The geometric interpretation of Gini’s absolute and relative mean difference has a straightforward interpretation in terms of inequalities: the further two points are away from each other, the more the households represented by these points are unequal. However, there are three serious problems with using multi-dimensional versions of the AMD or the RMD which we will have to solve if we want to use them as measures for multi-dimensional inequality.

First, there is the problem of normalising the dimensions. In fact, if the three dimensions in our example differ significantly with respect to their range, mean or standard deviation, the overall mean distance might be highly distorted. Imagine, for instance, the first dimension represents income (with values ranging from 1000 e to 1000000 e), the second dimension represents wealth (with values between 0 e to 5000000 e), and the third dimension a coefficient of economic security (with values between 0 and 1). The contribution of the third dimension to the mean difference would all but disappear compared to income and wealth. Since we are interested in the inequality in all three dimensions, this is clearly a problem. A possible solution would be to normalise all dimensions. We could, for instance, create an index and apply the transformation T = (xi − xmin )/(xmin − xmax ) to all values (xmin and xmax could be the highest and lowest value in each dimension). This would normalise all dimensions to the interval [0, 1]. However, this normalisation would automatically eliminate all absolute differences: the highest absolute distance in each dimension would always be equal to one, no matter how big the absolute difference in income or wealth between the rich and the poor. The fact that the dimensions are likely to have different scales is thus a serious problem and cannot be easily solved by normalisation if we are interested in absolute differences between economic positions.

3.3. ALTERNATIVE PROPOSALS TO MEASURE INEQUALITY

77

Second, the multi-dimensional versions of the AMD and the RMD we presented above have the disadvantage that they do not allow to analyse directly the contribution of each dimension to total inequality. For instance, it is unclear how much of the value for the AMD of 0.576 in our example above is due to the first, second or third dimension. However, in the context of the IEWB it might be highly desirable to know which of the three dimensions is the main driver of total inequality. Furthermore, in certain cases the absolute mean difference might remain constant even if inequality changed in all three dimensions since these variations could cancel each other out (e.g. a 10 % increase in income inequality and a 10 % decrease in wealth inequalities could offset each other to a certain extent). We therefore need to look for an alternative measure of multidimensional inequality, without sacrificing the straighforward graphical interpretation contained in Fig. 3.7.

Third, the fact that the AMD computes N 2 differences is not intuitive. In fact, we see in Fig. 3.7 that in the case of N = 3 all relevant information can be obtained from only three differences. The trivial differences P1 P1 = P2 P2 = P3 P3 = 0 could be left out. The same holds for the redundant differences that appear twice: if we accounted already for P1 P2 , including P2 P1 seems unnecessary. If N = 3, the mean distance should be based on three distances, if N = 4 we should compute six differences, for N = 5 ten differences. In general, for N points we should thus compute only (N 2 − N )/2 instead of N 2 differences.

This would make the measure intuitively more appealing and corresponds better to the geometric representation: in the end, we are interested in the average gap between different households, and not in the average gap between all possible combinations.

We argue that a slight modification to the AMD could solve all of these problems. If we compute the average of the distances between household vectors for each dimension separately, we can define an index of multidimensional inequality that is a) decomposable into the different dimensions; b) insensitive to scale differences; and c) based only on the relevant information. Consider the following measure of uni-dimensional inequality: PN −1 PN q (pi,d − pj,d )2 j>i i=1 (3.3) ADd ≡ (N 2 − N )/2 Although this expression looks very similar to AMDd in equation (3.1), we argue that ADd is more attractive as an inequality measure. It is simply the average of all differences between N points in the dimension d. Sticking to our above example, we will illustrate the computation of ADd . For the three points in Fig. 3.7 the measure is: d

d

P 1 P2 + P1 P 3 + P 2 P3 ADd = 3

d

This is simply the average of three distances Fig. 3.7 in the dimension d and thus a quite intuitive measure for the inequality between the three households. In our example, the values of the average distance ADd for the three dimensions are: q

q q 1 2 2 2 AD1 = (p1,1 − p2,1 ) + (p1,1 − p3,1 ) + (p2,1 − p3,1 ) = 1/3 3

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CHAPTER 3. REVISION OF INEQUALITY IN THE IEWB 1 AD2 = 3

q

1 AD3 = 3

q

(p1,2 − p2,2

)2

q q 2 2 + (p1,2 − p3,2 ) + (p2,2 − p3,2 ) = 1/3

(p1,3 − p2,3

)2

q q 2 2 + (p1,3 − p3,3 ) + (p2,3 − p3,3 ) = 2/3

We can show without much difficulty the effect of an equiproportional increase of values in one or more dimensions. If we compare two points Pi and Pj with the respective vectors of (p1,1 , p1,2 , . . . , p1,n ) and (p2,1 , p2,2 , . . . , p2,n ), the distance between these two points in dimension d is: q d Pi Pj = (p1,d − p2,d )2 If we multiply both vectors by a scalar a to obtain the points Pi0 and Pj0 , then the distance in dimension d between them becomes: q q d d Pi0 Pj0 = (ap1,d − ap2,d )2 = a2 (p1,d − p2,d )2 = aPi Pj Hence, the effect of an equiproportional increase of all values in one dimension would be an increase of the average distance ADd by the factor a. If we inflate several dimensions by the scalar a, then all the average distances ADd will increase by a.

What about an equal absolute increases in the values of each dimension? We can add the amount b to our points Pi and Pj to obtain Pi00 and Pj00 . The distance between them becomes: q q d 00 00 d Pi Pj = ((p1,d + b) − (p2,d + b))2 = (p1,d − p2,d )2 = Pi Pj We see that equal additions to all household vectors do not alter the level of ADd since all distances remain unchanged. Of course, this result holds also if we add equal amounts to all values of several dimensions. The insensitivity to equal additions and the sensitivity to proportional additions confirm that ADd is indeed an absolute measure.

If we want to obtain a relative measure, we have to bring back in the average lengths of all household vectors in dimension d, which we will denote λd . The average vector length can be written as: N 1 Xq 2 λd = pi,d N i=1 In our example above, the values for the average vector lengths for the different dimensions are: 1 √ 2 √ 2 √ 2 1 √ 2 √ 2 √ 2 λ1 = 0.5 + 0.5 + 0 = 1/3; λ2 = 0.5 + 0.5 + 1 = 2/3 3 3 1 √ 2 √ 2 √ 2 λ3 = 0.5 + 0 + 1 = 1/2 3 Passing from ADd to a relative measure is similar to dividing the AMD by the mean income µ to obtain the RMD. A relative version of ADd is the measure RDd . We define a

3.3. ALTERNATIVE PROPOSALS TO MEASURE INEQUALITY

79

relative measure for each dimension by dividing the average distance by the average vector length: Ad RDd ≡ λd We thus obtain for each dimensions d the average distance relative to the average vector length. In our example, inserting the values for the average distances and the average vector lengths yields the following RDd : RD1 =

1/3 = 1; 1/3

RD2 =

1/3 = 1/2; 2/3

RD3 =

2/3 = 1 1/3 1/2

It can easily be verified that RDd is insensitive to equiproportional increases. A multiplication by the factor a of all values would raise all ADd and λd by this factor so that RDd would remain unchanged.

We have now derived two uni-dimensional measures, AD and RD, and it may be useful to sum up the similarities and differences between them. The common features are: • Both AD and RD correspond closely to the idea of representing inequalities as geometric distances like we have done in Fig. 3.7. They are based only on the relevant distances in that they neglect the trivial and redundant distances that the AMD and the RMD take into account.

• The two measures reach their minimum value of zero when all households are equal, i.e. the distances between all household points are zero.

• They both allow for analysing inequalities in many different aspects simultaneously since the Euclidean space can be extended to n dimensions.

By contrast, the two measures also differ in several important ways: • The higher the average distance in a dimension, the higher will be the value of AD. It is an absolute measure of inequality since it rises when absolute differences between households increase. Furthermore, AD remains constant when the values of a dimension are increased by the same amount for all households. Therefore, the measure retains the most characteristic features of absolute measures of inequality like the AMD or Kolm’s index of absolute inequality (cf. Kolm, 1976).

• RD is a relative measure of inequality since it is insensitive to proportional increases: it has the kind of mean independence that characterises measures like the Gini concentration coefficient, the Theil measure or Atkinson’s index of inequality. The measure RD increases only if the average distance rises more than the average vector length.

• The use of absolute distances in the AD implies that economic data has to be adjusted for differences in inflation or purchasing power. This introduces a problem absent in relative measures: inflation would increase both the average distance between points and the average length of household vectors, thereby leaving the ratio RD unchanged.

This is, of course, only a technical and not a conceptual difference between AD and RD.

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CHAPTER 3. REVISION OF INEQUALITY IN THE IEWB

...to an index of multi-dimensional inequality We have argued that AD and RD are more attractive than the similar pair AMD and RMD since our measures correspond closer to the graphical representation of distances.

Nevertheless, we are left with the problem of scale differences between the dimensions.

Since the IEWB is an index and hence only meaningful if we trace its evolution over time, we would like to obtain an index that reflects the evolution of the ADd and RDd over time.

The scaling problem can be solved by converting the two measures into indices: IAt =

1 ADnt 1 AD1t + · · · + n AD1t−m n ADnt−m

(3.4)

IRt =

1 RDnt 1 RD1t + · · · + n RD1t−m n RDnt−m

(3.5)

We can interpret these two indices as averages of the inequality changes in the different dimensions. Since the average distance in each dimension in year t, ADtd , is compared to a base-line value in period (t − m) of the same dimension, the different scales disappear.

Only the evolution in per cent over time of each dimension enters the index. Each ratio (ADdt /ADdt−m ) captures the percentage change of the average distance in the dimension d. The overall index is simply the average of these percentage changes over time. IAt is the index that corresponds to the evolution of absolute inequality over time, while IRt is a relative index since average distances in period t are adjusted for the average vector length during the same period. Hence, IAt and IRt retain the characteristic features of AD and RD, but solve the problem of different scales since the indices are based on changes measured in percentages.

An example will illustrate the logic behind equations (3.4) and (3.5). We take the three points P1 = (0.5, 0.5, 0.5), P2 = (0.5, 0.5, 0) and P3 = (0, 1, 1) from our example above as starting points. Imagine that during period t the position of the household points in the three-dimensional space is modified in several ways: 1) all values in the first dimension grow by 10%; 2) we add the amount 2 to all values in the second dimension; 3) the third household obtains an absolute increase of 0.5 in the third dimension. We leave it up to the reader’s imagination what these modifications might represent (e.g. the equiproportional increase in the second dimension could represent economic growth; the absolute increases a higher business profit or a rise in real estate values etc.). The combined result of these modifications defines a new set of household vectors P10 = (0.55, 2.5, 0.5), P20 = (0.55, 2.5, 0) and P30 = (0, 3, 0.5) which is illustrated alongside the original points in Fig. 3.8. We will denote the values of ADd , RDd and λd etc. with a superscript t − 1 if they correspond to the original points P1 , P2 and P3 . We have seen above that AD1t−m = 1/3, AD2t−m = 1/3 and AD3t−m = 2/3. The relative measures were equal to RD1t−m = 1, RD2t−m = 1/3 and RD3t−m = 1 1/3. The values that correspond to P10 , P20 and P30 are marked with t. The average distances in period t are: AD1t

11 √ 1 √ 2 √ 2 2 0 + 0.55 + 0.55 = ; = 3 30

AD2t

1 √ 2 √ 2 √ 2 1 = 0 + 0.5 + 0.5 = 3 3

3.3. ALTERNATIVE PROPOSALS TO MEASURE INEQUALITY

81

dimension 2 P20 P30

P10 P2 P3 P1

dimension 1

dimension 3 Figure 3.8: Combined effect of modifications in the household vectors.

1 √ 2 √ 2 √ 2 0.5 + 1 + 1.5 = 1 3 The ratios (ADdt /ADdt−m ) in our example are equal to: AD3t =

AD1t = 1.1; AD1t−m

AD2t = 1; AD2t−m

AD3t = 1.5 AD3t−m

These numbers tell us that inequality, interpreted as geometric distances, in the first dimension increased by 10 %. This is in line with the result derived above that proportional increases by the factor a increase the average distance by the same factor. In our example we multiplied all values in the first dimension with 1.1, so as to increase them by 10%.

Consequently, the average distance in the first dimension increased by the same factor. The average distance in the second dimension remained the same. This, too, is not surprising since we added an equal amount of 2 to all values in this dimension. Since all distances are insensitive to this addition, the average distance did not change. The third dimension displays an increase of the average squared distance by 50%. The addition of 0.5 to the third dimension of only one household had thus a big impact on the inequality in this dimension, since it increases two of the three distances in our example. To get the overall picture of the evolution of absolute inequality, we simply insert the three changes into the formula for IAt , which yields: IAt =

1 (1.1 + 1 + 1.5) = 1.2 3

This means that the overall effect of the three hypothetical changes in our example increased absolute inequality by 20% .

In order to compute our relative measure of inequality, we have to calculate the new average vector lengths in period t: √ √ 1 √ 1 √ 2 √ 2 √ 2 t t 2 2 2 λ1 = 0.55 + 0.55 + 0 = 11/30; λ2 = 2.5 + 2.5 + 3 = 8/3 3 3

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CHAPTER 3. REVISION OF INEQUALITY IN THE IEWB

1 √ 2 √ 2 √ 2 0.5 + 0 + 1.5 = 2/3 3 We obtain the relative distances by dividing the ADd by the corresponding λd : λt3 =

RD1t =

11/30 = 1; 11/30

RD2t =

1/3 = 1/8; 8/3

RD3t =

1 = 1 1/2 2/3

Next, we compute the evolution of the Rd over time: RD1t 1 = 1; t−m = 1 RD1

RD2t 1/8 = 3/8; t−m = 1/3 RD2

RD3t 1 1/2 = 1 1/8 t−m = 1 1/3 RD3

These figures illustrate the main differences between the absolute measure ADd and the relative measure RDd . The proportional increase of 10% we affected to the first dimension leaves RD1 unchanged since the average vector length also increased proportionnally. The relative measure of the second dimension, RD2 , in which we added an equal amount of 2 to the values of each houshold, indicates a decrease of inequality of 62.5%. While this is in stark contrast to the constant value of AD2 we computed above, it is the expected effect of an equal absolute increase on a relative inequality measure. To see why, we simply have to think of the second dimension in terms of concentration. In period t − m, the sum of all values in this dimension was 0.5 + 0.5 + 1 = 2, of which 50% was concentrated in the hands of the third household (households one and two each holding 25%). In period t, the sum of all values is 2.5 + 2.5 + 3 = 8, of which the third household holds only 37.5% (with households one and two each holding 31.25%). The second dimension is thus considerably less concentrated in period t than in period t − m, even though all distances remained completely the same. The third dimension, in which we added 0.5 to the third household, shows a relative increase of inequality of 12.5%. This is the only dimension for which the absolute and the relative measure point in the same direction, although the increae in RD3 is less pronounced than the 50% increase of AD3 . The overall evolution of relative inequality is the weighted sum of the three dimensions. The index IRt in our example is therefore equal to: 1 IRt = (1 + 3/8 + 1 1/8) = 5/6 3 Relative multi-dimensional inequality decreased by 16.7%. Comparing this figure to the 20% increase of the absolute index IAt , we see that our two alternative measures often yield rather contradictory results. If our interest was merely to order the different distributions, we could use Sen’s instrument of intersection quasi-orderings (see our discussion in Section 2.1.7 on p. 55). In this case, the non-conflicting orderings are isolated from those that do not allow for an unambigous decision. If we think that AD and RD are both acceptable measures of inequality, we have seen that the only unambigous judgement we can make is that inequality in the third dimension increased. Hence, if we cannot decide whether we think of inequality in a relative or an absolute way, the changes in the two other dimensions cannot be judged upon.

Chapter 4 Empirical application 4.1

Data treatment

The data source we employed to evaluate economic inequalities is the household survey Budget des Familles, or BdF, which is compiled roughly every five years since 1956 by the French National Institute for Statistics and Economic Studies (INSEE). The most recent years covered by the survey have been 1979, 1984/1985, 1989, 1994/1995, 2000-2001.1 The survey covers all civilian non institutional households in metropolitan France and overseas departments, thus excluding the population living in prisons, the armed forces etc. Overseas territories are not in the scope of the survey. The metropolitan sample is obtained using as a sample frame the 1990 Census housing files, completed by a file containing new dwellings. The data collection unit is the household. No group or category is over-represented in the sample, since the main objective is to draw a global picture of the budget of all households living in France. Only the main residences are surveyed as other residences (vacant, secondary or occasional) are excluded from the survey scope.

Although the BdF is the most comprehensive data source covering our research questions available for researchers not attached to the INSEE, the survey bears some serious disadvantages. First of all, its primary purpose is not to evaluate wealth or risk inequalities, but includes similar variables only as complementary information. According to the INSEE, the main objective of the BdF â&#x20AC;&#x153;is to measure with utmost accuracy expenditures, consumption and income of French householdsâ&#x20AC;? (INSEE, 2000, p. 6). The information on wealth and economic risk are thus not very detailed since they are not the prime focus of the survey. The standard technique of oversampling the few but extremely rich households in order to obtain an accurate picture of overall wealth distribution is thus not employed in the BdF. Second, we already mentioned that the population living in institutions is not covered. In addition, the most deprived stratum of the society is also excluded from the sample: since the interviewees are selected from housing files, the population of homeless men and women does by definition not enter the BdF. A third drawback is related to the calculation of taxes. To establish a series of disposable income, the annual amount of tax 1

The results for 2006/2007 are expected in autumn 2007 and could unfortunately not yet be analysed.

83

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CHAPTER 4. EMPIRICAL APPLICATION

paid by the household has to be known. However, taxes are imperfectly captured by this survey. In fact, the survey records the income of year t and the taxes of year t − 1. In the surveys before 1995 the time lag between income and taxes was even two years. Fourth, income from property is systematically underestimated in the BdF, a feature that the survey shares with other data sources like the Enquête Revenus Fiscaux (cf. Legendre, 2004).

In an earlier application of the IBEE (op. cit.), we have developed a correction method to overcome this bias which we will employ again in this work. Other inconveniences of the household survey are all statistical errors common to this type of data collection: non-response bias, sample errors and (voluntary or involuntary) false responses.

Data on all three dimensions of inequality under analysis is only contained in the last two editions of the BdF, namely the 1994/1995 and the 2000/2001 survey. We therefore decided to evaluate the inequality statistics we presented in Chapter 3 for these two points in time. We hope to extent the series a soon as the 2006/2007 edition of the survey becomes available. In the remainder of this section, we explain for each of the three dimensions identified earlier how the underlying aspect of economic inequalities can be assessed via proxies from the household survey. As we anticipated in Section 3.2, none of the three data vectors we compute corresponds exactly to the definitions of the IEWB dimensions, but merely represents best estimates given the available information. Nevertheless, for each dimension we have tried to remain as close as possible to the well-being aspect as defined by the respective IEWB dimension.

4.1.1

Inequality of effective consumption per capita

From the definition of the first IEWB dimension we presented in Section 3.1 (p. 66), it can be seen that to accurately measure the inequality of effective consumption per capita we would have to estimate the differences between leisure consumption and the utilisation of governmental goods and services. Ideally, we should also take into account the inequality in life expectancy enjoyed by different people. Unfortunately, the available data does not allow for an analysis in such detail. Furthermore, serious conceptual issues would have to be clarified in order to measure the consumption of governmental production like military protection and others. However, we argue that a satisfactory proxy for inequality in the first IEWB dimension are adjusted disposable incomes per consumption unit, as this variable includes several of the relevant aspects as will be seen below.

In order to obtain this variable, we employ the same correction procedure presented in Jany-Catrice & Kampelmann (2007). The data correction addresses two problems: first, it includes information on an important — but unfortunately often absent — item of consumption, namely the housing services enjoyed by owners who reside in their property.

Based on a hedonistic pricing model, an econometric estimation is used to generate the imputed rent for all households of owner-occupiers (see Driant & Jacquot, 2005, for further discussion of this technique). The second problem addressed by our correction procedure is the underestimation of income from property in the household survey. Compared to the aggregate values in the national accountancy, it is safe to say that the household survey does not capture more than 50 % of this income category. We correct this underestimation with

4.1. DATA TREATMENT

85

the help of a somewhat restrictive hypothesis: we assume that the underestimation of each household is proportional to the household’s financial wealth as declared in the survey. This hypothesis allows us to ‘inflate’ the levels of property income in the household survey by a specific factor so as to make the total amount of this income type in the BdF correspond to the values observed in the national accountancy. By substracting the amount of taxes declared by each household, we thus obtain an estimate for the adjusted disposable income.

Next, we would like to retain the idea that household sizes have an important effect on scale economies in consumption enjoyed by their inhabitants. As can be seen in Fig. 3.1, Osberg and Sharpe proposed to multiply the values of per capita consumption by an index of household size to take the economies of scale in consumption into account. We can achieve a similar adjustment by dividing the disposable income per household by the corresponding number of consumption units. Technically, each household income is divided by the Oxford equivalence scale which assigns a value of 1 to the first household member, of 0.7 to each additional adult and of 0.5 to each child. This yields a vector of adjusted disposable incomes per consumption unit which we will evaluate below in terms of inequality. The distribution of this vector is illustrated in Table 4.1 for the two years in our data set.

100% Max 99% 95% 90% 75% Q3 50% Median 25% Q1 10% 5% 1% 0% Min Std Deviation

1994/1995 2000/2001 290599.33 480455.61 63654.08 67657.19 36309.79 39964.40 28027.12 30818.17 19484.61 21011.03 13591.35 14499.49 9733.18 10298.72 7142.20 7594.07 5866.64 6187.83 3811.23 4133.22 0.00 0.00 597122 647890

Table 4.1: Percentile distribution and standard deviation of adjusted disposable income per consumption unit (all values in 1995 euros) — Data source: BdF.

