INTRODUCTION

Few people would question that well-being results from many different attributes of human life, and the level of income, or expenditure, is only a crude proxy of the quality of living that a person enjoys.^[97] Should we then account for the multiple facets of well­being in the social evaluation of inequality and poverty? If so, how can we do it?

Acknowledging the multidimensional nature of well-being does not necessarily imply that the social evaluation must also be multidimensional.

At the opposite extreme are those who argue, on philosophical or practical grounds, that dimensions must be kept distinct in the social evaluation. If well-being domains are characterized by specific criteria and arrangements, some might adhere to Walzer’s (1983, p. 19) view of “complex equality” whereby “no citizen’s standing in one sphere or with regard to one social good can be undercut by his standing in some other sphere, with regard to some other good.” Ifinequalities in certain domains (e.g., basic life necessities or health) are less acceptable than they are in others (e.g., luxury goods), it might be jus­tifiable to adopt a piecemeal approach informed by the “specific egalitarianism” advo­cated by Tobin (1970).^[99] The intrinsic incommensurability of domains may then imply that “no simple ordered indicator of level of living can be constructed, either on an individual or on an aggregate level,” as asserted by Erikson (1993, p. 75) in sum­marizing the Swedish approach to welfare research. Or, the need to avoid the “ad hoc aggregation” and the unexplained trade-offs between domains, which are implicit in any composite or “mashup” index, might advise us “to derive the best measure possible for each of a logically defensible set of grouped dimensions—such as ‘income poverty,’ ‘health poverty’ and ‘education poverty’” (Ravallion, 2011a, p. 240; see also Ravallion, 2012a). In all these cases, the recognition of the inherent autonomy of each dimension, however motivated, leads to a piecewise social judgement that does not need any unitary measurement of human well-being.

There are reasons to take an intermediate position between these two extremes, how­ever. This may be because the above-described conditions for reducing well-being to a single variable may not hold: we might differ in our views about the appropriate equiv­alence scale or the weights to be placed on different goods, we might not have access to individual well-being measures, or we might reject the individual valuations altogether. Or, we may worry that the inequalities in different spheres accumulate and that the com­bination of multiple deprivations makes life much harder than just the sum of such dep­rivations. In these cases, we may need a social evaluation of poverty and inequality that is multidimensional and accounts for the joint distributions of all the elements of the vector x of well-being attributes.

Our aim in this chapter is to explore this intermediate route. We do not argue further whether we should or should not have a multidimensional social evaluation. We take it for granted, and we concentrate on how we can carry it out in a sound way. More pre­cisely, we examine the analytical and ethical foundations of methods for the multidimen­sional measurement of inequality and poverty, whether it be for descriptive, normative, or policy-making purposes. All these methods require numerous arbitrary, and hence debatable, assumptions. Elucidating their foundations helps us to unveil these assump­tions and understand their normative content. Taking this perspective, we pay little attention to the many multivariate techniques that have been developed in statistics and efficiency analysis. They provide valuable information, but their aggregation of mul­tiple attributes is based on empirically observed patterns of association among the vari­ables, and, thus, it lacks any clear ethical interpretation.

The theoretical literature on the multidimensional measurement of inequality and poverty has been growing very rapidly in the last quarter of a century, and it is still far from consolidation. Rather than engaging in a systematic rationalization of this literature, we provide a selective reading of it with the twofold objective of, first, identifying areas worthy of further investigation and, second, offering some guidance on how to use the rich and sophisticated machinery now available for empirical and policy-oriented appli­cations. As the multidimensional view of well-being has gained momentum in the policy discourse, its practical implementation has turned into an active battlefield where con­tenders passionately argue for opposing approaches—a good example being the Forum on multidimensional poverty in the 2011 volume of the Journal of Economic Inequality (see Lustig, 2011, for an introduction). We attempt to provide a balanced account of alter­native positions, as well as their strengths and weaknesses.

The chapter is divided into three parts, plus a closing section. In the next section, we briefly review three questions that are preliminary to any multidimensional analysis of well-being: the selection of the relevant dimensions, the indicators used to measure them, and the procedures for their weighting. These questions are theoretically intriguing and of considerable importance in empirical analyses, but we only outline their main features. Importantly, the choice made with regard to these issues may condition the analytical methods reviewed later. For instance, the fact that many variables used in multidimen­sional poverty analysis are dichotomous suggests paying particular attention to methods based on counting deprivations. The assumption that inequality does not change after proportionate variations of the variable under examination (scale invariance) may be rea­sonable for income, but much less so for life expectancy, impinging on the axiomatic measurement of multidimensional inequality.

3.1.1 Historical Developments and Main Themes

The multidimensional literature in economics began with seminal articles by Kolm (1977) and by Atkinson and Bourguignon (1982) on the dominance conditions for ranking mul­tivariate distributions. A few years later, Atkinson and Bourguignon (1987) developed sequential dominance criteria for the bivariate space of income and household composi­tion. Their aim was to impose weaker assumptions on social preferences than those implicit in the standard method of constructing equivalent incomes. Whereas the standard approach entails specifying how much a family type is needier than another one, sequen­tial dominance criteria only require ranking family types in terms of needs, although at the cost of obtaining an incomplete ordering. This application paved the way for a specific and fertile strand of research which focuses on the possibility that one attribute (e.g., income) can be used to compensate for another nontransferable attribute (e.g., needs, health).