4.1.2

Inequality of accumulation of productive resources per capita

The second IEWB dimension, stocks of wealth, also poses not only data, but also conceptual problems for the evaluation of inequality. All of the five components we listed in Fig. 3.2 are not easily assessed: the amount of human capital per capita is difficult to measure and data insufficient in the household survey; the stock of investment in R & D can hardly be

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CHAPTER 4. EMPIRICAL APPLICATION

linked to individual ownership and differences therefore difficult to evaluate; the foreign debt is owed to a large extent by institutional or public investors and can also not be evaluated in terms of household or individual inequality; and the cost of environmental degradation are perhaps not caused by all consumers, but in the end borne by the society as a whole.

However, a first picture on wealth inequality can be drawn by analysing the differences in per capita assets. This indicates the wealth levels which can relatively easily be linked to households. The household survey, although this source is not designed to be the most accurate description of wealth differences as we mentioned above, includes several questions on the composition and level of household wealth. For us, the most interesting question is the assessment of the financial value of all assets. In fact, both surveys contain the variable “total value of everything that the household owns”. This is not identical to the “stocks of productive assets” that the second IEWB dimension assesses as it does not only include productive assets that could produce future consumption, but also assets that could be exchanged for future consumption: diamonds or rare paintings are obviously included in “everything the household owns”, but they do not constitute ‘capital’ in the sense that they cannot be used for the production of other consumption goods. We argue, however, that this departure from productive assets would be more problematic on a societal level than in our case of interpersonal comparisons. As a matter of fact, the second dimension measures the consumption that will be available in the future for the society as a whole.

By contrast, the individual consumption in the future may very well be determined by the individually owned stocks of productive and unproductive assets, since in general both can be turned into future consumption. The financial value of households can therefore be regarded as a satisfactory proxy for wealth inequality.

In order to use the data of the household survey in this sense, two modifications have to be made. First, the worth of household assets is indicated in brackets so that they can not be directly evaluated in terms of differences. In 1994/1995 and 2000/2001 the overall value of assets is split up in eight brackets as can be seen in Table 4.2.

Value in brackets (in euros) 0 - 3,049 3,049 - 7,622 7,622 - 15,245 15,245 - 30,490 30,490 - 76,225 76,225 - 152,450 152,450 - 304,898 304,898 and more

1994/1995 (% of hhlds.) 2000/2001 (% of hhlds.)

7.69 7.11 10.45 9.23 9.41 9.71 8.27 8.71 16.64 14.04 26.69 24.76 15.21 19.07 5.65 7.37 100 100

Table 4.2: Financial value of all household assets — Data source: BdF. We therefore worked with the assumption that all households in a given bracket possess

4.1. DATA TREATMENT

87

the value corresponding to the centre of the bracket. If the true values are roughly normally distributed within the brackets, this assumption does not distort significantly the average differences between households as positive and negative deviations from the centre of the bracket would cancel each other out. The upper limit of the highest bracket was fixed at FRF 4,000,000 (or e 609,796).

Contrary to the definition of the first IEWB dimension, Osberg and Sharpe did not propose an adjustment for changes in household size in the wealth dimension but instead compute all values on a simple per capita basis. We therefore divided the value of all assets of each household by the number of persons living in it to obtain a corresponding estimate of per capita wealth of households.

4.1.3

Inequality of exposure to economic risks

The third dimension of inequality, differences in risk exposure, can only be evaluated for unemployment risks given the data at our disposal. The poverty risks for single mothers and the elderly are difficult to compare between people with different socio-demographic profiles. The risk of uncovered health expenditures can be evaluated ex post for the society as a whole with the help of total amounts from the national accounts. At the individual level, however, health risk is difficult to measure. In general, we face the problem of measuring retrospectively a risk at time t that was by definition uncertain at t. Uncertainty is a constituent element of any economic risk and makes its evaluation cumbersome, especially at the individual level as personal circumstances differ widely and influence the individual risk exposure.

And yet, we argue that the household survey contains a proxy that allows us to estimate the inequality in the risk exposure to unemployment. In fact, the BdF contains several variables concerning the subjective assessment of unemployment risk, notably the question of the likelihood of getting (or remaining) unemployed in the 12 months following the interview. Since this information is available both for the reference person and, where applicable, the partner of the reference person, we can thus construct an estimator for the individual risk exposure based on the subjective assessments communicated by the interviewees. We think that the reliance on this subjective risk estimation suits our purpose since it is easily communicable and does not require complex computations. An alternative estimation of unemployment risk would be to specify a maximum likelihood function to measure the probability of being unemployment given certain household or individual characteristics. However, this method is probably not only more difficult to communicate, but it is also uncertain whether we could thereby establish a better estimate of the situation at the individual level than the subjective opinions expressed by the interviewees. While the subjective risk assessment obviously opens the door to potential differences in â&#x20AC;&#x2DC;realâ&#x20AC;&#x2122; and perceived unemployment risk, we argue that it may reflect satisfactorily the inequalities in economic well-being from unequal exposure to unemployment risk. Table 4.3 presents the seven modalities of the relevant variable.

Before we can evaluate the inequalitiy in unemployment risk, we have to formulate a hypothesis of how to aggregate the individual risks borne by the different members of the

88 Degree of unemployment risk Not active No, there is no risk at all Possible, but the risk is low Possible, and the risk is intermediate Possible, and the risk is high Yes, and it is almost inevitable Refusal to answer the question

CHAPTER 4. EMPIRICAL APPLICATION Grade 1 2 3 4 5 -

1994/1995 (in %) 8.4 38.87 30.25 14.37 4.96 3.14 0 100

2000/2001 (in %) 10.86 43.61 25.65 10.70 4.27 4.73 0.18 100

Table 4.3: Distribution of subjective household unemployment risk in the 12 months following the survey â&#x20AC;&#x201D; Data source: BdF.

household. This is particularly difficult since households vary in their composition: some households consist of two income-earners, some of only one, others are single households and include only the reference person. For instance, a husband with high individual unemployment risk may still be better off than a single with the same risk if his wife has a very safe job. We argue that in the majority of cases the most obvious estimate for the household risk is probably the unemployment risk of the reference person (the statistics show that the reference person is by far the one who earns a higher income, is more active on the labour market and thus the highest risk driver for the household). We therefore evaluate inequality in unemployment risk for the reference person; only if the reference person is not active or refused to answer the question, we replaced it with the unemployment risk of the partner.

In both years, the data shows many inactive households: in 1995, 32 % of all reference persons were not active and in 2001 this proportion grew to 35 %. These housholds consist mainly of retirees without exposure to unemployment risk. In the fourth dimension of the IEWB, Osberg and Sharpe proposed to weigh the risk from unemployment by the proportion of the population between 15 and 64 years. We applied the same reasoning and excluded all household for which the age of the reference person does not fall within this interval. This eliminates most of the not active households in our data set and brings the inactive households down to 8.4 and 10.86 % for 1995 and 2001, respectively. We interpreted the remaining not active households as mostly reflecting early retirement, widowhood, handicapped persons or other parts of the population not directly exposed to risk from unemployment. Consequently, the inequality statistics we present in the next section are based on the porportion of households for which we could compute a grade of unemployment risk from 1 (no risk at all) to 5 (certainty that job will be lost).

4.2. RESULTS FOR ALTERNATIVE INEQUALITY STATISTICS

4.2

89

Results for alternative inequality statistics

In this section, we evaluate the three data vectors derived above with respect to their inequality. We will compute the measure AD â&#x20AC;&#x201D; the average absolute distance between all points â&#x20AC;&#x201D; as well as its relative version RD (both are presented in Section 3.3.1).

Since AD should reflect genuine differences in income and wealth, the monetary values are adjusted for inflation. We simply deflate all amounts with the French Consumer price index, which rose by 7.3 % between 1995 and 2001. To compare the results of these two measures with traditional inequality statistics, we also compute two additional sets of inequality measures: first, we evaluate the standard descriptive indicators, namely the Gini coefficient of concentration (cf. Section 2.1.3), the Theil measure (cf. Section 2.1.5), and another frequently employed indicator, the ratio of the ninth over the first decile; second, we calculate the inequality measures based on a welfare criterion, namely the Dalton measure (cf. Section 2.1.4) and the Atkinson index (cf. Section 2.1.6). In order to obtain numerical values for these two welfare measures, we have to specify their respective parameters. For the Dalton measure, we have chosen c = 1/6000 as the minimum income, and c = 1/10000 as the minimum wealth that yield positive welfare. The Atkinson index is evaluated for two values of inequality-aversion, namely the low aversion = 0.5, and a higher one corresponding to = 1.5.

The results for the first aspect of economic inequality â&#x20AC;&#x201D; inequality in adjusted disposable income per consumption unit â&#x20AC;&#x201D; are presented in Table 4.4. All descriptive indicators show an increase in inequality between 1994/1995 and 2000/2001, although the extent of this development differs greatly: while G increased by only around 2 %, the average absolute distance AD indicates a plus of over 10 %. Given the 7 % rise of the mean income, a considerable proportion of this difference can probably be explained by the improvement of the average living standard: the relative measures RD, G, T and D9 /D1 are all insensitive to proportional increases of all incomes. If we assume that at least part of the income growth was spread throughout the entire population via proportional adjustments of incomes, these measures would indicate lower inequality than the mean sensitive measure AD. Since France is a country in which wages in many sectors tend to be frequently adjusted for productivity gains and economic growth, it might be reasonable to assume that different parts of the population received increases with a growth rate close to the one of average income. As a consequence, the concentration of total income is likely to increase less than the average absolute distance. This leads to the differences between AD and the other descriptive inequality measures we observe in Table 4.4. As regards the welfare-based inequality statistics D and A, the particular numerical values displayed in the table depend on the values of the respective parameters c and . We have chosen these values arbitrarily and should therefore be careful with the interpretation of the numerical results for D and A. However, the decrease of D is in line with Daltonâ&#x20AC;&#x2122;s second principle: proportionate additions to incomes should lead to a decrease in inequality (cf. our discussion p. 35). If the growth of the average income is more or less spread throughout the population, we indeed expect â&#x20AC;&#x201D; ceteris paribus â&#x20AC;&#x201D; the diminution of inequality we observe for D in Table 4.4.

The Atkinson index, by contrast, is hard to interpret since the evolution over time changes

90

CHAPTER 4. EMPIRICAL APPLICATION

sign as we go from a low level of inequality aversion ( = 0.5) to a higher one ( = 1.5). It is arguably unclear what degree of inequality aversion should be applied in our case since it is hard to measure the convexity of the indifference curves of the social welfare function.

Sample size N Mean income (in 1995 euros) AD (in 1995 euros) RD G (Gini coefficient) T (Theil measure) D9 /D1 D (Dalton measure, c = 1/6000) A (Atkinson index, = 0.5) A ( = 1.5)

1994/1995 2000/2001 change in % 11294 10305 â&#x2C6;&#x2019;8.8 16619.66 17784.29 +7.0 10808.34 11913.62 +10.2 0.65 0.67 +3.0 0.32 0.33 +1.9 0.19 0.20 +3.4 3.92 4.06 +3.6 1.34 1.24 â&#x2C6;&#x2019;5.4 0.08 0.09 +11.3 0.46 0.23 â&#x2C6;&#x2019;49.4

Table 4.4: Statistics for income inequality â&#x20AC;&#x201D; Data source: BdF. The second aspect, inequality in wealth per capita, displays an even stronger opposition between absolute and relative descriptive measures. Here, AD even points in the opposite direction as it shows a 7.5 % increase in inequality, while all relative measures decrease over time (see Table 4.5). The average wealth per capita grew by 8.5 %, partly a result of the prolonged investment boom referred to as the â&#x20AC;&#x2DC;internet bubbleâ&#x20AC;&#x2122;.2 The remarkable range between AD and the lowest descriptive measure, the ratio D9 /D1 , of over 14 percentage points underlines that the choice of inequality statistics is far from being neutral. As can be seen in Table 4.5, it modifies completely our vision on the empirical observations: if we think of it as concentration, wealth inequality decreased by around 3-7 %. By contrast, if we believe inequality is the average difference per capita, it increased by over 7 %. As for the welfare-based measures, D again moves in the expected direction if we assume that at least part of the increase in average wealth can be interpreted as a proportional increase of the assets of many households. In that case, Daltonâ&#x20AC;&#x2122;s second principle again holds and inequality should, ceteris paribus, decrease. Once more the Atkinson measure is difficult to interpret as the evolution changes sign with different degrees of risk aversion.

Finally, we evaluate the inequality in the subjective assessment of unemployment risk in the two years. Since this risk is measured in grades from 1 to 5, the inequality indicators based on the concept of concentration, i.e. the Gini coefficient and the Theil measure, do not make sense in this case. Also the concepts of â&#x20AC;&#x153;equally distributed equivalent incomeâ&#x20AC;?

(Atkinson) and â&#x20AC;&#x153;welfare if the current income was equally distributedâ&#x20AC;? (Dalton) cannot be applied to the distribution of unemployment risk. We therefore only compute the absolute and relative average distance and the interdecile ratio, which give indications about how 2

Since our sample was collected in 2000/20001, the ensuing stock market bust is not yet reflected in our data.

4.2. RESULTS FOR ALTERNATIVE INEQUALITY STATISTICS

Sample size N Mean wealth (in 1995 euros) AD (in 1995 euros) RD G (Gini coefficient) T (Theil measure) D9 /D1 D (Dalton measure, c = 1/10000) A (Atkinson index, = 0.5) A ( = 1.5)

91

1994/1995 2000/2001 change in % 11294 10305 â&#x2C6;&#x2019;8.8 48965.10 53110.30 +8.5 57059.21 61327.73 +7.5 1.17 1.15 â&#x2C6;&#x2019;0.9 0.58 0.56 â&#x2C6;&#x2019;3.0 0.59 0.56 â&#x2C6;&#x2019;6.4 64.29 59.99 â&#x2C6;&#x2019;6.7 1.20 1.16 â&#x2C6;&#x2019;2.3 0.29 0.28 â&#x2C6;&#x2019;4.4 0.77 0.78 +0.2

Table 4.5: Statistics for wealth inequality per capita â&#x20AC;&#x201D; Data source: BdF. the unemployment risk is spread throughout the population. Before we interpret the evolution of these measures, we note that the average risk actually decreased by 3 % between 1994/1995 and 2000/2001. This result seems to confirm the sumultaneous drop in unemployment rates from 11.4 % in 1995 to 8.7 % in 2001 communicated by the INSEE. Interestingly, this decrease in the average unemployment risk did not translate into a smaller average distance between households. The average difference increased during the same period from a gap in grades of 1.09 to 1.13. Since the measure RD is the ratio of the absolute average risk difference and the average risk, it rose by even more than AD, namely by 7 %. The considerable increase of the interdecile ratio (+33.3 %) should note be taken to seriously. The jump is due to the fact that there are no intermediate values between the risk grades: all values are integers from 1 to 5 and the interdecile ratio can thus only change in rather big steps. During the obervation period, the ninth decile moved from grade 3 to grade 4, while the first decile remained unchanged at grade 1. This automatically led to a considerable increase in the interdecile ratio.

Sample size N Average risk AD RD D9 /D1

1994/1995 2000/2001 change in % 11294 10305 â&#x2C6;&#x2019;8.8 1.94 1.89 â&#x2C6;&#x2019;3.0 1.09 1.13 +3.7 0.56 0.60 +7.0 3 4 +33.3

Table 4.6: Statistics for inequality of exposure to economic risk â&#x20AC;&#x201D; Data source: BdF. The ranking of partial intersections orderings proposed by Sen (cf. Section 2.1.7) is easily applied to the results in the three inequality dimensions we just presented. We can define the partial ordering Q as the the non-conflicting ordering of all descriptive

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CHAPTER 4. EMPIRICAL APPLICATION

measures AD, RD, G, T and D9 /D1 . In this case, we see that Q ranks income inequality in 2001/2000 higher than five years earlier. However, wealth inequality cannot be ordered due to the opposed evaluation of absolute and relative measures. A partial ordering of inequality from unemployment risk can only be based on AD, RD and D9 /D1 : in this case, the intersection ordering indicates an increase in inequality of risk exposure.

We can now compute the aggregate development of economic risk. To do so, we employ the indices IAt and IRt we defined in Section 3.3.1. The first index is the weighted average of the changes in the absolute measure AD in each of the three dimensions of economic inequality. It is therefore equal to: IA2001 =

1 (1.102 + 1.075 + 1.037) = 1.071 3

(4.1)

Overall economic inequality as measured by IAt therefore increased by roughly 7 % during the period from 1994/1995 to 2000/2001. Similarly, we can evaluate the relative index: IR2001 =

1 (1.03 + 0.991 + 1.07) = 1.03 3

(4.2)

The multidimensional inequality based on the evolution of RD thus increased by only 3% during the same period. In the following section we insert these results into the IEWB in order to analyse the impact of inequality on overall economic well-being.

4.3

Evolution of the IEWB including the modified equality dimension

In the final step of our analysis we evaluate the IEWB for the alternative inequality measures we computed in the preceding section. The indices IA and IR will be combined with data already presented and discussed in an earlier application of the IEWB to the case of France for the period 1980 to 2003 (op. cit.). Since we had to restrict the analysis of three-dimensional inequality to the years 1994/1995 and 2000/2001, the IEWB will also be evaluated for this shorter period. We hope to extend our analysis of three-dimensional inequality to a more recent date with the 2006/2007 edition of the BdF. For the time being, multi-dimensional inequality measurement is thus restricted to the two years in our dataset and a linear interpolation for the years between them. The IEWB dimension ‘equality and poverty’ is a weighted average of two items: an equality index (weighted with 0.25) and a measure poverty intensity (weighted with 0.75). According to the data presented in our earlier version of the French IEWB, poverty intensity decreased between 1994 and 2000 by 2.3 %. This is the combined result of two factors: first, we noted a decrease in the relative monetary poverty rate from 9.4 to 9.1 % (ibid., figure 11); second, the poverty gap increased slightly from 22.1 to 22.31 %. The poverty gap indicates the ‘depth’ of poverty and gives an additional information on the extent of poverty: when µP L is the average income of the poor population, the poverty gap is defined as P G =

4.3. IEWB WITH MODIFIED EQUALITY DIMENSION

93

(µP L − P L)/P L. It thus measures the overall distance of the poor to the poverty line.

In our sample, this distance remained almost constant between 1994 and 2000 in France.

Table 4.7 shows the combined effect of this 2.3 % decrease in poverty intensity and three alternative equality indices. Since a higher value of the IEWB indicates an improvement in well-being, the 2.3 % decrease of poverty intensity enters the ‘poverty and equality’ dimension as an increase of 2.3 %. The same holds for the inequality indices which are turned into equality indices.

Evolution from 1994 to 2000 (in %) Original version (Osberg & Sharpe) Standard income equality (Gini coefficient) Alternative three-dimensional measure Economic equality (absolute index IA ) Economic equality (relative index IR )

Equality index

IEWB dimension ity & poverty’

‘equal-

+0.7

0.75×2.3+0.25×(+0.7) = +1.9

−7.1 −3.0

0.75×2.3+0.25×(−7.1) = −0.1 0.75×2.3+0.25×(−3.0) = +1.0

Table 4.7: Impact of alternative inequality measurement on equality and poverty dimension — Data source: INSEE, Enquête Revenus Fiscaux; BdF. In our earlier application, we have used the standard Gini concentration coefficient as a measure of inequality. This corresponds to the original IEWB definition proposed by Osberg and Sharpe. The Gini coefficient we used is the series published by the INSEE and thus based on standard income inequality as observed in the Enquête Revenus Fiscaux (ERF), an administrative source of French employers’ fiscal declarations and the INSEE’s preferred source for calculations of the Gini coefficient.3 . According to the INSEE, the Gini coefficient decreased between 1994 and 2000 slightly by 0.7% (i.e. a 0.7% increase of income equality). Together with the 2.3 % decrease of poverty intensity this amounts to a 1.9% improvement of the IEWB dimension ‘poverty and equality’ as can be seen in Table 4.7.

The multi-dimensional indices we have derived in this text show a slightly different picture. If the relative index IR is included in the equality and poverty dimension, the improvement is only 1 %. If we think of inequality as average absolute distances, the IEWB dimension displays stagnation is this aspect of economic well-being (−0.1 %).

It is obvious that the process of aggregation in the IEWB will make the impact of alternative inequality measures less visible. The weight of the inequality index in the third IEWB dimension is 25 %: consequently, the deterioration of 7.1 % of equality translates into a decrease of the corresponding dimension of only 0.25 × 7.1% = 1.775 %. If we employ the standard weighting sheme in which all four dimensions of well-being are weighted equally, the impact on the overall IEWB is of course even smaller, namely only 0.25×0.25×7.1% = 0.44%. The effect of aggregation can be seen in the evolution of the IEWB for the three 3

Similarly to the BdF, the ERF is also object of vivid debate: see, for instance, the discussion in Conseil d’analyse économique (2001).

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CHAPTER 4. EMPIRICAL APPLICATION

different inequality indices from 1994 until 2000 (Table 4.8 and Fig. 4.1). Not surprisingly, the overall indicator of economic well-being is relatively insensitive to the choice of the equality index: the change during our observation period is 9.7 % if we include the standard income Gini coefficient, 9.2 % for the index IA , and 9.4 % for the index IR . Although the ‘big picture’ undergoes only slight modification if we move from one concept of inequality to another, this does not mean that our reflections are unimportant cosmetics.

Consumption Flows Wealth Stocks Poverty intensity index Gini index (Source: INSEE, ERF) Multidim. equal. (based on IA ) Multidim. equal. (based on IR ) Poverty & Equality (incl. Gini) Poverty & Equality (incl. IA ) Poverty & Equality (incl. IR ) Economic Security IEWB (incl. income Gini) IEWB (incl. IA ) IEWB (incl. IR )

1994 100 100 100 100 100 100 100 100 100 100 100 100 100

1995 100.8 103.3 100.4 100.5 98.8 99.5 100.4 100.0 100.2 99.0 100.9 100.8 100.8

1996 101.1 105.5 100.8 101.1 97.6 99.0 100.8 100.0 100.3 95.6 100.8 100.5 100.6

1997 100.7 109.6 101.1 101.5 96.5 98.5 101.2 100.0 100.5 99.5 102.8 102.5 102.6

1998 102.9 110.9 101.5 102.2 95.3 98.0 101.7 99.9 100.6 105.0 105.1 104.7 104.8

1999 105.2 111.8 101.9 101.8 94.1 97.5 101.9 99.9 100.8 107.1 106.5 106.0 106.2

2000 107.6 115.7 102.3 100.7 92.9 97.0 101.9 99.9 101.0 113.5 109.7 109.2 109.4

Table 4.8: Evolution of the French IEWB and its components 1994-2000.

First of all, we have to bear in mind that we are unfortunately restricted to an extremely short observation period. If the trend in our data continues, the impact of alternative inequality measures would become more and more visible through time. We have some reason to believe that the gap we observed between relative and absolute measures is not only temporary, but might reflect forces which are deeply embedded in the economic system of progressive societies. If average real monetary values grow, and if different parts of the population benefit from this growth via proportional increases of income and wealth, we expect a systematic divergence between measures based on concentration on the one hand, and absolute differences on the other. The longer the observation period, the stronger would be the impact on the overall IEWB — a hypothesis we have to test as more recent (or more reliable past) data becomes available.

Second, even if the overall impact is small, the IEWB provides not only the general vision, but also the draws some details of the the ‘big picture’ of economic development.

Hence, even if alternative inequality measures hardly modify the synthetic well-being indicator, it is a useful heuristic in its own right that inequality actually increased if we adopt the concept of average absolute differences. The decomposability is an important feature of the IEWB as it allows to contrast positive and negative developments side by side in the same methodological framework. For instance, we can compare the evolution of the

4.3. IEWB WITH MODIFIED EQUALITY DIMENSION

95

value of index

108

106

104

Standard IEWB

IEWB incl. IA

102

1994 1995 1996 1997 1998 1999 2000

time

Figure 4.1: Evolution of standard IEWB and the alternative IEWB with multidimensional inequality measurement based on average absolute distances. two dimensions which employ the concept of typical citizens and heterogeneous citizens we presented in Tab. 3.1. The former corresponds to the dimensions ‘effective consumption’ and ‘accumulation of wealth’, while the latter groups together ‘equality and poverty’ and ‘economic risk’. Figure 4.2 illustrates the evolution of the ‘typical’ and ‘heterogeneous’ well-being in our sample: the dimensions based on average values grew strongly and almost linearly. By contrast, once we take the heterogeneity of the French society into account, the development is more volatile and less positive. If we substitute the standard income Gini coefficient with our index IA , the difference is even more pronounced. This example illustrates the usefulness of the IEWB’s decomposability, since the contrast between ‘typical’ and ‘heterogeneous’ citizens would have been less visible in the overall IEWB. Similarly, despite the fact that the global development is less sensitive to the changes we proposed in this text, we argue that they contain relevant information for the interpretation of economic well-being.

Concluding remarks Inequality measurement has come a long way. The apparent constants in Pareto’s répartition de la richesse were replaced by different concepts used to measure changes in the distribution of economic resources. The analytical methods were improved significantly, not least thanks to a cross-fertilization between the theory of choice under uncertainty, information theory and inequality analysis. In this text, it has been argued that the so-

96

CHAPTER 4. EMPIRICAL APPLICATION value of index

110

Consumption & wealth

108 106 104 102 Equality & security (incl. Gini) 100 Equality & security (incl. IA ) 98 1994

1996

1998

2000

time

Figure 4.2: Evolution of homogeneous and heterogeneous dimensions.

phistication of methods has not necessarily led to a more accurate vision of economic inequality. Some conventions embedded in the use of standard inequality statistics such as the Gini coefficient or the welfare-based index proposed by Atkinson may be questionable.

As a matter of fact, these conventions are in opposition with the fact that many people think of inequality as absolute differences between economic positions, an observation introduced by Kolm (1976). Furthermore, the technical complexity of the academic debate has rendered these hidden controversies somewhat inaccessible to non-experts. The absence of feedback loops between the academic circles and the users of inequality statistics might be a serious problem as regards the legitimacy of these measures.

To overcome these issues, two alternative ways to assess multi-dimensional inequalities have been introduced within the framework of the IEWB. Both methods are based on the same simple graphical interpretation and therefore suit the public debate. Each of the two measures corresponds to a different assumption on what inequality is: the first one, the absolute average difference, takes into account the gap in real terms between the economic positions of individuals; the second, the relative average difference, continues the traditional assumption that inequality should be insensitive to proportional increases in all monetary values. These measures not only have the advantage of corresponding to the wide-spread conception of inequality as a ‘gap between rich and poor’, but also allow to integrate negative values for incomes or wealth — something that has been consistently

4.3. IEWB WITH MODIFIED EQUALITY DIMENSION

97

ignored by measures based on the concept of concentration.

In our data set the impact on the overall economic well-being of these alternative measures is rather limited. However, the question whether inequality is regarded as an absolute or a relative concept has profound consequences of great importance. The most obvious consequence is probably the ambiguous impact of economic growth. Traditionally, it has been assumed that economic growth — as long as it is spread throughout the population via wage adjustments and other mechanisms — has no effect whatsoever on economic inequality. The reason for the insensitivity of inequality to economic growth is that the former has been regarded as identical to the notion of concentration. Once this identity is questioned, as was done with the concept of the average absolute gap, economic growth might actually lead to increasing inequality. In progressive societies this is obviously a problematic issue calling for a re-assessment of the focus on economic growth as a means to overcome societal problems. In fact, if people think of inequality as absolute gaps, economic growth itself might be the cause of a societal problem, and not its solution.

The results presented in this text could be extended in at least two directions: first, the empirical findings could be tested as new data becomes available (a new edition of the BdF is currently in preparation). This would allow to test our hypothesis that in progressive societies such as France, an increasing difference between relative and absolute measures of inequality should be observed. In addition, we noted that the BdF is not the most reliable source to evaluate variables such as wealth and unemployment. Other sources focussing on theses issues (e.g. the ERF) could help to test the robustness of our empirical results, possibly allowing to bring the analysis from the household down to the individual level. We believe that the concepts developed in this text can easily be applied to other sources and variables. A second extension could focus on inequality and the analysis of the BdF data. In our empirical discussion we looked at the average gap between households in different dimension. However, the BdF data also allows to analyse the same household in all three identified dimensions of inequality. It is therefore possible to verify to what extent unfavourable or favourable positions in one dimension are correlated with the other dimensions. This would lead to an even finer picture of the extent of economic inequalities — beyond the IEWB framework — and answer the question whether gaps between households in the different dimensions either reinforce or offset each other. Again, the concepts developed in this text could be useful for such an extension.

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Compte rendu du mémoire en français Avant propos Le choix de rédiger le mémoire en langue anglaise est entièrement le résultat de considérations pratiques. A l’exception des travaux de Vilfredo Pareto et d’Alain Desrosières, l’ensemble des textes sur lesquels nous avons pris appui est issu de la littérature anglophone. Nous voudrions donc éviter des confusions créées par des traductions incorrectes ou maladroites d’une langue étrangère (l’anglais) à une autre (le français). De plus, la plupart des interlocuteurs de l’auteur de ces lignes n’a que des connaissances de base du français.

Les nombreuses discussions amicales et/ou professionnelles — pourtant extrêmement utiles pour l’achèvement du texte final — n’auraient donc pas pu s’effectuer si le mémoire était réservé à un public purement francophone.

Cependant, il nous semble être utile de présenter un compte rendu en français du texte entier. Dans les pages qui suivent, nous tentons de retenir non seulement les conclusions, mais aussi une partie des raisonnements qui en étaient à l’origine. Ainsi, est proposé pour chacun des quatre chapitres de la version anglaise un compte rendu de plusieurs pages.

D’une part, nous espérons que ceci sert à raccourcir la lecture pour ceux qui ne retrouvent leurs intérêts que dans une ou plusieurs parties. Ils pourraient alors compléter la lecture de ces chapitres avec les comptes rendus fournis ci-dessous. D’autre part, les lecteurs francophones peuvent s’y procurer une vision générale de notre approche méthodologique, de nos objets d’étude et des résultats de notre recherche.

Avant de commencer par le compte rendu par chapitre quelques remarques d’ordre pratique : toutes les références aux numéros de pages, à un certain passage du texte ou à un chapitre particulier renvoient au document complet en anglais. Une partie des graphiques et des formules a été reproduite dans les comptes rendus afin d’éviter des allers et retours non nécessaires entre les deux documents. Par ailleurs, dans la version en Portable Document Format (.pdf) toutes les références chiffrées peuvent être utilisées comme des liens en cliquant sur le numéro de la page, de la section ou du chapitre de la référence (les références dans la version anglaise qui renvoient à d’autres passages du même document peuvent d’ailleurs être utilisées de la même façon). Nous espérons que ceci facilite la lecture ainsi que les comparaisons des deux documents.

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Chapitre 1 Introduction et méthodologie Pourquoi la mesure des inégalités est toujours pertinente Les inégalités, et de manière générale la distribution des ressources, représentent des problèmes fondamentaux en économie. Nous sommes d’avis qu’il est important et utile d’analyser si et à quel degré des configurations économiques peuvent être caractérisées comme « inégales ». On peut même suivre R.H. Tawney dans son opinion que « la science économique devrait surtout traiter l’inégalité » (Tawney, 1964). Les inégalités économiques ont une influence importante sur un large éventail de préoccupations sociales et peuvent être analysées sous un angle philosophique (les questions d’équité et de justice), économique (les problèmes d’incitations et de l’allocation des ressources), ou bien sociologique (la fonction et le rôle des inégalités socio-économiques). Le mémoire se limite à une question très spécifique : comment mesurer les inégalités économiques dans le cadre de l’indicateur de bien-être économique (IBEE), un instrument proposé par les chercheurs canadiens Lars Osberg et Andrew Sharpe. Cet indicateur a été conçu comme une heuristique permettant à ses divers utilisateurs de faire des jugements sur plusieurs aspects du bien-être économique.

Il contient des informations statistiques concernant quatre dimensions : 1) la consommation effective ; 2) l’accumulation des stocks de richesses ; 3) les inégalités et la pauvreté ; et 4) la sécurité économique.

L’IBEE est désormais reconnu comme un instrument utile pour l’analyse économique.

Par ce dernier terme, nous désignons non seulement une activité exercée par des spécialistes de la science économique, mais aussi des analyses effectuées par d’autres acteurs qui s’intéressent aux aspects politiques, éthiques ou sociologiques liés aux résultats économiques.

Les usagers de l’IBEE dépassent donc le cercle des « experts » économiques (comme les statisticiens, les théoriciens de l’économie du bien-être etc.), et l’IBEE doit prendre en considération les conceptions et les représentations des autres acteurs qui en font usage.

Cependant, nous avons remarqué dans une application antérieure de l’IBEE aux données françaises (cf. Jany-Catrice & Kampelmann, 2007) que la place des inégalités économiques à l’intérieur de l’architecture de l’IBEE n’est pas entièrement satisfaisante. En effet, Osberg et Sharpe ont retenu les quatre dimensions du bien-être économique que nous venons de citer. Et pourtant, les inégalités n’y sont évaluées qu’en termes des revenus disponibles. Ceci ne correspond pas à l’idée que le bien-être repose sur l’ensemble des quatre 103

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dimensions. Autrement dit, si nous décidons que le bien-être économique est lié à la consommation, aux stocks de richesses et à la sécurité économique, nous devrions considérer les inégalités dans l’ensemble de ces dimensions et non seulement l’inégalité des revenus. Cette inconsistance dans la structure interne de l’IBEE a inspiré le thème du mémoire ainsi que les principales questions de recherche : sont les inégalités correctement prises en compte dans l’indicateur de bien-être économique ? Et si cela n’est pas le cas, comment l’indicateur peut-il être amélioré sans perdre sa transparence et son approche intuitive ?

A partir de ces questions sur la place des inégalités au sein de l’IBEE, le premier chapitre développe des réflexions sur le concept des inégalités en général. Ceci nous amène à une interrogation plus fondamentale et à poser une question encore plus importante : comment les inégalités économiques devraient-elles être mesurées ?

Etant donné la disponibilité d’une large gamme de mesures d’inégalités — y compris des indicateurs apparemment très légitimes tel que le coefficient de concentration de Gini — on pourrait penser que cette question ne mérite pas l’attention que nous sommes disponibles à lui prêter. En fin de compte, Osberg et Sharpe ont également opté pour la simple utilisation du coefficient de Gini pour rendre compte des inégalités de revenus dans la version originale de l’IBEE. Nous présentons trois arguments pour souligner la pertinence de la question comment les inégalités économiques devraient être mesurées : 1. Il n’existe pas un seul, mais une panoplie d’indices, de coefficients ou d’autres instruments statistiques qui visent à rendre compte des inégalités. Comme ces indicateurs sont souvent en contradiction les uns avec les autres, le choix d’une mesure spécifique n’est pas neutre et devrait être basé sur des arguments légitimes et plausibles (nous traitons les contradictions entre les différentes mesures disponibles dans le chapitre 2).

Evidemment, le choix d’une statistique doit correspondre directement à la problématique donnée (dans notre cas celle d’une application de l’IBEE). Par conséquent, il nous semble être nécessaire de vérifier si les mesures traditionnelles d’inégalités correspondent effectivement à l’usage que nous en voulons faire au sein de l’IBEE. Pour une telle analyse, nous devons regarder de plus près les différences entre les mesures alternatives, ainsi que les jugements et conventions encastrés dans leurs usages.

2. Le texte argumente que la mesure des inégalités contient des questions controverses qu’on retrouve de manière similaire dans les débats autour de la mesure de la pauvreté. A titre d’exemple, le point de vue que la pauvreté est un phénomène absolu est souvent contrasté à celui qu’il s’agit là essentiellement d’une position défavorisée d’une partie de la population relative à la situation du gros de la société. La question « qu’est-ce la pauvreté ? » ne semble pas permettre une réponse claire. Le texte argumente que la mesure des inégalités contient de façon pareille plusieurs aspects controverses qui sont analysés tout au long du mémoire. Par ailleurs, nous formulons l’hypothèse selon laquelle l’absence de ces controverses des débats peut être expliquée en partie par la complexité technique qui caractérise désormais cette mesure. Le débat semble être dominé par des spécialistes qui mobilisent des outils inaccessibles à une partie importante des usagers. Ceci pose des problèmes pour le propos transparent et plutôt démocratique d’un instrument comme l’IBEE. Par conséquent, le mémoire

105 vise à faire de la lumière dans les controverses liées à la mesure des inégalités et veut proposer des issues plausibles et bien fondés.

3. La troisième raison pour laquelle une discussion plus fondamentale sur la mesure des inégalités est pertinente est de nature pragmatique. Si l’idée d’une multidimensionnalité de l’inégalité est acceptée — comme le cadre d’analyse de l’IBEE le suggère — ceci introduit la difficulté de rendre compte des inégalités dans plusieurs espaces en même temps et d’agréger ces multiples espaces dans une mesure globale. Les statistiques traditionnelles tel que le coefficient de concentration de Gini ne sont pas directement applicables à des problèmes à plusieurs dimensions. De nouveau, la solution de cette difficulté doit correspondre à l’objectif général qui est défini par les usages qu’on peut faire de l’IBEE. Au centre de notre approche est donc le souci de joindre la logique générale de l’IBEE, d’un coté, et la mesure statistique des inégalités au sein de l’IBEE de l’autre. Par conséquent, les revues d’indicateurs d’inégalités disponibles dans la littérature ne nous peuvent servir que partiellement traitant souvent des interrogations plus techniques (comme le thème de la décomposabilité) ou font référence à des cadres d’analyse distincts (comme l’approche utilitariste). En revanche, notre réflexion est d’avantage axée sur la cohérence entre l’usage du concept de l’inégalité dans la communication normale et son opérationnalisation statistique.

A la suite du choix de problématique l’analyse exclut certaines interrogations importantes. Dans le texte anglais, nous fournissons les explications pourquoi les questions suivantes ne sont pas traitées explicitement dans le mémoire : 1) le problème général « Inégalité de quoi ? » inspiré par le livre Inequality Re-Examined de Sen ; 2) les interrogations spécifiquement lié à la mesure du bien-être, comme celui de la dichotomie entre l’utilité cardinale et ordinale ou la comparabilité interpersonnelle ; 3) les multiples causes des inégalités ne seront non plus analysées de manière explicite.

Le plan du mémoire est divisé en quatre chapitres. Le premier chapitre propose une analyse de la nature du concept de l’inégalité. L’analyse mobilise des éléments de réflexion issus de la sociologie de la connaissance, de l’approche conventionnaliste, ainsi que de la histoire de la raison statistique d’Alain Desrosières. Le chapitre vise à expliciter le cadre méthodologique du mémoire en fournissant en même temps une terminologie des concepts qui réapparaissent à plusieurs reprises dans le texte.

Le deuxième chapitre applique la méthodologie introduite dans le premier chapitre au discours académique sur la mesure des inégalités en sciences économiques. Dans un essai de rassembler des éléments d’une histoire interne des statistiques de l’inégalité, interprétées comme des conventions, nous discutons les contributions scientifiques dans ce domaine jugées comme les plus pertinentes pour notre propos : ce sont les travaux de Vilfredo Pareto, Max O. Lorenz, Corrado Gini, Hugh Dalton, Henri Theil, Anthony B. Atkinson et d’Amartya Sen. Le chapitre donne à part cela une revue des développements plus récents de la littérature. Cette histoire interne vise à mettre en relief l’évolution chronologique des conventions les plus importantes et montre de cette façon aussi le processus de légitimation

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des mesures les plus utilisées. Comme les conventions et leur légitimation sont souvent liées, une compréhension de l’établissement de certaines conventions au profit d’autres est importante pour notre question de recherche.

Le troisième chapitre propose deux manières alternatives de mesurer l’inégalité économique dans le cadre de l’IBEE. L’argumentation s’appuie sur l’acceptation des conventions qui nous semblent être légitimes et une remise en question des conventions qui sont moins plausibles étant donné le propos de l’IBEE. Cette argumentation confronte l’histoire interne du discours académique avec l’usage de l’IBEE. Ce dernier introduit alors dans la discussion d’une part les représentations et les usages du concept de l’inégalité par des acteurs externes au discours académique et d’autre part la logique interne de l’IBEE. Pour ceux qui ne sont pas familiers avec l’IBEE, le troisième chapitre donne au début une brève introduction à sa méthodologie.

Le dernier chapitre contient une application empirique des résultats du chapitre précèdent au cas de la France. La sensibilité des mesures alternatives de l’inégalité y est testée et discutée. Ce chapitre utilise une application antérieure de l’IBEE (cf. op. cit.) et mobilise des données issues de l’enquête Budgets de Familles.

Dans le texte anglais, l’introduction se termine par une description de la place de ce mémoire dans le projet personnel de l’auteur.

Discuter l’indiscutable : l’inégalité comme une convention Cette section situe le concept d’inégalité dans le cadre de plusieurs approches théoriques complémentaires, à savoir celle de la sociologie de la connaissance, celle de la théorie de conventions, ainsi que celle de l’approche historique d’Alain Desrosières. Ces théories sont mobilisées puisque les inégalités sont considérées comme des faits sociaux et qu’elles rentrent donc étroitement dans le champs d’analyse de ces théories. Loin de proposer une réflexion sociologique approfondie, le texte fait recours à ces approches pour éclaircir deux points importants qui sont liés à la discussion des inégalités : 1. La nature du concept « inégalité » provoque des questions épistémologiques que nous ne pouvons pas ignorer dans une discussion scientifique. Existe-t-il une « vraie » définition des inégalités économiques ? Et dans le cas contraire, comment émergent des définitions de ce concept ?

2. Il y a une relation importante entre la discussion scientifique à propos de l’inégalité et l’usage de ce même concept dans le langage courant. L’utilisation du terme « inégalité » au sein du discours académique correspond-elle à la manière telle qu’il est employé à l’extérieur du monde scientifique ? Un écart sémantique poserait-il des problèmes sérieux ou simplement une inconvenance négligeable ?

La nature du concept de l’inégalité Dans l’interrogation sur la nature du concept de l’inégalité un certain nombre de résultats de la sociologie de la connaissance est utilisé. Notamment le principe de la construction

107 sociale développé par Berger & Luckmann (1966) sert à souligner que les objets sociaux tels que les inégalités ne peuvent pas être classifiés comme des faits qui sont « vrais » indépendamment de toute communication interpersonnelle. A tout moment, plusieurs points de vue alternatifs sur un même objet peuvent concourir pour le statut d’être la vraie représentation de la réalité sociale. Ceci fournit des arguments à une position épistémologique, à savoir la position d’un relativisme selon lequel toutes les opinions ont la même validité.

L’issue classique d’une épistémologie relativiste est celui adoptée dans le texte : toutes les opinions ne sont pas valables, car, les conceptions partagées par une communauté donnée de personnes représentent une forte contrainte à ce qui peut compter comme des représentations correctes des objets sociaux. A l’intérieur de cette communauté de personnes la réalité sociale est créée par une co-construction entre les différents participants de la communication interpersonnelle. Ceci est un résultat important de notre discussion car le concept de l’inégalité rentre clairement dans la catégorie des constructions sociales. Par conséquent, il est impossible de vérifier la validité d’une définition quelconque des inégalités économiques sans tenir compte de la co-construction du terme au niveau de la société.

Pour s’assurer de la scientificité de notre propos, la nature du concept de l’inégalité nous force à analyser notre problème du point de vue d’une co-construction. Ceci constitue une partielle remise en question de l’approche traditionnelle qui domine la littérature spécialisée, qui a plutôt tendance à construire des mesures statistiques des inégalités de manière « unilatérale », c’est-à-dire sans l’intervention des acteurs externes au débat scientifique.