With the partial exception of Maasoumi (1986, 1989), who recasted multidimen­sional analysis into a unidimensional space by means of a utility-like function, it was not until the mid-1990s that Tsui (1995, 1999) moved on to the axiomatic approach to inequality indices in order to achieve complete orderings. The bases of the axiomatic analysis of partial and complete poverty orderings were laid down at about the same time by Chakravarty et al. (1998), Bourguignon and Chakravarty (1999, 2003, 2009), and Tsui (2002). Notably, multidimensional indices of inequality and poverty associate real numbers to each multivariate distribution, as does the univariate analysis of a composite well-being indicator, but with the important difference that they do not need to go through the aggregation of well-being attributes at the individual level.

At the turn of the twentieth century, the literature on multidimensional poverty and inequality was still in its infancy. The first volume of this Handbook (Atkinson and Bourguignon, 2000) did not feature any specific chapter on the topic, and the com­prehensive analytical chapter on the measurement of inequality by Cowell (2000) devoted only three pages to multidimensional approaches. Ever since, the theoretical lit­erature has grown conspicuously, however, and we can identify two main lines of research.

The first line devotes considerable effort to developing the axiomatic approach to both poverty and inequality measurement. Researchers delve into the different ways to model the patterns of association (correlation) between the variables, which is the single feature that distinguishes multidimensional from unidimensional analysis, and elaborate alternative axioms. They have also come to realize that a mechanical transpo­sition of the properties typically adopted in the univariate analysis of income distribu­tion may not be straightforward, and sometimes it may not even be appropriate. A case in point is the extension to life expectancy of the scale invariance property of inequality measures just mentioned. An even more cogent example is the Pigou-Dalton principle of transfers, a central tenet of income inequality measurement (Atkinson and Brandolini, forthcoming). This principle states that a mean-preserving transfer of income from a richer person to an (otherwise identical) poorer person decreases inequality. On the one hand, an interpersonal transfer might be unfeasible and even ethically debatable for a dimension such as the health status, despite being acceptable for income. On the other hand, the generalization of the principle to a multivariate framework is far from univocal, as explained in detail in Section 3.4.1.

The second line of research focuses on what Atkinson (2003, p. 51) labels the “counting approach.” This multidimensional approach is at the same time the newest (in terms of theoretical elaboration) and the oldest (in terms of empirical practice). For example, the main poverty statistic adopted by a parliamentary commission of inquiry over destitution in Italy in the early 1950s was a weighted count of the number of households failing to achieve minimum levels of food consumption, clothing availabil­ity, and housing conditions (Cao-Pinna, 1953). Modern applied research on material deprivation owes much to the pioneering work by Townsend (1979) and Mack and Lansley (1985) in Britain.^[100] Ever since the publication of their studies, it has had a huge impact on the social policy debate in Ireland and the United Kingdom, and later in the European Union.^[101] Nevertheless, we lack a full-fledged theoretical treatment of the nor­mative basis of the counting approach. The recent work by Alkire and Foster (2011a,b) fills this gap, in part, by providing the axiomatic characterization of a family of multi­dimensional counting poverty indices. Yet, the difficulties illustrated by Atkinson (2003) in reconciling the counting approach with a social welfare approach are still unsettled. In our view, part of the problem may derive from defining welfare criteria in terms of the distributions of the underlying continuous variables rather than in terms of the distribution of deprivation scores, which is the key variable considered in the counting approach. The distribution of deprivation scores contains all the relevant information in the counting approach, which by construction implies neglecting levels of achievement in the original variables. Disagreement on this point, and on the implicit loss of information, might have some part in the recent controversies surrounding the counting approach.

The less developed analytical structure, in the face of the method’s popularity in applied research, is the main reason for devoting a relatively larger space to the counting approach in this chapter. However, counting deprivations is also the simplest way to embed the association between individual-level dimensions into an overall index of deprivation. It is useful to illustrate two aspects of multidimensional measurement that recur throughout the chapter. The first is the order of aggregation. In the count­ing approach, the synthesis of available information begins with aggregation across single dimensions for each individual, and then across individuals. Inverting the order of aggregation by first computing the proportions of people suffering from depriva­tion in each dimension and then aggregating these proportions into a composite index of deprivation yields the same result only if the dimensions of well-being are “independent.” If this is not the case, the composite index of deprivation misses the impact of cumulating failures in more than one dimension. The second aspect is the contrast between the “union criterion” and the “intersection criterion,” which plays a fundamental role in the measurement of multidimensional poverty, as stressed by Atkinson (2003). The occurrence of deprivation in some dimensions does not neces­sarily entail a condition of overall poverty: we may define people to be poor when they are deprived in at least one dimension (union criterion) or in all dimensions (inter­section criterion), or else in some fraction of the dimensions considered in the analysis. The choice of a critical number of dimensions to identify poverty status introduces an additional threshold relative to those already set for defining deprivation in each dimen­sion, which is a central feature of the “dual cut-off” approach proposed by Alkire and Foster (2011a,b).

In Sections 3.3 and 3.4, we discuss first, the counting approach, then the axio­matic treatment of poverty, and finally, the axiomatic treatment of inequality. This sequence reflects the growing complexity of data requirements, rather than a chronolog­ical order. In this chapter we pay no attention to the assessment of data quality and the elaboration of inference tools, although they are admittedly two crucial issues in empirical analyses.

3.2.