Un regard inspirateur sur le processus de construction sociale est celui développé par Desrosières (1993) dans son histoire de la raison statistique. En analysant les mécanismes de l’objectivation dans le domaine statistique, Desrosières met l’accent sur les éléments construits — et in fine arbitraires — de ces références apparemment indiscutables comme par exemple les statistiques officielles. Il montre que le processus d’objectivation des faits sociaux repose avant tout sur des conventions, une vision ici adoptée pour la discussion des inégalités.

En sciences économiques, des interrogations autour du thème des conventions ont donné naissance aux théories économiques de conventions. Le texte ne retient que quelques éléments de base de ces théories comme par exemple celui de l’interprétation des conventions comme un dispositif cognitif collectif, proposée par O. Favereau (1989, p. 295). Cette interprétation des conventions nous permet d’analyser la construction sociale des représentations statistiques (l’aspect cognitif ) ainsi que le processus de co-construction (l’aspect collectif ). Dans ce contexte nous citons l’étude de Gadrey & Jany-Catrice (2007), qui applique le concept de conventions au débat sur la mesure de richesse. En effet, ces auteurs montrent que dans la discussion des mesures alternatives au PIB, certains acteurs prennent conscience du caractère conventionnel des mesures traditionnelles et mettent en question la légitimité du PIB pour des évaluations du bien-être. Selon Gadrey et Jany-Catrice les conventions qui perdent en légitimité sont ceux qui « concernent la représentation globale de ce qui compte et de ce qui devrait être compté au titre de la richesse d’une nation, et de la contribution de diverses activités ou patrimoines » (ibid., p. 103). Dans le contexte d’une discussion des inégalités au sein de l’IBEE, les conventions sur ce qui devrait être compté sont déjà encastrées dans l’architecture même de l’indicateur, notamment dans le choix des

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variables et des dimensions. Par conséquent, le mémoire se concentre sur les conventions plutôt d’ordre technique : ces derniers doivent correspondre aux représentations et l’usage du concept des inégalités partagés par les utilisateurs de l’IBEE. Pour analyser la correspondance entre conventions techniques de mesure et conceptions des utilisateurs il nous semble être utile de nous approprier l’approche historique de Desrosières. Ceci permet de comprendre à quel moment et pour quelles raisons les conventions les plus importantes ont émergé dans la littérature. Cependant, la méthodologie plus ambitieuse de Desrosières est modifiée. En effet, ce dernier pouvait s’appuyer sur son ample expérience comme chercheur-sociologue et fonctionnaire-statisticien pour présenter à la fois une histoire interne (des méthodes, théorèmes etc.) et une histoire externe (des acteurs, des laboratoires, des processus opératoires etc.). Bien que cette double vision nous semble être très instructrice, il nous n’est pas possible d’en fournir l’équivalent pour la genèse des mesures d’inégalité, notamment à cause d’un manque d’expérience et de temps. Par conséquent, le texte ne rassemble dans un premier temps que des éléments d’une histoire interne du discours scientifique sur la mesure statistique des inégalités économiques. Cependant, nous confrontons dans le troisième chapitre cette vision interne des inégalités avec les représentations des acteurs externes et la logique de l’IBEE. Après avoir présenté des arguments en faveur d’une vision du concept de l’inégalité comme une convention, le texte prolonge le raisonnement en y introduisant un aspect normatif. En effet, nous sommes d’avis que pour être légitime, le processus d’objectivation doit reposer sur une co-construction des mesures d’inégalité plutôt que sur un monologue des spécialistes techniques. Ceci nous mène à penser qu’une absence de communication entre les usagers des résultats statistiques et les experts techniques est susceptible d’aboutir dans certains cas à une pseudo-légitimité des mesures des inégalités. La co-construction des statistiques qui s’appuie sur des allers et retours entre les différentes sphères d’acteurs (feedback loops), est nécessaire pour établir la légitimité des mesures et une cohérence sémantique entre l’utilisation scientifique et non scientifique. Dans ce contexte nous introduisons le terme de la communication normale employé par Sen (1975), qui a été le premier à insister sur le fait que la science économique n’est pas complètement libre de définir le terme de l’inégalité de manière arbitraire (ibid., pp. 47-78). En effet, Sen indique qu’il est problématique que les différentes définitions d’inégalité qui ont émergé dans la littérature puissent aboutir à des résultats contradictoires. A titre d’exemple, il est possible que la même modification d’une distribution de revenus soit interprétée simultanément comme une diminution de l’inégalité en terme d’utilité, une stagnation de l’inégalité en terme de revenus et une augmentation de l’inégalité évaluée par l’indice d’Atkinson (cf. la discussion p. 45). Le constat de ces contradictions possibles conduit Sen à traiter la communication normale comme une contrainte au débat scientifique, qui doit rester « raisonnablement proche » au langage courant. En nous appuyant sur les résultats de Desrosières, le texte argumente qu’une telle contrainte ne rend pas suffisamment compte du principe de co-construction car elle impose uniquement que le monde de la science observe la communication normale sans qu’il y ait une communication bilatérale (absence de feedbacks). L’importance d’une telle communication pour la légitimité du processus d’objectivation est très claire dans l’analyse de Desrosières, comme le montrent ses observations citées en p. 12. Nous interprétons ces

109 remarques de Desrosières comme des arguments indiquant que le langage scientifique doit correspondre au langage commun pour obtenir sa légitimité. Dans notre contexte, ce langage commun peut être délimité par celui des usagers de l’IBEE décrit dans le chapitre 3.

En somme, notre interrogation sur la nature du concept de l’inégalité et sur la relation entre le discours scientifique et la communication normale nous conduit à traiter la mesure de l’inégalité comme une convention. Tandis que dans le deuxième chapitre le texte examine la genèse des conventions dans le domaine scientifique, ce dernier est confronté à d’autres considérations plus externes dans le chapitre 3. Ceci aboutit à deux mesures alternatives qui visent non seulement à être plus consistantes vis-à-vis la logique interne de l’IBEE, mais aussi à mieux correspondre aux emplois du terme « inégalité » dans la communication normale.

Etant donné les conclusions du paragraphe précédent, il serait incohérent de commencer la discussion par une définition précise de l’inégalité. Le concept a été construit au fil du temps et par des acteurs différents et c’est justement ce processus qui est analysé dans le chapitre 2. A la fin, le premier chapitre donne encore quelques remarques sur des termes liés au concept de l’inégalité qui réapparaissent à plusieurs reprises dans le texte.

Premièrement, une distinction est faite entre les concepts tels que la concentration, la diffusion, la dispersion, l’entropie, la variation d’une part et celui de l’inégalité d’autre part. Bien que la littérature utilise de manière récurrente des analogies entre ces différents termes qui sont d’ailleurs certainement liés, nous pensons que le terme de l’inégalité possède un contenu sémantique indépendant et qu’il n’est donc pas identique à la concentration ou encore à la dispersion. L’usage synonyme de ces termes risque d’ignorer des éléments importants du débat à propos de la mesure de l’inégalité.

Une autre distinction soulignée dans cette section est celle entre la pauvreté et l’inégalité. Bien que la tendance vers une vision de la pauvreté comme un phénomène relatif plutôt qu’absolu ait vraisemblablement rapproché les significations respectives, une différence sémantique entre les deux concepts persiste. Tandis que la pauvreté reste une description de la situation de ceux qui se retrouvent en bas de la distribution des ressources, l’inégalité est concernée par des questions relatives à différentes parties de cette distribution.

Enfin, le texte rappelle une terminologie introduite par Rosenbluth, qui distingue entre deux types d’instruments descriptifs utilisés dans le contexte de la mesure de l’inégalité économique : 1) un tableau ou graphique, qui permet à analyser différentes parties de la distribution et 2) un indice qui compare des distributions entières (1951, p. 935). Nous rappelons également que chacune de ces deux formes offre à la fois des avantages et des inconvénients. En effet, tout indice est insensible à un nombre infini de modification de la distribution et ignore donc des variations jugées comme négligeables. En revanche, les instruments graphiques sont sensibles à un nombre plus élevé de modifications mais souvent, ils ne permettent pas d’en tirer des conclusions en ce qui concerne le développement global de l’inégalité.

Chapitre 2 Une histoire interne du discours académique sur la mesure des inégalités Ce deuxième chapitre applique l’analyse en termes de conventions au discours académique sur les inégalités économiques pour en fournir des éléments d’une histoire interne.

Ceci dit, il est évident que la littérature en sciences économiques qui traite la mesure d’inégalités est un vaste champ. Même si on restreint le sujet à celui de l’inégalité de revenus ou de capital, la littérature concernée a désormais rendu impossible toute tentative de synthèse cohérente. Une illustration de l’énorme quantité de textes fondateurs est la longueur de la biographie retenue dans On Economic Inequality d’A. Sen qui s’étend sur 31 ( !) pages. La maîtrise de cette littérature est clairement l’œuvre d’une vie entière et l’auteur de ces lignes est conscient de ses limitations à cet égard.

Le premier chapitre contient la proposition d’adopter une perspective chronologique pour discerner les conventions importantes dans la mesure des inégalités. Pour effectuer une telle analyse chronologique une sélection des textes s’impose. Par conséquent, le deuxième chapitre commence avec la présentation du critère de choix appliqué à la sélection. Le critère retenu est l’impact sur les conventions des différentes contributions scientifiques.

A l’aide de ce critère sont sélectionnés, dans un premier temps, les textes fondateurs de V. Pareto, C. Gini et M. O. Lorenz. En effet, la spécification mathématique des distributions proposée par Pareto, le coefficient de concentration de Gini, ainsi que la courbe de Lorenz sont devenus des dispositifs standard pour représenter des distributions empiriques et des référents communs dans l’analyse des inégalités. Dans un deuxième temps, sont retenues les contributions qui ont le plus marqué la mesure des inégalités en terme de bien-être (welfare), une approche qui est également devenue standard dans la littérature académique : ce sont les textes de H. Dalton, A. B. Atkinson et A. Sen. Une autre contribution qui a fortement influencé les méthodes scientifiques de la mesure de l’inégalité est celle de H. Theil, qui a mis en avant le thème de décomposition, et par là inspiré des nombreuses recherches théoriques et empiriques sur des questions liées au problème de rendre les statistiques d’inégalités décomposables.

Ces sept contributions sont donc analysées en vue de leur impact sur les conventions de mesure. Par conséquent, le deuxième chapitre est loin d’être une histoire complète et n’apporte peu d’information technique au lecteur déjà familier avec les auteurs sélectionnés.

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111 En revanche, le chapitre vise à rendre visible quelques conventions cruciales encastrées dans les mesures apparemment légitimes et fréquemment utilisées. Il est argumenté que ceci permettra une analyse critique de la légitimité de ces derniers dans le chapitre 3.

La Loi de Pareto : l’inégalité constante ou décroissante ?

La première contribution discutée est celle de Pareto. Tout d’abord, une distinction est faite entre, d’une part, l’analyse parétienne de la répartition de la richesse et, d’autre part, la définition et la mesure des inégalités de revenus de Pareto. Bien que ces deux éléments soient entremêlés dans l’analyse de Pareto, nous constatons que la définition des inégalités proposée par Pareto mérite une attention à part entière. Notamment le contraste avec les mesures d’inégalités proposées plus tard par Lorenz et Gini montre clairement l’évolution des conventions dans cette première période de l’analyse quantitative de l’inégalité, dont Pareto est le précurseur.

Avant Pareto, le problème de l’inégalité avait été traité presque exclusivement sous un angle qualitatif, notamment par K. Marx, qui contraste la situation économique des différentes classes. Ceci inspira les remarques suivantes du libéral français P. Leroy-Beaulieu : « L’influence des lois économiques sur la répartition des richesses est un sujet beaucoup moins exploré que l’influence des mêmes lois sur la circulation. [...] Sans doute les volumes sur ce qu’on appelle les questions ouvrières abondent, mais la plupart sont absolument vides, sans rien de précis, de positif et de scientifique » (citation dans Busino, 1964).

Pareto s’approprie cette critique de Leroy-Beaulieu et l’accorde avec sa propre préférence pour l’économie politique comme une « science dure » (cf. son Cours, publié en 1896).

Par conséquent, il analyse la répartition de la richesse comme un phénomène quantitatif et avec une approche inductive. Le résultat le plus connu de l’analyse parétienne est sans doute sa Loi de Répartition, qui peut être résumée de manière relativement simple. Après avoir observé des formes de répartition remarquablement semblables pour toutes les séries de données dont il dispose, Pareto propose la formule suivante qu’il proclame valide pour toutes les économies et à tout temps : log N≥y = log A − α log y où « y est un montant de revenu [individuel], N≥y est le nombre de personnes qui reçoivent un revenu de ce montant ou plus élevé, A et α sont constants, le premier variant avec le nombre total de revenus considérés, le dernier une vraie constante puisqu’elle apparaît d’être presque la même pour des pays différents, autour de 1.5 » (Edgeworth, 1926, pp. 712713 ; notation harmonisée avec le texte).

En plus de la Loi de Répartition, le texte discute la définition particulière d’inégalités de Pareto. Cette dernière est dérivée par Pareto indépendamment de sa Loi. Avant de la présenter, il est rappelé que Pareto avait longtemps refusé de formuler une définition de la notion « diminution des inégalités » : « Il vaut mieux éviter ce terme ambigu » était encore sa position en 1897. Puis, dans le deuxième volume de son Cours, il s’interroge enfin sur la question du mémoire : « Mais quelle est la vraie signification des termes : moindre inégalités des revenus [...] ? » (ibid, p. 318).

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Etant donné son postulat d’une « loi naturelle » de la répartition de la richesse, nous argumentons que la discussion de Pareto des changements d’inégalités peut sembler contradictoire. En effet, en opposition avec la constance des paramètres de sa spécification, Pareto constate : « Actuellement, dans nos sociétés, il parait bien que c’est ce dernier cas [une diminution des inégalités] qui se vérifie, et un grand nombre d’observations nous font connaître que le bien-être du peuple s’est, en général, accru dans les pays civilisés » (ibid, p. 323).

Quelle est la définition d’inégalités sur laquelle est basée cette observation ? La définition formulée par Pareto est influencée par les idées de Leroy-Beaulieu, qui avait proposé un concept proche à celui de la pauvreté relative : « Les progrès du bien-être de la classe inférieure de la population sont [...] plus rapides que ceux de la classe moyenne et de la classe élevée. Sans arriver à un nivellement des conditions qui est impossible [...] le mouvement économique actuel conduit à une moindre inégalité entre les fortunes. » A ceci Pareto rajout : « La diminution de cette inégalité sera donc définie par le fait que le nombre de pauvres va en diminuant par rapport au nombre des riches. [...] En général, lorsque le nombre des personnes ayant un revenu inférieur à x augmente par rapport au nombre des personnes ayant un revenu supérieur à x, nous dirons que l’inégalité des revenus augmente » (ibid., p. 320).

Cette définition est exprimée mathématiquement comme suivant : uy =

N≥y N

où N est la population totale. Lorsque uy augmente, l’inégalité au niveau y diminue. Il est donc nécessaire d’évaluer uy à tous les niveaux de revenu pour mesurer l’inégalité totale.

A partir de cette expression mathématique, le texte illustre le lien que Pareto établit entre sa Loi et sa définition d’inégalité. En effet, il montre qu’une valeur plus élevée du coefficient α indique une inégalité plus élevée, et vice versa. Dans ce contexte, il est observé que la mesure uy — un instrument du type 1 dans la terminologie de Rosenbluth présentée dans la section 1.3 — est ainsi transformé en indicateur synthétique. En d’autres termes, le coefficient α peut être utilisé comme un raccourci pour évaluer uy à tous les niveaux de y. Cependant, ce résultat ne peut être exploité que si la distribution en question suit la forme spécifiée par la Loi de Pareto. Si cela n’est pas le cas, le coefficient α n’a pas la même signification et ne peut pas être interprété comme un indice synthétique d’inégalité.

Néanmoins, la mesure uy reste a priori valide pour représenter les inégalités même si la Loi de Pareto n’est pas vérifiée.

Ceci nous conduit à identifier deux propriétés supplémentaires de la mesure uy : 1. Premièrement, cette mesure est sensible aux changements du revenu moyen : une augmentation de tous les revenus par une somme égale et une multiplication de tous les revenus par un scalaire positive conduisent à une moindre inégalité en termes de uy .

2. Deuxièmement, la définition de Pareto peut être interprétée comme faisant une distinction entre concentration et inégalité.

113 Le texte rappelle que le revenu total n’apparaît pas dans le calcul et que ce sont des nombres relatifs de personnes qui y sont analysés. De plus, notons la ressemblance entre uy et la formule standard utilisée aujourd’hui pour rendre compte de la pauvreté relative : K≡

N<P L N

où N<P L est le nombre de personnes qui vivent avec un revenu en dessous du seuil de pauvreté P L. En effet, la mesure K est un point particulier de uy , à savoir y = P L tel que N L . L’intuition pour cette mesure de l’inégalité est donc basée sur K = 1 − uP L = 1 − ≥P N une notion de pauvreté relative au lieu de concentration.

Après la présentation de l’analyse parétienne de la répartition et de l’inégalité de la richesse, le mémoire résume l’impact de Pareto sur les conventions dans le domaine en question. Pour ceci, un bref compte rendu de la réception de la Loi de Pareto est fourni.

Il semble que la disparition de la mesure d’inégalité uy de la littérature est vraisemblablement dû au fait que de nombreux auteurs postérieurs à Pareto ne font pas de distinction entre la définition du concept d’inégalité et la Loi de Répartition. La stratégie de Pareto d’interpréter le coefficient α en termes d’inégalité semble avoir largement contribué à cette confusion. Même si la définition particulière d’inégalité de Pareto n’a pas marqué longtemps le discours scientifique, nous rappelons trois points qui illustrent l’impact de Pareto sur les conventions : 1. Pareto, dans un objectif de rendre l’analyse des inégalités plus scientifique, était un précurseur des méthodes quantitatives dans ce domaine. Bien que ceci ne soit pas la seule approche à la question d’inégalités, elle semble être dominante en sciences économiques jusqu’à nos jours.

2. En liant sa mesure d’inégalité et sa Loi, Pareto a été un pionner dans l’identification de mesures sommaires ou synthétiques (summary measures) d’inégalité : si la Loi de Pareto est valide, le coefficient α synthétise toute l’information sur l’inégalité en un seul chiffre. Ceci est devenu la méthode standard et n’a pas été sérieusement mise en question jusqu’à la critique de Sen des classements complets (voir notre discussion dans la section 2.1.7, p. 55).

Ces deux points sont pertinents pour le problème de mesurer l’inégalité dans le contexte de l’IBEE. Pareto était inspiré par les demandes de Leroy-Beaulieu que la mesure des inégalités devrait être plus « précise », plus « positive » et plus « scientifique ». En optant pour une approche de quantification, l’IBEE vise également à contribuer des « chiffres solides » au débat sur le bien-être, sinon Osberg et Sharpe auraient adopté la forme d’un résumé littéraire sur le même sujet.

Par ailleurs, le mémoire travaille avec une mesure en forme d’indice au lieu d’une représentation graphique du bien-être car la dimension égalité et pauvreté de l’IBEE contient une mesure synthétique. La mesure α de Pareto nous avertit des dangers d’une telle procédure.

En effet, les mesures synthétiques risquent d’être décontextualisées et ont souvent tendance à s’autonomiser lors d’une utilisation par d’autres acteurs. La mesure synthétique α de Pareto n’a du sens que si on accepte aussi sa définition de l’inégalité. Cependant, des auteurs

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postérieurs (y compris Gini, 1915 ; Dalton, 1920 ; et Lorenz, 1905) ont jugé cette mesure synthétique sous l’angle de leurs définitions (respectives). Nous évoquons dans ce contexte l’importance d’une communicabilité et d’une intuitivité élevées des mesures d’inégalités.

La courbe de Lorenz : l’inégalité comme concentration La discussion de l’apport de Lorenz débute sur une brève présentation de la proposition relativement simple de cet auteur d’évaluer la concentration d’une distribution de revenus : « La méthode est la suivante : tracez selon une axe le pourcentage cumulé de la population du plus pauvre au plus riche, et selon l’autre le pourcentage de la richesse totale détenue par ces pourcentages de la population » (Lorenz, 1905, p. 217). Les courbes qui résultent de cette méthode peuvent être évaluées facilement en termes de concentration : « la règle d’interprétation sera que plus le courbe est coudée, plus la concentration est élevée » (ibid., p. 217).

Le texte illustre cette approche à l’aide des exemples et des graphiques pour les cas des courbes avec et sans intersection. Sont également évoqués les avantages de la méthode proposée par Lorenz : elle peut être appliquée à des distributions de tailles et de valeurs totales différentes ; les graphiques ne font pas recours aux logarithmes et sont donc plus intuitives ; les courbes peuvent facilement être transformées en mesure synthétique.

Cependant, est souligné que le succès qui connut la courbe de Lorenz est non seulement le résultat de l’éloquence d’une méthode graphique. Lorenz marque la discussion des inégalités surtout en déclarant que la concentration et l’inégalité devraient être vues comme des synonymes. Le titre de sa communication originale indique clairement que son auteur est intéressé par une « méthode de mesurer la concentration de la richesse ». Lorenz utilise le terme de l’égalité comme étant l’opposé de la concentration, c’est-à-dire comme synonyme de « diffusion ». Il est alors un de premiers auteurs à proposer une dichotomie entre les extrêmes de l’égalité d’un coté, et la concentration complète, de l’autre.

A l’aide de cette définition de l’inégalité, il réfute facilement presque toutes les mesures qui avaient été discutées avant : les méthodes proposées par Wolf, Soetbeer, Holmes et Pareto ne passent pas son test puisqu’elle ne font pas l’égalisation entre la concentration et l’inégalité, qui semble être la seule vision possible pour Lorenz. Au lieu d’argumenter contre les définitions alternatives de l’inégalité, il y voit des « erreurs » et des « raisonnements fallacieuses » étant donné sa vision que l’inégalité est identique à la concentration.

En ce qui concerne l’impact de la méthode de Lorenz sur les conventions, deux points importants sont discutés : 1. Lorenz est à l’origine d’une équivalence entre les termes de la concentration et de l’inégalité. Son raisonnement est basé sur l’agrégat de revenus : ce ne sont pas les montants absolus de revenu de chaque individu qui entrent dans le calcul, mais que leurs parts dans le revenu total. En remplissant la notion de l’« inégalité de revenus »avec le contenu du concept de la « diffusion du revenu total », Lorenz exprime plus ou moins explicitement que les différences de revenus entre individus en valeurs absolues ne sont pas importantes : seules les différences de leurs revenus relatifs se-

115 raient pertinentes. A l’exception de quelques contributions isolées (voir Kolm, 1976 ; Blackorby & Donaldson, 1980), ceci est devenu l’approche standard dans la littérature.

2. Plus clairement que Pareto, Lorenz évoque le problème des comparaisons ambiguës entre distributions différentes. Lorsque les courbes de concentrations affichent une intersection, un jugement immédiat quant à leurs degrés d’inégalité est très difficile.

La distinction entre les comparaisons qui permettent des décisions claires et celles qui nécessitent une analyse plus extensive est devenue un thème récurrent dans la littérature.

En vue du problème d’identifier une mesure des inégalités satisfaisante dans le cadre de l’IBEE, ces deux points doivent être pris en compte. Comme il est souligné dans la section 1.1, un critère important qu’une telle statistique devrait satisfaire est son adéquation avec les représentations de ses usagers potentiels. L’idée de l’inégalité économique comme concentration est-elle représentative pour les conceptions de non experts ? Il semble que la plupart de personnes — et pas seulement les égalitaristes radicaux — assignerait au moins une importance faible aux différences absolues entre les revenus individuels. Ce point est approfondi dans la section 2.3 et dans le chapitre 3.

Le coefficient de Gini : un complément à la courbe de Lorenz Dans cette section le double apport de Corrado Gini dans le domaine de l’analyse des inégalités est résumé. D’une part, il utilise comme Pareto des spécifications logarithmiques pour approximer la distribution des revenus. Mais sa contribution la plus importante pour nous est un catalogue de mesures de la variabilité, de la concentration et de la mutabilité qui est résumé dans son livre Variabilità e Mutabilità, publié en 1921. Nous discutons deux de ses mesures synthétiques : la différence moyenne absolue (DMA) et la différence moyenne relative (DMR), qui peuvent être écrites mathématiquement comme suivant : PN PN i=1 j=1 |yi − yj | DMA = N2 PN yi DMA DMR = où µ = i=1 µ N La DMR est devenue la méthode standard dans la mesure empirique de l’inégalité, notamment grâce au lien avec la courbe de Lorenz. Le lien est l’indice de Gini, qui est défini comme le ratio entre l’aire de concentration (l’aire entre la droite d’équi-répartition et la courbe de concentration) et l’aire de concentration maximale (qui correspond au cas ou le revenu total est concentré dans une seule main) : aire de concentration DMR = aire de concentration maximale 2 Le coefficient de Gini peut être appliqué à des distributions de toutes sortes de variables quantitatives, comme le souligne son auteur dans une réponse à Dalton (1920). La mesure G≡

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de concentration de Gini n’est pas le résultat d’un raisonnement en termes de bien-être ou d’utilité et peut être employé pour l’analyse de la distribution de revenu, de chaussures ou de puces. Cette universalité de l’indice de Gini contribue sans doute à sa réputation d’être une mesure objective et neutre.

Au fur et à mesure que la courbe de Lorenz est devenue la méthode standard pour la représentation graphique des distributions de valeurs monétaires, le coefficient de Gini apparaît comme la mesure synthétique la plus utilisée. Ceci a eu un impact considérable sur les conventions dans le domaine de l’analyse des inégalités. Deux points sont discutés : 1. Le coefficient de Gini a renforcé la pratique de résumer toute l’information sur une distribution dans un seul chiffre. L’avantage de la statistique de Gini consiste dans le fait qu’elle ne nécessite pas d’estimation d’une spécification mathématique de la forme de la distribution (comme le faisait le coefficient α de Pareto). Le coefficient de Gini permettait donc de décider sans ambiguïté laquelle de deux distributions est la plus inégale et était donc une solution au problème d’intersections entre deux courbes de concentration.

2. L’utilisation répandue de la DMR a consolidé le point de vue qu’inégalité et concentration sont essentiellement les mêmes concepts. Puisque la courbe de Lorenz est un instrument pour représenter graphiquement la concentration d’une distribution, le coefficient de Gini est aussi une mesure de concentration. La dominance de la DMR dans le débat scientifique est d’autant plus intéressante que Gini présentait dans son livre Variabilità e Mutabilità aussi des mesures de dispersion absolue, notamment la DMA. Implicitement, l’IBEE prolonge la tradition de Gini dans la mesure où l’indicateur d’Osberg et Sharpe vise à résumer toute information sur le bien-être économique (ou ses différentes composantes) dans un seul chiffre. La validité de cette approche est analysée dans la section 2.1.7 (p. 55).

Une autre conclusion du succès du couple Gini/Lorenz concerne le processus de réception et pénétration des mesures d’inégalités. Il semble que la communicabilité et intuitivité sont aussi importantes pour qu’une mesure soit acceptée que la pureté conceptuelle et l’élégance mathématique. Dans ce contexte Lars Osberg est cité, qui dans un article de 1985 rappelle que ce ne sont pas forcément les statistiques les plus « correctes » qui s’imposent au débat publique.

Dalton et l’effet de l’inégalité sur le bien-être Nous présentons ici la rupture épistémologique introduite par Hugh Dalton en 1920.

Cet auteur a été le premier à insister sur une évaluation directe de l’effet de l’inégalité des revenus sur le bien-être (welfare). Selon Dalton, l’économiste est avant tout concerné par les conséquences de la répartition des revenus. Dans son approche, la description de l’inégalité n’est donc plus un objectif de la recherche : Dalton passe directement à l’évaluation à l’aide du critère d’une fonction de bien-être à maximiser. L’inégalité économique entre dans cette

117 maximisation par la « coïncidence très spéciale » (Sen, 1973, p. 16) que la fonction de bienêtre utilisée par Dalton atteint sa valeur maximale lorsque tous les individus sont dotés avec le même montant de la variable analysée.

Les hypothèses sur la fonction de bien-être et le raisonnement de Dalton sont illustrés dans le graphique ci-contre qui représente le bien-être économique d’une société composée de deux individus. Les courbes regroupent les points pour lesquelles le bien-être de la société est au même niveau. La droite Y Y 0 représente toutes les repartions possibles du revenu total Y = Y 0 entre les deux individus. Par conséquent, le point A est une distribution inégale puisque individu 1 reçoit une part plus importante de Y qu’individu 2. Il est à noter que plus on s’approche sur la droite Y Y 0 du point A vers la droite d’équi-répartition EE 0 , plus les courbes d’indifférence indiquent des niveaux de bien-être plus élevés. Le bienêtre maximal pour un niveau de Y donné est atteint dans le point B, où les deux individus reçoivent la même part de Y . Plus la distribution observée de Y s’éloigne du point B, plus l’inefficience en termes de bien-être est importante. revenu indiv. 2 Y E0 µ B

I2 A

I1

45˚ E

Y0

revenu indiv. 1

Fig. 3 – Bien-être économique dans le cadre utilitariste simple.

Le raisonnement de cet exemple peut être aisément généralisé à des sociétés de tailles plus élevées. Ceci conduit Dalton à définir l’inégalité en termes de l’inefficience distributionnelle qu’elle entraîne : nw(µ) Bien-être total maximal D≡ = Pn Bien-être total observé i=1 w(yi )

n

où

1X µ= yi n i=1

Cette mesure d’inégalité requiert une spécification plus précise de la fonction de bienêtre pour permettre une application empirique. Le texte présente la spécification proposée par Dalton et montre que l’approche de Dalton pose le problème d’un paramètre libre qui doit également être spécifié pour obtenir des valeurs numériques pour D. Un autre apport important de Dalton a été sa méthode de tester l’acceptabilité des mesures alternatives d’inégalités à l’aide d’une liste de « principes ». Tous ces principes

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sont des conséquences du cadre utilitariste et résultent donc directement des hypothèses sur la forme de la fonction de bien-être. Nous nous contentons ici de nommer les principes retenus par Dalton et reportons le lecteur aux explications dans le texte (p. 35). Les quatre principes de Dalton sont : 1) le principe de transferts ; 2) le principe d’additions proportionnelles aux revenus ; 3) le principe d’additions égales aux revenus ; 4) le principe d’additions proportionnelles aux personnes.

Cette liste définit comment une mesure acceptable d’inégalité devrait réagir aux modifications contenues dans les différents principes. Appliquée à une batterie d’indicateurs alternatifs d’inégalité, cette approche conduit Dalton à classer le coefficient de Gini comme la mesure la plus acceptable en vue du critère de maximisation du bien-être.

Par ailleurs, le texte souligne deux aspects de l’héritage de Dalton : 1. L’apport le plus important de Dalton est sa modification réussie de l’objectif de la mesure d’inégalité : selon Dalton, elle ne devrait pas décrire les inégalités observées ou essayer de répondre à des questions comme « les inégalités dans ce pays croissentelles ? ». La tradition initiée par Dalton exclut l’élément descriptif de la discussion et passe directement au jugement normatif des distributions. Suivant ce précurseur, une partie importante de la littérature sur les inégalités économiques discute des formes alternatives de la fonction de bien-être au lieu de décrire les inégalités.

2. La méthode de Dalton d’employer un test sous forme d’une liste de principes est devenue une convention dans la littérature (cf. Theil, 1967 ; Atkinson, 1970 ; Sen, 1972 ; et Kolm, 1976, qui l’utilise comme un ensemble d’axiomes). Il est souligné que cette approche est une conséquence du caractère flou du concept inégalité et permet d’énoncer de manière transparente les desiderata utilisées pour décider sur l’acceptabilité des mesures. Il est à noter que non seulement la méthode de lister les différentes caractéristiques d’une mesure acceptable est devenue conventionnelle. Aussi certains des éléments de la liste de Dalton se sont autonomisés du cadre utilitariste et apparaissent dans les desiderata d’autres auteurs. Ceci est certainement le cas du principe de transferts utilisé par Atkinson (1970), Kolm (1976) et Theil (1967), ce qui montre que ce principe est perçu comme un dispositif cognitif collectif par les spécialistes de l’analyse d’inégalité.

L’analogie de Theil et le thème de décomposabilité La statistique d’inégalité proposée par Henri Theil est fondée sur une analogie entre probabilités et la distribution d’un montant d’argent. Cette section contient une explication de cette analogie en exposant les principaux concepts de la théorie d’information utilisés dans le programme de Theil. Ensuite, l’impact de l’apport de Theil est identifié et discuté.

Selon la théorie d’information, on distingue différents messages par rapport à leur contenu en information. De manière générale, ce dernier dépend de l’utilité et de la nouveauté de l’information du message, dans le sens que le message peut modifier la connaissance du récipient de la réalité. Si le message est déjà connu, son contenu en information est faible. Supposons que nous ignorons si un certain évènement a eu lieu. Puis, suppo-

119 sons que le message en question contient l’information que cet évènement a effectivement eu lieu. Il est clair que le contenu informationnel de ce message dépend de la probabilité d’occurrence de l’évènement. Lorsqu’il est absolument certain que l’évènement a été réalisé (i.e. la probabilité d’occurrence est égale à 1), le contenu en information du message est zéro. En revanche, un message qui nous dit qu’un évènement avec une faible probabilité a eu lieu contient un haut degré d’information.

Dans la théorie d’information cette relation négative entre probabilité (x) et contenu informationnel (h(x)) est formalisée de manière particulière : 1 x Un autre concept nécessaire pour la compréhension de la statistique de Theil est le contenu informationnel espéré. Supposons que nous observons un système complet qui consiste de N évènements indépendants E1 , · · · , EN , et qu’un seul parmi ces évènements aura lieu. Les probabilités de ces évènements sont : h(x) = log

xi , i = (1, ..., N )

avec

N X

xi = 1

xi ≥ 0

et

i=1

Nous définissons maintenant un message particulier : après un de ces N évènements a lieu, un message définitif et fiable sera reçu contenant l’information quel Ei a effectivement été réalisé. Il est possible de former une opinion sur le contenu informationnel espéré de ce message avant qu’il ne soit reçu. De nouveau la valeur de ce contenu dépend des probabilités des évènements : si l’occurrence d’un évènement du système est certaine, le contenu informationnel espéré du message est zéro. Ce raisonnement est formalisé dans la théorie d’information en définissant le contenu espéré comme la somme pondérée de l’ensemble des h(xi ). Les pondérations sont simplement les probabilités xi . Avant que le message soit transmis et qu’il nous informe lequel des N évènement a eu lieu, sa valeur informationnelle espérée H est alors : H(x) =

N X i=1

xi h(xi ) =

N X

xi log

i=1

1 xi

où x à gauche est le vecteur des N probabilités. La valeur minimale de H(x) est zéro, son maximum est atteint dans le cas où tous les évènements ont la même probabilité, i.e. les N événements ont la probabilité 1/N . Dans ce cas, la valeur de H(x) est log N .

Le passage du contenu informationnel espéré à la mesure d’inégalité de Theil est la similitude formelle entre probabilités et les parts dans une distribution de revenus : les deux sont toujours positifs, et leur somme est égale à 1. Il est donc techniquement possible de calculer une valeur numérique pour H(x) en remplaçant le vecteur des probabilités par un vecteur des parts dans le revenu total. Ceci aboutit à la mesure d’inégalité de Theil, qui est définie comme la différence entre H(x) et sa valeur maximale log N : T ≡ log N − H(s) = log N −

N X i=1

si log

1 si

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Cette mesure d’inégalité T varie entre zéro (égalité complète) et log N (inégalité complète).

L’apport original de Theil consiste dans la décomposabilité de l’inégalité globale mesurée par T en inégalité entre différents groupes et inégalité au sein de ces groupes. Dans le texte l’arithmétique de cette décomposition est récapitulée pour arriver à l’expression suivante : T = log N −

X i=1

si

1 =B+W si

où B = log N −

k X

Sg log

g=1

W =

k X g=1

" Sg log Ng −

X si i=1

1 Sg /Ng

1 log Sg si /Sg

(inégalité entre groupes) # (inégalité au sein des groupes)

On peut observer que B a la même forme que T — la mesure globale d’inégalité —, avec la différence que les parts dans le revenus des individus sont remplacées par le ratio (Sg /Ng ).

Ce ratio reflète les différences du revenu par tête entre les k groupes. B est donc interprétée par Theil comme l’inégalité entre les groupes.

Theil propose W comme mesure d’inégalité totale à l’intérieur des groupes. La différence entre crochets dans l’expression W est également de la forme de H(x), sauf que cette fois-ci sont remplacées les parts des individus dans le revenu total par les parts respectives dans le revenu du groupe. L’expression entre crochets est donc interprétée comme l’inégalité à l’intérieur du groupe g. L’inégalité intra-groupe est alors W , la somme pondérée des k inégalités. Les pondérations sont les parts respectives des groupes dans le revenu total Sg .

La mesure T repose sur la ressemblance formelle de probabilités et parts dans le revenu total. Mais rien n’assure que le remplacement du vecteur de probabilités dans la valeur du contenu informationnel espéré par le vecteur de parts individuelles dans le revenu aboutisse effectivement à une mesure d’inégalité. Pour argumenter que ceci est le cas, Theil mobilise — à l’image de Dalton — une liste de caractéristiques qu’une mesure d’inégalité doit posséder pour qu’elle soit une statistique acceptable. Cette liste contient les éléments suivants : 1. La mesure doit atteindre sa valeur minimale lorsque la distribution est caractérisée par une égalité complète, définie comme la situation dans laquelle tous les individus reçoivent la même proportion du revenu total.

2. Le maximum de la mesure d’inégalité correspond à l’inégalité complète, définie comme la situation dans laquelle une personne reçoit l’intégralité du revenu global et le reste de la population ne touche rien. Cette exigence est moins évidente que la première.

Elle indique que pour Theil, l’argument de Lorenz de définir l’inégalité complète comme concentration complète peut être traité comme un dispositif cognitif collectif. En effet, Theil ne fournit aucune justification élaborée concernant cette vision

121 d’inégalité complète sauf le simple constat que « concentration et inégalité sont essentiellement les mêmes concepts » (ibid., p. 128).

3. La mesure doit indiquer une diminution des inégalités si une somme est transférée d’une personne plus riche à une personne plus pauvre jusqu’au point où les deux revenus sont égaux. Ceci est équivalent au Pigou-Dalton principe de transferts (cf. p. 37). De nouveau ce principe est « un test évident » pour Theil, tandis que Dalton et Pigou avaient encore fait recours aux mathématiques pour prouver la validité de cette exigence.

4. Le degré maximal d’inégalité dans une situation d’inégalité complète dépend de la taille de la population N . En effet, la mesure T n’a pas de borne supérieure fixe et la valeur maximale log N dépend clairement de N . Selon Theil, plus une population est importante, plus l’inégalité potentielle est grande : une économie de N membres dans laquelle une personnes possède toutes les richesses contient moins d’inégalité qu’une économie avec N + k membres (k étant un nombre entier positif) dans laquelle une personne possède tout.

5. Une modification proportionnelle de tous les revenus (qui donc n’affecte pas les parts dans le revenu total) ne change pas la valeur de la mesure d’inégalité. Ceci est un corollaire du point deux sur cette liste et une propriété héritée de Lorenz (voir la discussion p. 26).

6. La mesure doit être facilement décomposable en inégalités inter- et intra-groupes.

Ceci veut dire que la statistique doit être indépendante des choix alternatifs pour diviser la population en groupes. La somme des inégalités des groupes doit toujours être égale à l’inégalité total.

Theil montre que sa mesure d’inégalité T passe tous ces tests et qu’elle apparaît donc comme une mesure acceptable.

L’introduction du thème de décomposition élargit les exigences posées aux mesures d’inégalités. Theil rajoute une demande supplémentaire que les statistiques candidates doivent satisfaire pour être acceptables. Cependant, il est à noter que la justification pour ajouter cette propriété est d’ordre pratique, et pas nécessairement de nature conceptuel.

Il peut être utile dans le cadre de différentes études de pouvoir décomposer une mesure, par exemple en fonction des groupes sociodémographiques. Par conséquent, la propriété de décomposabilité a inspiré une multitude de contributions dans la littérature spécialisée (voir la partie 2.2, p. 59).

Pour le problème lié à l’IBEE, la réflexion doit se positionner par rapport à aux moins deux questions qui seront traitées dans le chaptre 3. Premièrement, la décomposabilité estelle utile pour notre propos ? Et deuxièmement, la mesure de Theil est-elle une statistique acceptable dans le contexte de l’IBEE ?

Ensuite, le texte note que point 4 sur la liste ci-dessus ne va pas de soi. Il n’est pas évident qu’une économie complètement concentrée de deux personnes soit moins inégale qu’une économie complètement concentrée de trois, quatre ou N personnes. Car lorsque une personne monopolise le revenu total, il est vrai que tous les individus sauf un sont

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complètement égaux par rapport à leurs revenus. Plus nous rajoutons des personnes à une distribution complètement concentrée, plus le nombre de personnes complètement égales augmente. Bien qu’une telle distribution soit vraisemblablement caractérisée par une profonde malaise en termes de pauvreté, de justice ou d’autres considérations, il semble inadéquat de définir l’inégalité complète comme une situation dans laquelle tous les individus sauf un sont complètement égaux.

L’indice d’Atkinson : une focalisation sur les méthodes analytiques Dans cette section l’indice d’inégalité développé par Anthony B. Atkinson en 1970 est discuté. Par la proximité de l’analyse d’Atkinson aux idées de Dalton (voir section 2.1.4), une comparaison de ces deux contributions s’impose et leurs similitudes et différences sont analysées.

Comme Dalton, Atkinson part de l’hypothèse d’une fonction de bien-être sociétal de la forme suivante : W ≡

N X

U (yi )

avec

0 ≤ yi ≤ k

et

i = 1, ..., N

i=1

La forme de la fonction U (y) est caractérisée par deux inégalités : dU (y) >0 dy

avec

d2 U ≤0 dy 2

Est discuté par la suite comment le Théorème d’Atkinson résulte de ces définitions. En effet, Atkinson montre qu’avec les spécifications ci-dessus, il existe un lien entre l’information contenue dans la courbe de Lorenz et les comparaisons entre distribution en termes de W . Le Théorème d’Atkinson dit que si, et seulement si, des courbes de Lorenz n’ont pas d’intersection la question de l’inégalité des différentes distributions peut être tranchée sans ambiguïté. Sans intersection des courbes de Lorenz, toutes fonctions qui vérifient la spécification ci-dessus résultent dans le même rangement des distributions en termes d’inégalité.

Une conséquence importante de ce théorème est que lorsqu’on observe une intersection des courbes de Lorenz, il est nécessaire de spécifier la fonction U avec plus de précision pour pouvoir décider quelle distribution est la plus inégale. La solution proposée par Atkinson au problème de spécification de U introduit la notion du revenu équivalent distribué également.

Nous exposons cette notion à l’aide d’un exemple d’une économie hypothétique de deux personnes.

Le diagramme ci-contre montre les courbes d’indifférences qui correspondent au bienêtre total de différentes distributions alternatives du revenu total Y entre deux individus.

Si la distribution initiale est le point A, le niveau de bien-être généré par cette distribution est celui qui correspond au bien-être de la courbe d’indifférence Iw . Il est clair que ce niveau de bien-être peut être atteint avec un revenu total plus faible si sa répartition est modifiée.

Le graphique montre que le montant le plus faible qui génère le même niveau de bien-être

123 que la distribution A est égal à 2 × yede . En effet, si chacun des deux individus reçoit exactement yede , comme c’est le cas de la distribution C, le niveau du bien-être total serait identique à celui du point A. Dans ce cas, yede est appelé le revenu équivalent distribué également de la distribution A. Il est à noter que le revenu moyen µ de la distribution A revenu indiv. 2 Y

µ B

yede

C Iw

A 45˚

Y0

0

revenu indiv. 1

Fig. 4 – Illustration du revenu équivalent distribué également. est plus élevé que yede . Seul au point C µ et yede coïncident. Ceci conduit Atkinson à définir sa mesure d’inégalité comme suit : A≡1−

yede µ

En fonction de la différence entre µ et yede , cette mesure est 0 (égalité complète), 1 (inégalité complète), ou prend des valeurs intermédiaires. Suite à cette définition d’inégalité, Atkinson peut préciser directement la forme fonctionnelle de yede . Ceci rend une comparaison de toutes distributions possible (même en cas d’intersection des courbes de Lorenz), et aboutit à des valeurs numériques pour la mesure A. En effet, yede peut être exprimé en fonction de U (y) : N U (yede ) =

N X

U (yi )

i=1 N 1 X U (yi ) U (yede ) = N i=1

L’inverse de la fonction U (y) donne une expression pour yede en fonction de y : yede (y) = U −1

!

N 1 X U (yi ) N i=1

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Lâ&#x20AC;&#x2122;indice dâ&#x20AC;&#x2122;inĂŠgalitĂŠ devient :

A=1â&#x2C6;&#x2019;

U yede (y) =1â&#x2C6;&#x2019; Âľ

â&#x2C6;&#x2019;1

P N 1 N

i=1 U (yi )

Âľ

Lorsque U (y) est spĂŠcifiĂŠe et peut ĂŞtre inversĂŠe, lâ&#x20AC;&#x2122;indice dâ&#x20AC;&#x2122;inĂŠgalitĂŠ A peut ĂŞtre ĂŠvaluĂŠ empiriquement.

La spĂŠcification particulière pour la fonction U (y) proposĂŠe par Atkinson est explicitement basĂŠe sur les conventions dans le domaine dâ&#x20AC;&#x2122;indicateurs dâ&#x20AC;&#x2122;inĂŠgalitĂŠ. En effet, il dĂŠfinit sa mesure A dâ&#x20AC;&#x2122;une telle façon quâ&#x20AC;&#x2122;elle reste inchangĂŠe si tous les revenus sont augmentĂŠs proportionnellement. Pour que lâ&#x20AC;&#x2122;indice A ait cette propriĂŠtĂŠ, Atkinson donne la forme suivante Ă la fonction U (y) : y1â&#x2C6;&#x2019; si 6= 1 et â&#x2030;Ľ 0 1â&#x2C6;&#x2019; U (y) = log y si = 1 Dans le texte lâ&#x20AC;&#x2122;insensibilitĂŠ de A Ă des multiplications proportionnelles de tous les revenus est illustrĂŠe. Cette propriĂŠtĂŠ conduit Ă un indice A de la forme suivante pour toutes les valeurs positives de :  1 P  ( N1 yi1â&#x2C6;&#x2019; ) 1â&#x2C6;&#x2019; 1â&#x2C6;&#x2019; si 6= 1 et â&#x2030;Ľ 0 Âľ A= 1 P exp ( N logyi )  si = 1 1â&#x2C6;&#x2019; Âľ Lâ&#x20AC;&#x2122;avantage de lâ&#x20AC;&#x2122;approche dâ&#x20AC;&#x2122;Atkinson est quâ&#x20AC;&#x2122;elle permet de restreindre Ă lâ&#x20AC;&#x2122;aide dâ&#x20AC;&#x2122;un nombre très faible dâ&#x20AC;&#x2122;hypothèses toutes les spĂŠcifications possibles de A Ă la classe dĂŠfinie ci-dessus. Cependant, il existe une infinitĂŠ de A Ă cause du paramètre qui doit ĂŞtre prĂŠcisĂŠ pour toute application empirique. Atkinson propose lâ&#x20AC;&#x2122;interprĂŠtation suivante pour ce paramètre : ÂŤ Dans ce cas, la question est limitĂŠe Ă choisir , qui est clairement une mesure du degrĂŠ dâ&#x20AC;&#x2122;aversion dâ&#x20AC;&#x2122;inĂŠgalitĂŠ â&#x20AC;&#x201D; ou la sensibilitĂŠ relative Ă des transferts Ă diffĂŠrents niveau de revenu. Lorsque augmente, nous attachons plus dâ&#x20AC;&#x2122;importance Ă des transferts dans le bas de la distribution et moins dâ&#x20AC;&#x2122;importance Ă des transferts dans le haut de la distribution. Le cas limite dans un extrĂŞme est â&#x2020;&#x2019; â&#x2C6;&#x17E; qui correspond Ă la fonction mini {yi }, qui prend uniquement en compte des transferts au groupe des revenus les plus faibles (et qui nâ&#x20AC;&#x2122;est donc pas strictement concave) ; dans lâ&#x20AC;&#x2122;autre extrĂŞme, nous avons = 0 qui correspond Ă la fonction dâ&#x20AC;&#x2122;utilitĂŠ linĂŠaire qui ĂŠvalue les distributions uniquement selon le revenu total Âť (Atkinson, 1970, p. 257).

Le texte illustre comment cette interprĂŠtation est dĂŠrivĂŠe dâ&#x20AC;&#x2122;une analogie entre lâ&#x20AC;&#x2122;aversion contre le risque dans la thĂŠorie de choix. A lâ&#x20AC;&#x2122;aide dâ&#x20AC;&#x2122;un exemple lâ&#x20AC;&#x2122;impact du choix de sur les courbes dâ&#x20AC;&#x2122;indiffĂŠrences et montrĂŠ (p. 52). Par ailleurs, est rĂŠsumĂŠ comment lâ&#x20AC;&#x2122;indice dâ&#x20AC;&#x2122;Atkinson a ĂŠtĂŠ prĂŠsentĂŠ comme une alternative attractive au coefficient de Gini : avec

125 son théorème, Atkinson a d’abord établi une relation avec la courbe de Lorenz ; ensuite, il introduit le concept d’aversion contre l’inégalité et montre que le coefficient de Gini contient un degré d’aversion plus élevé au milieu de la distribution qu’aux franges. L’interprétation claire de A en termes de bien-être et son paramètre d’aversion contre l’inégalité ont permis de mettre en cause la position dominante du coefficient de Gini dans le débat académique.

En ce qui concerne l’impact sur les conventions d’Atkinson dans le domaine de la mesure d’inégalité, les points suivants sont analysés : 1. L’indice d’Atkinson continue la tradition d’employer des mesures sommaires pour synthétiser toute l’information sur l’inégalité dans un seul chiffre. La raison pour opter pour une mesure sommaire semble être purement conventionnelle : « L’approche conventionnelle dans presque tous les travaux empiriques est d’adopter une statistique sommaire d’inégalité comme [...] » (ibid., p. 244).

2. La décision de retenir la propriété d’insensibilité de A à une augmentation proportionnelle est également basée sur une référence explicite aux conventions : « Maintenant, nous avons vu que presque toutes les mesures conventionnelles sont définies relative à la moyenne [...] » (ibid., p. 257). Ceci est un point très important car sans cette propriété il serait considérablement moins évident d’obtenir des valeurs numériques pour A (cf. Kolm, 1976).

3. En plaçant son approche dans le cadre d’analyse de Dalton, Atkinson pouvait présenter sa mesure comme étant en continuité à une contribution plus ancienne et presque classique. En enrichissant le raisonnement en terme de bien-être, Atkinson a éloigné le discours académique des indicateurs purement descriptifs et renforcé l’idée d’évaluer directement les conséquences des inégalités au lieu de leur ampleur.

4. Atkinson a modifié le principe de transferts de Dalton en exigeant qu’une mesure acceptable d’inégalité soit plus sensible en bas de la distribution. L’argument en faveur de cette modification est l’intuition d’Atkinson qu’une sensibilité constante aux transferts ne serait pas acceptée par la plupart des gens.

5. Avec son refus de l’approche descriptive, son raisonnement approfondi en terme de fonction de bien-être et son exigence d’une sensibilité aux transferts variable, Atkinson a sans doute contribué à la complexification des instruments analytiques. Nous rappelons d’ailleurs qu’Atkinson a exprimé explicitement ses préférences pour des considérations théoriques au profit de la communicabilité des mesures d’inégalité.

Le tour de force conceptuel de Sen Après un bref commentaire sur l’apport global de Sen au débat sur l’inégalité économique, le texte discute quatre points qui sont importants pour la présente discussion.

Tout d’abord, il est rappelé que Sen préconise une position intermédiaire entre d’un coté, l’approche descriptive ou « objective » à la mesure de l’inégalité et, de l’autre coté, les évaluations normatives en terme de bien-être. Sen propose que le concept d’inégalité possède un caractère dual qui mélange ces deux éléments. Ceci attribut une place plus importante à

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la description d’inégalité et, par là, à la correspondance des mesures d’inégalité à la « communication normale ». Pour Sen, un indicateur acceptable doit non seulement satisfaire des considérations d’ordre technique, mais également être « raisonnablement proche » à l’usage normale du concept d’inégalité. Dans ce contexte, le texte montre comment Sen utilise le concept de « communication normale » à plusieurs reprises comme une contrainte à la validité des mesures alternatives d’inégalité (p. 56). Il semble que Sen emploie ce terme comme étant un objet que l’économiste peut identifier de manière relativement simple.

Malgré ce rappel en faveur de l’intégration d’éléments descriptifs, la méthode de Sen reste encastrée dans l’évaluation de l’inégalité en termes du bien-être. Cependant, il argumente que l’approche utilitariste n’est pas adaptée à l’analyse de l’inégalité car elle associe le maximum de la fonction de bien-être à la notion de l’égalité qu’à cause d’une « coïncidence très spéciale » . Sen propose d’introduire la préférence pour l’égalité directement par des axiomes explicites.

Le problème de décider si l’inégalité est un concept relatif est également abordé par Sen, mais apparemment sans une décision claire en faveur ou contre l’insensibilité des mesures d’inégalité à une multiplication proportionnelle. Sen juge que la question si l’inégalité est un concept absolu ou relatif est un dilemme conceptuel. Cependant, dans la version élargie et mise à jour de On economic inequality, publiée en 1997, Sen souligne que « traditionnellement on pense de l’inégalité comme un concept relatif » et rejette l’introduction d’un « élément absolutiste » dans les statistiques d’inégalité.

Enfin, un autre élément pertinent de la réflexion théorique de Sen est l’observation que la complétude des mesures statistiques est problématique. En effet, Sen remarque que le concept d’inégalité n’est pas suffisamment clair pour toujours permettre des jugements sans ambiguïté. Pour décider laquelle de deux distributions est la plus inégale, il serait nécessaire qu’un contraste relativement marqué entre elles soit observable. Cette réflexion conduit Sen à proposer l’instrument de rangement par intersection, qui vise à identifier les situations plus faciles à évaluer. Le principe d’un rangement par intersection est d’évaluer toute une batterie de statistiques d’inégalités pour isoler les cas conflictuels de ceux où toutes les statistiques conduisent au même rangement des distributions. En termes techniques, on définit d’abord un ensemble de k rangements, C j , for j = 1, ..., k, qui sont tous a priori plausibles. L’intersection de ces k rangements complets est annotée Q et peut être écrite comme : yQx si, et seulement si ∀j = 1, ..., k : yC j x Ceci veut dire que la distribution y obtient une place plus élevée dans le rangement Q que la distribution x si le même rangement est observé pour toutes les mesures d’inégalité C j .

Par exemple, si le rangement par intersection est défini comme l’intersection du coefficient de concentration de Gini et le ratio interdécile D9 /D1 , le rangement Q permet d’ordonner les distributions si le rangement C G du coefficient de Gini ne contredit pas le rangement C ID du ratio interdécile. Cependant, force est de constater que l’approche de rangements par intersection contient toujours un élément arbitraire à cause de la sélection nécessaire de mesures qui y sont intégrées.

Ensuite, le texte approfondie la réflexion sur les deux apports de Sen les plus importants

127 pour nos questions : la position particulière de la communication normale dans l’œuvre de Sen et l’hypothèse de l’incomplétude du concept d’inégalité.

En effet, l’utilisation de la communication normale comme une contrainte ne rend pas suffisamment compte du fait que les conceptions sont les résultats d’une co-construction : elles ne peuvent pas être construites de manière unilatérale par la science. Il semble que le problème de rester « raisonnablement proche » à la communication normale est plus difficile à résoudre que l’approche de Sen ne le laisse penser.

L’hypothèse d’incomplétude du concept d’inégalité introduit un problème central des mesures sommaires. Les rangements complets issus des statistiques synthétiques pourraient en effet être plus précises que le concept d’inégalité lui-même. Par conséquent, la précision des statistiques introduit forcément un élément arbitraire dans l’analyse. Néanmoins, une solution à cette ‘sur-précision’ des mesures sommaires doit rester communicable et transparente. La panoplie d’indicateurs contenue dans un rangement par intersection risque d’introduire une multitude de concepts différents dans une seule mesure et accroître la complexité informationnelle et l’opacité des statistiques.

Les généralisations récentes des méthodes Malgré le fait que le tour de force conceptuel de Sen soit la dernière contribution discutée en détail dans ce chapitre, l’analyse scientifique de l’inégalité économique a connu des évolutions importantes depuis la première édition de On economic inequality de 1973, et même après la version actualisée de 1997. Une exemple de développements récents est la tendance d’insister sur des approches multidimensionnelles, dont l’IBEE est indirectement le résultat. D’autres innovations conceptuelles et théoriques peuvent être identifiées. Cependant, il est argumenté que la littérature académique sur la mesure empirique des inégalités économiques fait d’avantage preuve d’un approfondissement des méthodes conventionnelles déjà en place à partir des années 1970. Le résumé de développements récents par Jenkins & Micklewright (2007) indique également que nous sommes en présence d’un procès de consolidation et d’amélioration des approches conventionnelles. Cinq axes de recherche peuvent être cités pour illustrer la plausibilité de cette hypothèse : 1. La courbe de Lorenz et le théorème d’Atkinson ont été les bases pour les concepts de dominance généralisée de Lorenz et de dominance stochastique de Shorrocks (1983), Foster & Shorrocks (1987) et d’autres.

2. L’indice d’Atkinson a été généralisé et a donné naissance à la « famille d’Atkinson » (Jenkins & Micklewright, 2007, p. 13). La dérivation de classes d’indices paramétriques avec des caractéristiques normatives explicites a été systématisée et approfondie.

3. L’indice décomposable de Theil a inspiré une classe d’entropie généralisée de mesure d’inégalité développée par Bourguignon (1979) et d’autres.

4. D’autres développements, comme le traitement systématique des erreurs d’échantillonnage et la dérivation des intervalles de confiance pour les mesures d’inégalités

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COMPTE RENDU DU MÉMOIRE EN FRANÇAIS proposés par Beach and Davidson (1983) sont directement basés sur la méthode de Lorenz et les courbes généralisées de Lorenz.

5. La critique de Paglin (1975) de ne pas confondre l’inégalité entre et à l’intérieur de familles est essentiellement une extension du coefficient de concentration de Gini et de l’approche de Lorenz.

Cette section termine avec la conclusion que la focalisation sur la concentration — initiée par Lorenz et Gini — et l’approche basée sur la fonction de bien-être par Dalton et Atkinson sont devenues des conventions acceptées dans ce domaine. Suite à des généralisations et extensions de ces concepts, la mesure de l’inégalité semble aujourd’hui d’être plus complexe et technique qu’auparavant.

Proche de la ‘vérité’ ou du citoyen ?

Pour terminer ce chapitre sur l’histoire interne du discours académique sur la mesure de l’inégalité, trois aspects sont discutés qui semblent indiquer que le progrès scientifique dans ce domaine ne se traduit pas forcément par un rapprochement aux conceptions de la communication normale.

Premièrement, la présence gênante d’un arbitrage entre communicabilité et complétude analytique de la discussion scientifique est rappelée. Cet arbitrage n’a pas été suffisamment pris en compte et a rendu les mesures plus difficilement communicables. Par conséquent, un nombre élevé d’acteur est désormais exclu du débat ; leurs conceptions risquent d’être ignorées ou au moins sous-représentées. Or, une mesure avec l’objectif d’assister au débat public comme l’IBEE doit confronter ce problème et s’assurer d’une communicabilité acceptable afin de permettre l’intégration des conceptions des usagers dans la construction des mesures statistiques.

Deuxièmement, l’arbitrage entre communicabilité et complétude théorique crée une conséquence directe pour la distinction entre d’une part, les mesures descriptives et, d’autre part, les approches plus normatives basées sur l’évaluation d’une fonction de bien-être. Il semble que ces dernières augmentent le degré de complexité de la mesure empirique et risquent de la rendre plus opaque pour les utilisateurs non spécialistes.

Enfin, malgré le progrès méthodologique considérable, ils restent des problèmes conceptuels qui sont loin d’être résolus de manière satisfaisante. La focalisation de la littérature scientifique sur les mesures relatives n’est pas suffisamment balancée et ignore les points de vue alternatives observables dans la communication normale.

Chapitre 3 La mesure des inégalités dans l’IBEE Une brève introduction à l’IBEE d’Osberg et Sharpe Ce chapitre contient des éléments de contexte sur l’indicateur de bien-être économique proposé par L. Osberg et A. Sharpe. Tout d’abord, nous présentons les objectifs et les usagers de cet instrument, ensuite la genèse de l’IBEE est esquissée.

Quant aux objectifs, l’IBEE a été conçu comme un outil d’évaluation de la performance économique des politiques gouvernementales et, de manière générale, comme une heuristique sur l’état économique d’une société. En analysant les explications des auteurs canadiens dans leurs articles sur l’IBEE, le texte montre que l’usager ciblé n’est pas l’expert technique qui base une recherche sophistiquée sur les résultats de l’indicateur. L’IBEE assiste plutôt le « citoyen », qui dans l’exercice de ses devoirs démocratiques nécessite une vision globale et informée sur la réalité (socio-) économique. L’objectif de l’IBEE est donc d’assister le débat public en fournissant un outil d’évaluation au citoyen. Il vise à résumer l’information pertinente dans un format compréhensible. Une conséquence de cet objectif est que les « perceptions populaires » jouent un rôle important dans l’analyse d’Osberg et Sharpe et que l’indicateur devrait permettre aux usagers d’y retrouver leurs systèmes de valeurs.

Les bases théoriques de l’IBEE ont déjà été établies en 1985 dans un article de Lars Osberg. Cependant, la première application empirique n’a été effectuée qu’en 1998 pour l’économie canadienne. Le texte cite les applications qui ont suivi ce premier exemple et note que l’IBEE possède désormais une place dans le débat international autour de la mesure du bien-être économique (p. 64).

La méthodologie de l’IBEE Sont présentés dans cette section l’architecture générale de l’indicateur ainsi que les concepts qui y sont intégrés. Les quatre grandes dimensions du bien-être identifiées par Osberg & Sharpe (2005) sont les suivantes : « (1) Flux effectifs de consommation par tête (valeur monétaire à prix constants) — affectée d’un indice de progression de l’espérance de vie, et ajustée pour tenir compte des variations du temps de travail annuel par personne. A cette 129

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consommation s’ajoutent les dépenses gouvernementales par tête, hors service de la dette4 , ainsi que le travail domestique non rémunéré et bénévolat5 .

(2) Stock de capital national physique net par tête (valeur monétaire)6 — auquel sont ajoutés les stocks de R&D par tête (valeur monétaire7 ), les variations dans la valeur du patrimoine de ressources naturelles par tête (valeur monétaire)8 , les variations de stocks de capital humain9 . De ces stocks sont déduits la dette extérieure nette par tête, ainsi que les coûts des émissions polluantes10 .

(3) Indicateur synthétique de pauvreté combinant le taux de pauvreté monétaire et une mesure de l’intensité de cette pauvreté. Un indicateur d’inégalité de la distribution des revenus compose également une partie de cette dimension.

(4) Sécurité économique — des risques économiques liés au chômage, à la maladie, aux risques de rupture familiale (familles monoparentales) et à la pauvreté des personnes âgées. » Osberg et Sharpe sont conscients du fait que le bien-être économique n’est pas un objet homogène. Cependant, ils argumentent qu’il est nécessaire de « faire la somme des choses qui sont conceptuellement distinctes » afin d’aboutir à des évaluations du bienêtre économique global. Le tableau ci-dessous résume les différents concepts contenus dans l’IBEE. Concept « Citoyen typique » ou « agent représentatif » « Citoyens hétérogènes »

Temps Présent Futur Flux moyens de revenus Accumulation courants productifs

de

stocks

Distribution — inégalité des Insécurité des revenus furevenus et pauvreté turs

Tab. 9 – Concepts dans l’IBEE. Source : Osberg & Sharpe, 2005.

Le caractère hybride de la compilation de données et de leur synthèse statistique est une des forces de l’indicateur : plutôt que d’imposer une vision hégémonique en termes 4

Il n’y a pas ici de soustraction de dépenses jugées « défensives » comme cela était le cas dans les travaux initiaux de Nordhaus et Tobin.

5 L’estimation de la valeur de l’heure de travail domestique est effectuée sur la base d’un salaire horaire de personnel domestique.

6 Méthode dite d’inventaire permanent, appliquée à tous les stocks mesurables d’équipement productif privé ou public, locaux d’habitation, infrastructures...

7 Comptabilisé par attribution d’un taux d’amortissement de 20 % aux séries de flux.

8 En fonction des données nationales ou internationales existantes, on peut tenter d’inclure des valeurs estimées pour les ressources en minéraux principaux, forêts, et réserves d’énergie (ce que font les auteurs pour le cas du Canada).

9 Coûts de l’éducation de l’ensemble de la population, estimés sur la base des coûts par niveau d’études et de la répartition de la population par niveau.

10 Limité dans cet indicateur au coût social estimé des émissions de CO2 .

131 méthodologiques, il insiste sur les variables qui constituent l’indicateur, recourant ensuite à la méthode jugée la plus appropriée : monétarisation pour les dimensions consommation et accumulation de stocks productifs, moyennes normalisées à partir d’indices connus (une version simplifiée du Sen-Shorrocks-Thon Index et le coefficient de Gini) pour les inégalités et la pauvreté, méthode originale de calcul de risques économiques pour la dimension sécurité économique.

Pour chacune des quatre dimensions de l’IBEE sont présentées la structure générale et les pondérations des différentes variables qu’elles contiennent.

Premier pilier de l’indicateur synthétique, la consommation ajustée repose sur l’hypothèse que le bien-être économique est directement corrélé aux volumes de biens et services consommés. Les ajustements procédés tiennent compte de la taille des ménages, par le biais de l’utilisation des échelles d’équivalence, des dépenses publiques, de l’espérance de vie, et de la valeur du loisir. Le graphique suivant présente le contenu ainsi que le mécanisme d’agrégation de la dimension consommation effective.

Consommation personnelle par tête

*

Indice de revenu équivalent

+

Valeur du loisir

+

Dépenses publiques (biens & services)

*

Indice d’espérance de vie

=

Consommation réelle totale par tête

Fig. 5 – Dimension 1 : consommation.

La deuxième dimension — l’accumulation des stocks de richesse productive— , considérée comme une estimation des flux de consommations futures, est susceptible d’influencer le bien-être pour deux raisons. D’une part, les individus sont soucieux de leur propre condition matérielle dans le futur. D’autre part, il est raisonnable de supposer que la plupart des individus est sensible à la situation matérielle des générations futures. La dimension contient les principales catégories de facteurs productifs, à savoir le capital fixe, le stock des investissements en matière de recherche et développement, ainsi qu’une estimation de la valeur du capital humain de la population. Tandis que la mesure des deux premiers facteurs repose sur des conventions statistiques traditionnelles, la valeur monétaire du capital humain est estimée sur la base des coûts par niveau d’études et de la répartition de la population par niveau.

Les individus sont concernés par un certain niveau de redistribution des richesses : quelle part leur sera-t-elle attribuée ? Quelle part sera attribuée aux autres ? Osberg et

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COMPTE RENDU DU MÉMOIRE EN FRANÇAIS Stock réel de capital p. t.

+

Stock réel de R&D p. t.

+

Dette extérieure nette p. t.

+

Stock réel de capital humain p. t.

+

Coût de la dégradation environnementale

=

Stocks de richesses productives

Fig. 6 – Dimension 2 : stocks de richesse.

Sharpe considèrent de ce point de vue que le bien-être dépend des revenus moyens, certes, mais aussi du degré de pauvreté et d’inégalité. Le fondement théorique de cette dimension repose en partie sur les travaux de John Rawls car elle met l’accent sur le bas de la distribution des revenus. Concrètement, la pauvreté y est intégrée à travers le produit du taux de pauvreté relatif et du taux d’intensité de pauvreté. Ce produit est une version simplifiée de l’Indice Sen-Shorrocks-Thon. Cependant, la dimension vise à élargir l’approche rawlsienne en attribuant une valeur non nulle aux inégalités entre les individus qui ne sont pas considérés comme pauvres. Ces inégalités sont mesurées à travers l’outil de mesure le plus répandu, à savoir le coefficient de Gini sur la distribution des revenus disponibles annuels (voir notre discussion p. 27).

0.75*

Intensité de la pauvreté

+

Indice d’inégalité de revenu 0.25* (Coefficient de Gini) =

Egalité et pauvreté

Fig. 7 – Dimension 3 : Egalité et pauvreté.

La dimension sécurité économique, certainement la plus originale de l’IBEE, repose sur l’idée que si le futur est incertain, les individus seront concernés par le degré de sécurité économique auquel ils peuvent prétendre. Les auteurs proposent d’identifier quatre risques économiques désignés, les considérant comme des proxy de risques matériels liés à la maladie, au chômage, aux ruptures familiales et à la vieillesse. Dans chaque cas, le risque de

133 perte économique lié à l’événement en question est évalué comme une probabilité conditionnelle, elle-même représentée comme le produit de diverses circonstances. La potentialité de chaque risque est pondérée par la part de la population concernée. L’hypothèse fondamentale est que les variations du niveau subjectif d’anxiété qui résulte d’une insécurité (variations de bien-être subjectif ) sont proportionnelles aux variations du risque objectif.

(Risque économique lié au chômage) *(P1 /P )

+

(Risque financier lié à la maladie) *(P2 /P )

+

(Risque économique lié à la pauvreté monoparentale) *(P3 /P )

+

=

(Risque écon. de pauvreté (personnes agées))*(P4 /P ) Sécurité économique

Fig. 8 – Dimension 4 : sécurité économique. Légende : P1 = Population entre 15-64 ; P2 = population entière ; P3 = population de femmes mariées avec enfants ; P4 = population entre 45-64. P = P1 + P2 + P3 + P4 .

Les pondérations utilisées pour permettre l’agrégation des dimensions sont ex natura rei arbitraires mais transparentes, permettant d’une part d’évaluer la sensibilité du choix de ces pondérations, d’autre part de les modifier en fonction de systèmes de préférences collectives, qu’il reste à construire. Dans la partie applicative suivante, nous utiliserons la pondération la plus fréquemment suggérée par les auteurs, à savoir des poids égaux attribués à l’ensemble des dimensions.

Quatre dimensions, trois inégalités Après avoir présenté l’IBEE dans sa forme originale, une proposition de modification de la dimension égalité et pauvreté est proposée. En effet, nous notons une inconsistance dans la structure interne de l’IBEE qui résulte de la manière avec laquelle l’indicateur rend compte de l’inégalité. D’un coté, l’IBEE identifie quatre dimensions comme étant pertinentes pour l’évaluation du bien-être économique. De l’autre coté, l’inégalité est évaluée pour une seule variable, à savoir pour le revenu par tête à travers le coefficient de Gini. Nous constatons que si nous admettons que le bien-être économique est composé de multiples dimensions, l’inégalité économique devrait aussi consister de ces éléments multiples. Le graphique cicontre illustre la proposition d’analyser trois aspects de l’inégalité économique. Les trois dimensions consommation par tête, stocks de richesses par tête et sécurité économique engendrent trois espaces d’inégalités.

Cette proposition nous conduit à la modification de la dimension égalité et pauvreté illustrée sur le graphique suivant.

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COMPTE RENDU DU MÉMOIRE EN FRANÇAIS Consommation réelle totale

Stocks de richesses

Sécurité économique

Distribution de la consommation

Distribution des stocks de richesse

Inégalités face aux risques économiques

Egalité économique Fig. 9 – Proposition de mesurer l’égalité au sein de l’IBEE. 0.75*

Intensité de la pauvreté

+

0.25*

=

Egalité économique

Egalité et pauvreté

Fig. 10 – Dimension 3 modifée : égalité et pauvreté.

Propositions alternatives de mesurer l’égalité dans l’IBEE Après avoir énoncées les dimensions des inégalités économiques à évaluer, cette section traite l’opérationnalisation de la mesure. Afin de pouvoir mesurer les inégalités relatives à la consommation, aux stocks de richesse et à la sécurité économique, il est nécessaire de choisir une statistique qui reflète de manière satisfaisante l’objectif de l’IBEE. Les principales conventions identifiées dans le chapitre 2 sont donc discutées dans le contexte spécifique de cette question. Ceci conduit à une confrontation des considérations internes issues du discours académique avec les conceptions des acteurs externes (les usagers de l’IBEE).

Lors de la discussion des contributions les plus importantes dans le domaine de la mesure empirique des inégalités, un consensus assez large sur les points suivants a été identifié : a) dans le cas d’un conflit entre complétude théorique et simplicité des mesures, l’approche conventionnelle semble donner la priorité aux considérations théoriques (cf. Atkinson, 1970, p. 253) ; b) l’analyse quantitative est sans doute l’approche dominante à la mesure des inégalités économiques ; c) l’acceptabilité des statistiques d’inégalité est testée

135 conventionnellement de manière indirecte à l’aide d’une liste de caractéristiques désirées ; d) malgré la critique de Sen que les rangements complètes aboutissent à une précision en partie arbitraire, la plupart des analyses continuent à employer des statistiques sommaires pour comparer le degré d’inégalités de différentes distributions ; e) concentration et inégalité sont souvent regardées comme « essentiellement le même concept » (Theil, 1964, p. 128).

La légitimité de chacun de ces cinq aspects est discutée dans le contexte de la mesure multidimensionnelle des inégalités au sein de l’IBEE. Etant donné que l’objectif de cet indicateur consiste à représenter les conceptions du « citoyen », nous argumentons que plusieurs de ces méthodes conventionnelles doivent être mise en question. Il est à noter qu’il est impossible de définir avec exactitude le concept d’inégalité à l’aide du critère de la « communication normale ». Cependant, il semble la légitimité de l’IBEE dépend du degré avec lequel la mesure est co-construite et prend en compte des considérations internes et externes.

Ceci nous conduit à identifier la communicabilité des mesures comme une conditio sine qua non de la co-construction des statistiques (p. 70). Si la technicité des statistiques dépasse un certain seuil de complexité, le « citoyen » ne sera pas en mesure de vérifier si ses valeurs sont effectivement reflétées par les instruments empiriques.

Un aspect normatif qui n’a pas été suffisamment pris en compte par les méthodes conventionnelles est la distinction entre les concepts relatif et absolu d’inégalité qui a été remarqué à plusieurs reprises tout au long du texte. Si l’inégalité est perçue comme étant relative, alors les augmentations proportionnelles ne modifient pas l’ampleur de l’inégalité.

Toutes les mesures basées sur la notion de concentration possèdent cette caractéristique d’être insensible à des augmentations proportionnelles car elle prennent uniquement en compte les ratios entre les différents revenus. En revanche, si l’inégalité est perçue comme un concept absolue, l’écart absolu entre les positions des individus est un facteur d’inégalité.

Bien que la relation entre différences absolues et la notion d’inégalités soit traitée par des contributions isolées (notamment par Kolm, 1976), l’approche orthodoxe est de considérer l’inégalité comme un concept relatif.

Selon notre analyse, il n’est pas légitime de donner préférence à un de ces deux visions alternatives des inégalités (p. 73). Le texte présente des arguments en faveur de l’hypothèse que l’inégalité est souvent associée avec les différences en termes absolues et cite les travaux de Kolm (1976) et Ravallion (2003). Ces auteurs indiquent que l’opposition en question peut être pensée comme des positionnements politiques alternatives. En effet, Kolm associe une attitude « de gauche » aux mesures absolues d’inégalité et réserve la vision relative aux attitudes « de droite ». Selon Ravallion, une partie du débat autour de la globalisation des marchés reflète la différence entre une vision absolue et relative des inégalités.

En conclusion, deux façons alternatives de penser les inégalités sont présentés. Elles sont moins sophistiquées que les statistiques qui intègrent directement une quelconque fonction de bien-être, mais ont l’avantage d’être plus faciles à communiquer et à représenter graphiquement. Comme l’opposition entre inégalité absolue et inégalité relative semble refléter des opinions politiques divergentes, nous proposons deux concepts alternatifs afin de laisser le choix au citoyen d’évaluer le bien-être en fonction de son système de valeurs.

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COMPTE RENDU DU MÉMOIRE EN FRANÇAIS

Les deux concepts alternatifs proposés pour représenter les inégalités multidimensionnelles sont les suivants : 1. Un concept facilement communicable et intuitive est de penser des inégalités comme des écarts moyens. Si nous imaginons chaque dimension des inégalités comme étant une dimension d’un espace, chaque ménage peut être traité comme un point dans cet espace. L’inégalité entre deux ménages est simplement l’écart entre leurs points respectifs. L’inégalité totale est alors l’écart moyen entre tous les ménages. Ceci est un concept absolu d’inégalité.

2. Si on divise toutes les écarts par la moyenne de différentes dimensions, nous obtenons un écart moyen relatif, qui reste cependant multidimensionnel. Ceci est donc similaire à un coefficient de concentration de Gini à plusieurs dimensions.

Mesurer les écarts : une approche géométrique Les deux concepts issus de la réflexion sur la légitimité des méthodes conventionnelles offrent l’avantage de permettre une représentation graphique simple. Cette section montre que les écarts moyens peuvent être pensés comme des distances moyennes dans un espace Euclidien. Ceci permet de préciser les formules à appliquer aux données empiriques et franchir la dernière étape de l’opérationnalisation de la mesure des inégalités dans l’IBEE. La première partie de cette section est dédiée à un rappel sur les distances Euclidiennes et montre comment les DMA et DMR proposées par Gini (voir section 2.1.3) peuvent être représentées comme des distances dans l’espace. Cependant, nous argumentons que les versions multidimensionnelles des mesures de Gini nous confrontent à des problèmes sérieux quant à leur utilisation comme des indicateurs d’inégalité multidimensionnelle. Ces problèmes sont, premièrement, les différences d’échelles entre les trois dimensions à évaluer.

L’écart moyen entre plusieurs points sera forcément plus sensible aux écarts dans une dimension qui varient entre 1000 e et 1000000 e qu’aux écarts dans une autre dimension pour laquelle les valeurs ne varient qu’entre 0 et 1. Une normalisation simple comme la transformation T = (xi − xmin )/(xmin − xmax ), appliquée à toutes les valeurs, ne peut pas résoudre ce problème (xmin et xmax pourraient être les montants maximaux et minimaux de chaque dimension). En effet, une telle transformation normalise toutes les dimensions à un intervalle [0, 1] et élimine une partie importante des écarts absolue entre les points.

La deuxième difficulté est qu’une version multidimensionnelle de la DMA ou de la DMR ne nous permet pas d’analyser les contributions des différentes dimensions à l’égalité totale.

Enfin, les mesures de Gini sont basées sur toutes les différences possibles entre tous les points, c’est-à-dire y compris les différences de tous les ménages avec eux-mêmes. De plus, chaque écart entre deux ménages différents apparaît deux fois dans le calcul. Nous pensons qu’il serait plus intuitif de baser la mesure d’inégalité que sur les différences entre des points différents et de réduire leur nombre de N 2 à (N 2 − N )/2.

A cause de ces trois aspects problématiques une modification de DMA est proposée. La stratégie est de formuler une mesure d’inégalité unidimensionnelle basée sur (N 2 − N )/2 différences et de la transformer en indice pour surmonter la difficulté lié aux différentes

137 échelles des dimensions. La mesure unidimensionnelle proposée est définie comme : PN −1 PN q (pi,d − pj,d )2 i=1 j>i ADd ≡ (N 2 − N )/2 La computation de ADd est illustrée à l’aide d’un exemple qui montre qu’il s’agit effectivement d’une mesure absolue dans le sens définie ci-dessus (p. 78). Pour obtenir une mesure relative d’inégalité, nous introduisons la notion de la longueur moyenne des vecteurs dans la dimension d, qui sera annotée λd et qui peut être écrite comme suivant : N 1 Xq 2 pi,d λd = N i=1

Passer de la mesure ADd à un indicateur relatif est semblable à diviser la DMA par le revenu moyen pour obtenir la DMR. Une version relative de ADd est donc la mesure RDd , qui est définie comme : Ad RDd ≡ λd De nouveau le calcul de la mesure est illustrée à l’aide des exemples et les différences et similitudes entre les deux statistiques sont discutées (p. 79).

Enfin, pour synthétiser les ADd et RDd de chaque dimension d dans une mesure de l’évolution globale des inégalités, nous proposons les indices suivants : IAt =

1 AD1t 1 ADnt + · · · + n AD1t−m n ADnt−m

IRt =

1 RD1t 1 RDnt + · · · + n RD1t−m n RDnt−m

Nous pouvons interpréter ces deux indices comme la moyenne des changements d’inégalités dans les différentes dimensions. Puisque la distance moyenne de chaque dimension à la date t, ADtd , est comparée à la valeur de base de la même dimension, les différences d’échelles entre les dimensions disparaissent. Seules les évolutions temporelles en pourcentage de chaque dimension sont retenues dans l’indice. La logique de ces indices est illustrée à l’aide d’un exemple (p. 80).

Chapitre 4 Application empirique Traitement de données Avant de passer à l’application des mesures d’inégalité, la source utilisée pour la partie empirique est décrite en début de ce chapitre : l’enquête Budget des Familles, BdF. Les limites de cette source sont également discutées. Notamment le champ restreint, la faible fréquence et le fait que les questions de sécurité économique et du patrimoine économique ne sont pas des objectifs principaux de l’enquête sont identifiés comme des limites importantes.

Des données sur l’ensemble des trois dimensions d’inégalités de l’IBEE ne sont malheureusement disponibles que pour les deux éditions les plus récentes de l’enquête BdF, à savoir les éditions des années 1994/1995 et 2000/2001. Pour chacune des trois dimensions nous expliquons comment les définitions contenues dans l’IBEE peuvent être approximées par des variables issues de l’enquête BdF. Le constat s’impose qu’il est impossible d’évaluer les trois dimensions en termes d’inégalité si nous retenons les définitions exactes de ces dimensions. Ceci est dû à deux facteurs : d’une part, notre source (le BdF) ne contient pas de renseignements sur l’ensemble de facteurs qui font partie des dimensions du bien-être, comme par exemple l’espérance de vie au sein de chaque ménage. D’autre part, nous sommes confrontés à des problèmes d’ordre conceptuel car un certain nombre de variables (comme par exemple la consommation des biens et services produits par le gouvernement) ne sont pas facilement individualisable. Le texte présente pour chaque composante des trois dimensions les raisons de soit retenir la variable et de l’approximer à l’aide de l’enquête BdF, soit de ne pas l’intégrer. Ceci aboutit à des proxies des trois dimensions qui sont ensuite évalués en termes d’inégalité.

Inégalité de la consommation effective par tête Nous argumentons qu’un proxy satisfaisant pour l’inégalité en termes de consommation effective par tête est le revenu disponible ajusté par unité de consommation. Pour obtenir cette variable, nous mettons en œuvre la même procédure de correction que nous avons présentée dans Jany-Catrice & Kampelmann (op. cit.). La correction de données adressent deux problèmes : premièrement, nous incluions des informations sur une composante importante de la consommation privée, à savoir les loyers fictifs dont bénéficient les 138

139 propriétaires qui habitent dans leurs propres logements. Deuxièmement, nous corrigeons la sous-estimation des revenus issus du patrimoine des ménages. La sous-estimation des revenus de patrimoine est un problème bien connu et apparaît non seulement dans l’enquête BdF, mais aussi dans d’autres sources. Notre approche consiste à ‘gonfler’ les valeurs des revenus de patrimoine pour que leur montant total coïncide avec le montant issu de la comptabilité nationale, une source jugée comme plus fiable.

Comme Osberg et Sharpe insistent sur l’importance des économies d’échelle associées à la taille de ménages, nous divisons le revenu disponible par le nombre d’unité de consommation (défini par l’échelle d’équivalence d’Oxford). La distribution par percentiles qui résulte de ces traitements est illustrée dans le tableau ci-dessous.

100% Max 99% 95% 90% 75% Q3 50% Median 25% Q1 10% 5% 1% 0% Min Std Deviation

1994/1995 2000/2001 290599.33 480455.61 63654.08 67657.19 36309.79 39964.40 28027.12 30818.17 19484.61 21011.03 13591.35 14499.49 9733.18 10298.72 7142.20 7594.07 5866.64 6187.83 3811.23 4133.22 0.00 0.00 597122 647890

Tab. 10 – Distribution par percentiles et écart type du revenu disponible ajusté par unité de consommation (toutes les valeurs en 1995 euros) — Source de données : BdF.

Inégalité de stocks de richesses par tête A cause de problèmes conceptuels qui rendent une évaluation exacte de l’inégalité de stocks de richesses impossible, nous présentons un proxy : la valeur totale de tous ce que possède le ménage, exprimé en montant par tête.

L’enquête BdF contient une question de ce type, qui n’est cependant renseignée qu’en tranches au lieu de valeurs « exactes ». Mais comme les huit tranches sont les mêmes pour les années 1994/1995 et 2000/2001, il est possible d’obtenir une approximation de la valeur de tout ce que possède le ménage à l’aide des hypothèses relativement peu restrictives (p. 86). Les montants par ménage sont ensuite divisés par le nombre de personnes vivant dans le ménage pour pouvoir évaluer les inégalités par tête entre les ménages. Le tableau ci-dessous montre la proportion de ménages dans les différentes tranches pour les deux enquêtes de 1994/1995 et 2000/2001.

140

COMPTE RENDU DU MÉMOIRE EN FRANÇAIS Valeurs en tranches (en euros) 0 - 3,049 3,049 - 7,622 7,622 - 15,245 15,245 - 30,490 30,490 - 76,225 76,225 - 152,450 152,450 - 304,898 304,898 et plus

1994/1995 (% de mén.) 2000/2001 (% de mén.)

7.69 7.11 10.45 9.23 9.41 9.71 8.27 8.71 16.64 14.04 26.69 24.76 15.21 19.07 5.65 7.37 100 100

Tab. 11 – Valeur monétaire de tout ce que possède le ménage — Source de données : BdF.

Inégalité face aux risques économiques L’enquête BdF permet d’approximer un des risques désignés par Osberg et Sharpe, à savoir l’inégalité face au risque de chômage. Dans le cadre du BdF, chaque ménage enquêté est demandé d’évaluer le risque de chômage dans les prochains 12 mois pour les différentes personnes qui constituent le ménage. Ceci permet de calculer l’écart entre les risques subjectifs des ménages et d’analyser la distribution du risque au sein de la population.

Il est nécessaire de formuler une hypothèse comment les risques des différents membres du ménage peuvent être agrégés. Nous avons décidé de retenir que le risque de la personne de référence, sauf si cette personne n’est pas active ou si elle a refusé de répondre à la question. Que dans ces cas le risque de chômage du conjoint est utilisé comme approximation du risque économique du ménage. Le tableau ci-dessous présente la proportion des différents degrés de risque.

Degré de risque de chômage Non active Non, il n’y a aucun risque C’est possible, mais le risque est faible C’est possible, et le risque est moyen C’est possible, et le risque est élevé Oui, c’est quasiment inévitable Refus

Score 1 2 3 4 5 -

1994/1995 (en %) 8.4 38.87 30.25 14.37 4.96 3.14 0 100

2000/2001 (en %) 10.86 43.61 25.65 10.70 4.27 4.73 0.18 100

Tab. 12 – Distribution du risque subjectif de chômage dans le 12 mois qui suivent l’enquête — Source de données : BdF.

141

RĂŠsultats pour les statistiques alternatives dâ&#x20AC;&#x2122;inĂŠgalitĂŠ Cette partie prĂŠsente lâ&#x20AC;&#x2122;ĂŠvaluation des trois vecteurs de donnĂŠes â&#x20AC;&#x201D; qui dĂŠcrivent respectivement le revenu disponible, le patrimoine et le risque de chĂ´mage â&#x20AC;&#x201D; par rapport Ă leur inĂŠgalitĂŠ. Est calculĂŠe la mesure AD (la distance moyenne absolue entre tous les points) ainsi que la version relative de cette mesure RD (les deux sont prĂŠsentĂŠes dans la section 3.3.1). Pour comparer les rĂŠsultats avec les statistiques traditionnelles dâ&#x20AC;&#x2122;inĂŠgalitĂŠ sont ĂŠgalement calculĂŠs : le coefficient de Gini (cf. Section 2.1.3), la mesure de Theil (cf. Section 2.1.5), le ratio interdĂŠcile, la mesure de Dalton (cf. Section 2.1.4) et lâ&#x20AC;&#x2122;indice dâ&#x20AC;&#x2122;Atkinson (cf. Section 2.1.6). La mesure de Dalton est ĂŠvaluĂŠe pour une valeur du revenu minimum de c = 1/6000 et pour une valeur du patrimoine minimal c = 1/10000. Lâ&#x20AC;&#x2122;indice de Atkinson est calculĂŠ pour deux niveaux dâ&#x20AC;&#x2122;aversion contre lâ&#x20AC;&#x2122;inĂŠgalitĂŠ : une aversion faible de = 0.5, et une aversion forte de = 1.5.

Les rĂŠsultats pour le premier aspect dâ&#x20AC;&#x2122;inĂŠgalitĂŠ ĂŠconomique sont prĂŠsentĂŠs dans le tableau ci-dessous. Tous les indicateurs descriptifs indiquent une augmentation de lâ&#x20AC;&#x2122;ĂŠgalitĂŠ entre 1994/1995 et 2000/2001. Cependant, la taille de cette augmentation varie significativement : tandis que G nâ&#x20AC;&#x2122;augmente que par 2 %, la distance moyenne absolue AD indique un changement de plus de 10 %. Etant donnĂŠ lâ&#x20AC;&#x2122;augmentation de 7 % du revenu moyen, une proportion considĂŠrable de cette diffĂŠrence peut vraisemblablement ĂŞtre expliquĂŠe par lâ&#x20AC;&#x2122;amĂŠlioration du niveau de vie moyen. Les mesures relatives RD, G, T et D9 /D1 sont toutes insensibles Ă des augmentations proportionnelles de tous les revenus. Si nous supposons quâ&#x20AC;&#x2122;au moins une partie de la croissance du revenu moyen est distribuĂŠe parmi les diffĂŠrentes couches de la sociĂŠtĂŠ, ces mesures indiceraient une inĂŠgalitĂŠ plus faible que la statistique absolue AD. Concernant le dĂŠveloppement des mesures basĂŠes sur lâ&#x20AC;&#x2122;ĂŠvaluation dâ&#x20AC;&#x2122;une fonction de bien-ĂŞtre, les valeurs numĂŠriques observĂŠes dĂŠpendent des paramètres respectifs c et . Nous avons choisi ces valeurs de manière arbitraire et devons ĂŞtre prudents avec leur interprĂŠtation. Cependant, la diminution de D est en accord avec le deuxième principe proposĂŠ par Dalton : une addition proportionnelle Ă tous les revenus devrait aboutir Ă une diminution de lâ&#x20AC;&#x2122;inĂŠgalitĂŠ (cf. notre discussion p. 35). Si la croissance du revenu moyen affecte plusieurs parties de la distribution, nous anticipons â&#x20AC;&#x201D; ceteris paribus â&#x20AC;&#x201D; la diminution de lâ&#x20AC;&#x2122;inĂŠgalitĂŠ que nous observons pour D. En ce qui concerne la statistique A, nous observons quâ&#x20AC;&#x2122;une interprĂŠtation est difficile. Ceci ne vaut pas seulement pour lâ&#x20AC;&#x2122;inĂŠgalitĂŠ en termes de revenu, mais aussi pour la dimension suivante, les stocks de richesses.

La raison pour cette difficultĂŠ dâ&#x20AC;&#x2122;interprĂŠtation est que lâ&#x20AC;&#x2122;ĂŠvolution de lâ&#x20AC;&#x2122;indice A change de signe lorsque nous passons dâ&#x20AC;&#x2122;un degrĂŠ dâ&#x20AC;&#x2122;aversion contre lâ&#x20AC;&#x2122;inĂŠgalitĂŠ faible Ă un degrĂŠ plus ĂŠlevĂŠ. Comme il est problĂŠmatique de connaĂŽtre le degrĂŠ dâ&#x20AC;&#x2122;aversion dâ&#x20AC;&#x2122;une sociĂŠtĂŠ (ou mĂŞme dâ&#x20AC;&#x2122;une personne), il est difficile dâ&#x20AC;&#x2122;interprĂŠter ces chiffres.

La deuxième dimension, lâ&#x20AC;&#x2122;inĂŠgalitĂŠ de richesses par tĂŞte, affiche une opposition encore plus forte entre les mesures descriptives absolues et relatives. Ici, AD pointe mĂŞme dans la direction opposĂŠe avec une augmentation de 7.5 %, tandis que toutes les mesures relatives diminuent entre les deux dates (voir le tableau ci-dessous). En mĂŞme temps, le patrimoine moyen a augmentĂŠ par 8.5 %, et une partie de la diffĂŠrence entre mesures relatives et absolues peuvent ĂŞtre expliquĂŠe analogue Ă la première dimension. La mesure D ĂŠvolue de

142

COMPTE RENDU DU MĂ&#x2030;MOIRE EN FRANĂ&#x2021;AIS

Taille de lâ&#x20AC;&#x2122;ĂŠchantillon N Revenu moyen (en 1995 euros) AD (en 1995 euros) RD G (Coefficient de Gini) T (Mesure de Theil) D9 /D1 D (Mesure de Dalton, c = 1/6000) A (Indice dâ&#x20AC;&#x2122;Atkinson, = 0.5) A ( = 1.5)

1994/1995 2000/2001 change en % 11294 10305 â&#x2C6;&#x2019;8.8 16619.66 17784.29 +7.0 10808.34 11913.62 +10.2 0.65 0.67 +3.0 0.32 0.33 +1.9 0.19 0.20 +3.4 3.92 4.06 +3.6 1.34 1.24 â&#x2C6;&#x2019;5.4 0.08 0.09 +11.3 0.46 0.23 â&#x2C6;&#x2019;49.4

Tab. 13 â&#x20AC;&#x201C; Statistiques de lâ&#x20AC;&#x2122;inĂŠgalitĂŠ de revenu â&#x20AC;&#x201D; Source de donnĂŠes : BdF. nouveau en accord avec le deuxième principe de Dalton.

Taille de lâ&#x20AC;&#x2122;ĂŠchantillon N Patrimoine moyen (en 1995 euros) AD (en 1995 euros) RD G (Coefficient de Gini) T (Mesure de Theil) D9 /D1 D (Mesure de Dalton, c = 1/10000) A (Indice dâ&#x20AC;&#x2122;Atkinson, = 0.5) A ( = 1.5)

1994/1995 2000/2001 change en % 11294 10305 â&#x2C6;&#x2019;8.8 48965.10 53110.30 +8.5 57059.21 61327.73 +7.5 1.17 1.15 â&#x2C6;&#x2019;0.9 0.58 0.56 â&#x2C6;&#x2019;3.0 0.59 0.56 â&#x2C6;&#x2019;6.4 64.29 59.99 â&#x2C6;&#x2019;6.7 1.20 1.16 â&#x2C6;&#x2019;2.3 0.29 0.28 â&#x2C6;&#x2019;4.4 0.77 0.78 +0.2

Tab. 14 â&#x20AC;&#x201C; Statistiques de lâ&#x20AC;&#x2122;inĂŠgalitĂŠ de richesse â&#x20AC;&#x201D; Source de donnĂŠes : BdF. Enfin, nous ĂŠvaluons lâ&#x20AC;&#x2122;inĂŠgalitĂŠ en termes du risque subjective dâ&#x20AC;&#x2122;ĂŞtre au chĂ´mage dans les 12 prochains mois. Puisque ce risque est mesurĂŠ Ă lâ&#x20AC;&#x2122;aide dâ&#x20AC;&#x2122;un score entre 1 et 5, les indicateurs dâ&#x20AC;&#x2122;inĂŠgalitĂŠ basĂŠs sur le concept de concentration, Ă savoir le coefficient de Gini et la mesure de Theil, nâ&#x20AC;&#x2122;ont pas de sens dans ce cas. Aussi les concepts du ÂŤ revenu ĂŠquivalent ĂŠgalement distribuĂŠ Âť (Atkinson) et du ÂŤ bien-ĂŞtre si le revenu actuel serait distribuĂŠ ĂŠgalement Âť (Dalton) ne peuvent pas ĂŞtre appliquĂŠs Ă la distribution du risque de chĂ´mage. Par consĂŠquent, que les distances moyennes absolues et relatives ainsi que le ratio interdĂŠciles sont ĂŠvaluĂŠs. Avant dâ&#x20AC;&#x2122;interprĂŠter lâ&#x20AC;&#x2122;ĂŠvolution de ces mesures, nous notons que le risque moyen a diminuĂŠ par 3 % entre 1994/1995 et 2000/2001. Ce rĂŠsultat semble confirmer la chute simultanĂŠ du taux de chĂ´mage de 11.4 % en 1995 Ă 8.7 % en 2001 communiquĂŠe par lâ&#x20AC;&#x2122;INSEE. Il est intĂŠressant Ă observer que cette diminution du risque de

143 chômage n’a pas conduit à une distance moyenne plus faible entre les ménages. La différence moyenne a augmenté durant la même période d’un écart de 1.09 à 1.13. Puisque la mesure RD est le ratio entre la différence moyenne absolue des risques et le risque moyen, cette fois-ci RD a augmenté plus que AD, à savoir par 7 %. L’augmentation considérable du ratio interdécile (+33.3 %) ne doit pas être pris trop sérieux. Le bond de cette mesure est due au fait qu’il y n’a pas de valeurs intermédiaires entre les scores du risque : les scores étant des nombres entiers de 1 à 5, le ratio peut uniquement changer en étapes relativement larges. Durant la période d’observation, le neuvième décile a changé d’un score de 3 à un score de 4, tandis que le premier décile reste inchangé. Ceci a conduit automatiquement à l’augmentation considérable du ratio.

Taille de l’échantillon N Risque moyen AD RD D9 /D1

1994/1995 2000/2001 change en % 11294 10305 −8.8 1.94 1.89 −3.0 1.09 1.13 +3.7 0.56 0.60 +7.0 3 4 +33.3

Tab. 15 – Statistiques de l’inégalité face au risque économique — Source de données : BdF. IAt

Les résultats présentés ci-dessus peuvent facilement être insérés dans les deux indices et IRt , définis dans la section 3.3.1. Le premier indice est égale à : IA2000 =

1 (1.102 + 1.075 + 1.037) = 1.071 3

L’inégalité globale mesurée par IAt a donc augmenté par environ 7 % pendant la période de 1994/1995 à 2000/2001. L’indice relatif est égal à : IR2000 =

1 (1.03 + 0.991 + 1.07) = 1.03 3

L’inégalité multidimensionnelle basée sur l’évaluation de RD a augmenté par 3 % durant la même période.

Evolution de l’IBEE modifié Dans l’étape finale de l’analyse, l’IBEE est évalué pour les mesures alternatives d’inégalités que nous avons calculé dans la section précédente.

Jusqu’à la publication de l’édition 2005/2005 de l’enquête BdF, nous sommes obligés de restreindre notre analyse aux deux dates pour lesquelles nous disposons de données.

L’évolution de l’inégalité a été estimée par une interpolation linéaire et est ici combinée avec les résultats d’une application antérieure de l’IBEE (op. cit).

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Nous rappelons la structure de la dimension égalité et pauvreté, qui est une somme pondérée de deux éléments : un indice d’égalité (avec le poids 0.25) et une mesure de l’intensité de la pauvreté (avec le poids 0.75). Cette dernière a diminué par 2.3 % entre 1994 et 2000 (ibid., figure 11). Le tableau ci-dessous montre l’effet combiné de cette diminution et l’évolution de trois indices alternatives d’inégalité. Puisqu’une valeur plus élevée de l’IBEE indique une amélioration du bien-être, la diminution de l’intensité de la pauvreté de 2.3 % est transformée en une augmentation de 2.3 %. La même transformation est appliquée aux indices d’inégalité qui sont modifiés pour devenir des indices d’égalité.

Evolution entre 1994 et 2000 (en %) Version originale (Osberg & Sharpe) Egalité de revenu standard (Gini) Mesures trois-dimensionnelles Egalité économique (Indice absolu IA ) Egalité économique (Indice relatif IR )

Indice d’égalité

Dimension de l’IBEE ‘égalité et pauvreté’

+0.7

0.75×2.3+0.25×(+0.7) = +1.9

−7.1 −3.0

0.75×2.3+0.25×(−7.1) = −0.1 0.75×2.3+0.25×(−3.0) = +1.0

Tab. 16 – Impact de différentes mesures d’inégalités sur la dimension égalité et pauvreté — Sources de données : INSEE, Enquête Revenus Fiscaux ; BdF. Lors de notre application antérieure, le coefficient de Gini a été utilisé comme mesure d’inégalité. Ceci correspond à la définition initiale de l’IBEE proposée par Osberg et Sharpe.

Le coefficient de Gini est issu de la série publiée par l’INSEE et donc basé sur l’inégalité standard en terme de revenu mesuré par l’Enquête Revenus Fiscaux (ERF), une source administrative. Selon l’INSEE, le coefficient de Gini a légèrement diminué entre 1994 et 2000 par 0.7 % (ce qui signifie une augmentation de 0.7 % de l’égalité). Ensemble avec la diminution de 2.3 % de l’intensité de la pauvreté ceci résulte à une amélioration de 1.9 % de la dimension égalité et pauvreté, comme le montre le tableau ci-dessus.

Or, les indices multidimensionnels montrent une évolution différente. Si l’indice relatif IR est intégré dans la dimension égalité et pauvreté, l’amélioration est réduite à 1 %. Si nous pensons de l’inégalité comme une distance absolue moyenne, la dimension de l’IBEE stagne.

Il est évident que le processus d’agrégation de l’IBEE rend l’impact des mesures alternatives d’inégalités moins visible. Le poids de l’indice d’inégalité dans la troisième dimension de l’IBEE est 25 % : par conséquent, la détérioration de 7.1 % de l’égalité se traduit par une diminution de la dimension par seulement 0.25 × 7.1% = 1.775 %. Si nous utilisons le système de pondération standard qui associe le même poids aux quatre dimensions, l’impact sur l’IBEE global est encore plus faible, à savoir seulement 0.25 × 0.25 × 7.1% = 0.44%. Cet effet est visible dans les trois versions de l’IBEE associées aux différentes mesures d’inégalités (voir le tableau ci-dessous). Il n’est pas surprenant que l’indicateur global de bien-être économique est relativement insensible au choix de l’indice d’inégalité : le changement durant la période d’observation est 9.7 % si nous incluions le coefficient de concentration de

145 Gini, 9.2 % pour l’indice IA , et 9.4 % pour l’indice IR . Malgré le fait que la vision globale sur le bien-être n’est pas significativement modifiée en passant d’un concept d’inégalité à l’autre, ceci ne signifie pas que nos réflexions ne sont que des détails cosmétiques.

Flux de consommation Stocks de richesses Indice d’intensité de la pauvreté Indice de Gini (INSEE, ERF) Egal. Multidim. (basée sur IA ) Egal. Multidim. (basée sur IR ) Egalité & pauvreté (y.c. Gini) Egalité & pauvreté (y.c. IA ) Egalité & pauvreté (y.c. IR ) Sécurité économique IBEE (y.c. Gini de revenu) IBEE (y.c. IA ) IBEE (y.c. IR )

1994 100 100 100 100 100 100 100 100 100 100 100 100 100

1995 100.8 103.3 100.4 100.5 98.8 99.5 100.4 100.0 100.2 99.0 100.9 100.8 100.8

1996 101.1 105.5 100.8 101.1 97.6 99.0 100.8 100.0 100.3 95.6 100.8 100.5 100.6

1997 100.7 109.6 101.1 101.5 96.5 98.5 101.2 100.0 100.5 99.5 102.8 102.5 102.6

1998 102.9 110.9 101.5 102.2 95.3 98.0 101.7 99.9 100.6 105.0 105.1 104.7 104.8

1999 105.2 111.8 101.9 101.8 94.1 97.5 101.9 99.9 100.8 107.1 106.5 106.0 106.2

2000 107.6 115.7 102.3 100.7 92.9 97.0 101.9 99.9 101.0 113.5 109.7 109.2 109.4

Tab. 17 – Evolution de l’IBEE français et de ses composantes 1994-2000.

Tout d’abord, il est à noter que nous sommes malheureusement restreints à une période d’observation extrêmement courte. Si la tendance observée dans les données continue, l’impact des mesures alternatives d’inégalité va devenir de plus en plus visible au cours du temps. Plusieurs raisons nous laissent croire que l’écart observé entre les mesures relatives et absolues n’est pas seulement temporaire, mais reflète des forces profondément encastrées dans les systèmes économiques des sociétés progressives. Si les valeurs monétaires réelles croissent, et si différentes parties de la population profitent de cette croissance via des augmentations proportionnelles des revenus et de la richesse, alors nous anticipons une divergence systématique entre les mesures basées sur la concentration d’un coté, et les différences absolues de l’autre.

Deuxièmement, même si l’impact net est faible, l’IBEE offre non seulement une vision globale, mais informe également sur les différents aspects du développement économique.

Par conséquent, même si les mesures alternatives d’inégalité ne modifient que légèrement l’indicateur synthétique, l’information que les inégalités économiques ont augmentées si nous adoptons le concept des différences moyennes contient une valeur à part entière. La décomposabilité de l’IBEE est une caractéristique importante car elle permet de contraster les développements positifs et négatifs dans le même cadre analytique. A titre d’exemple, nous pouvons comparer l’évolution des deux dimensions qui emploient le concept du citoyen typique et citoyen hétérogène que nous avons présentés dans le tableau 3.1 (p. 65). Le premier correspond aux dimensions consommation effective et stocks de richesses, tandis que le dernier regroupe égalité et pauvreté et risque économique. Le graphique 4.2 en p. 96

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montre le contraste entre ces deux groupes et illustre l’utilité de la décomposabilité de l’IBEE. Remarques conclusives Depuis les constantes dans la répartition de la richesse de Pareto, la mesure des inégalités a connu une longe évolution. Les méthodes analytiques ont été améliorées de manière significative, notamment grâce à un échange fructueux entre la théorie de choix, la théorie d’information et l’analyse d’inégalité.

Ce mémoire montre que la sophistication des méthodes n’a pas forcément conduit à une vision plus claire sur les inégalités économiques. Plusieurs conventions encastrées dans l’usage des statistiques standard de l’inégalité — comme le coefficient de Gini ou l’indice de Atkinson — restent (au moins) discutables. En effet, ces conventions sont en oppositions avec l’idée qu’une partie de la population pense des inégalités comme étant des différences absolues entre les positions économiques, une idée déjà introduite au débat par Kolm (1976). Nous avons argumenté que la complexité technique du discours académique a éloigné ces controverses du citoyen non expert. Par ailleurs, l’absence d’une communication effective entre les cercles académiques et les usagers potentiels des statistiques pourrait causer un problème sérieux en vue de la légitimité des mesures de l’inégalité.

Pour surmonter ces lacunes, nous avons introduit — dans le cadre de l’IBEE — deux mesures alternatives pour évaluer des inégalités économiques multidimensionnelles. Les deux méthodes sont basées sur une interprétation graphique simple et sont donc adaptées au débat public. Chacune de ces deux mesures correspond à une hypothèse différente sur la nature des inégalités : la première, la différence absolue moyenne, prend en compte l’écart en termes réelles entre les positions économiques des individus ; la deuxième, la différence relative moyenne, continue l’hypothèse traditionnelle que l’inégalité devrait être insensible aux augmentations proportionnelles des valeurs monétaires.

Selon les données à notre disposition, l’impact sur le bien-être économique de ces mesures alternatives est limité. Cependant, la question si nous pensons de l’inégalité comme un concept absolu ou relatif entraîne des conséquences profondes d’une grande importance.

La conséquence la plus évidente est vraisemblablement l’effet ambigu de la croissance économique. Traditionnellement, la proposition dominante est que la croissance économique — si elle est répartie sur l’ensemble de la population via des ajustements des salaires et d’autres mécanismes similaires — n’a aucun effet sur l’inégalité économique. La raison pour cette insensibilité de l’inégalité à la croissance est que la première a été pensée comme identique à la notion de concentration. Une fois cette identité est mise en question, comme nous l’avons fait avec le concept de l’écart absolu moyen, la croissance économique est facteur des inégalités croissantes. Dans nos sociétés progressives, ceci est évidemment un effet problématique et met en question la focalisation sur la croissance économique comme un moyen d’attaquer un certain nombre de défis sociétaux. En effet, si une partie de la société pense de l’inégalité comme des différences absolues, la croissance économique pourrait être la cause d’un problème sociétal au lieu de sa solution.