3.1 Introduction

Poverty remains a burning problem in many countries of the world even in the early twenty-first century. Many people in such countries and in many relatively rich countries as well need to struggle in making ends meet. Consequently, removal of poverty continues to be one of the major economic policies for many people in the world.

We have argued explicitly in earlier chapters that well-being of a population is a multidimensional phenomenon (see Stiglitz et al., 2009). Hence, poverty can be regarded as a manifestation of insufficiency of achievements in different dimensions of well-being. The emphasis on multidimensionality of poverty arises from the cognizance that income by itself cannot capture many important factors that may downgrade a person to poverty. Reckless spending by a very wealthy person on consumption of unhealthy food items is likely to deteriorate health status. In other words, this income-rich person becomes deprived in health dimension of well-being. A wealthy person cannot increase the quantity of an inadequately supplied public good by spending money on his own.

In a study for Montevideo, the capital of Uruguay, Katzman (1989) noted that 13% of households were poor in the income dimension but did not encounter deprivation with respect to basic needs and the opposite happened for 7.5% of the households. Ruggeri Laderchi (1997) employed Chilean data to conclude that income alone is unable to furnish an all-inclusive picture of poverty. In their study, Stewart et al. (2007) noted that 53% of undernourished Indian children did not belong to income-poor households. It was noted as well that 53% of the children coming from the households that were poverty stricken in the income dimension were not undernourished. These studies clearly indicate income in itself does not convey deprivations in nonmonetary dimensions (see also Bradshaw and Finch, 2003 and Alkire et al., 2015).

Attempts to cluster dimensional achievements, using prices as respective weights, into a composite figure indicating overall achievement performance may not be unambiguously acceptable. The reason behind this is that there may not exist a suitable price system that enables us to aggregate individual achievements into a single number, which can be compared with an income poverty line to judge whether the person is poor or not. This is the well-known income method adopted in many earlier studies. This may be due to nonexistence of markets for some dimensions. Examples include public goods, consumption and production externalities, and informational asymmetry. This is true as well for rationed goods (see Tsui, 2002 and Bourguignon and Chakravarty, 2003).

A very important reason for assessing poverty from multidimensional perspective is based on a concrete recommendation made by the European Union. In March 2000, in the Lisbon European Union Council, it was decided to shift poverty measurement policy from income dimension to a multidimensional framework. Five objectives on employment, innovation, education, social exclusion, and climate/energy, to be achieved by 2020, were laid by the European Union.

Among different methods, suggested for the measurement of multidimensional poverty, are the basic needs, social exclusion, and capability approaches. The basic needs formulation to multidimensional poverty is concerned with the definitions of minimum resources required to maintain barely “physical efficiency” (Townsend, 1954, p. 131; see also Streeten et al., 1981). At the individual level, social exclusion can be regarded as a person's exclusion from participation in primary economic and social activities of human well-being (see, among others, Akerlof, 1997; Atkinson, 1998; Atkinson et al., 2002, 2017). According to Sen (1993), the capability approach relies on an outlook of living as a mixture of different “doings” and “things”. Quality of life needs to be evaluated with respect to attainment of valuable functiongs. (See also Sen, 1987). In this method, functionings and capability are two integral aspects of poverty measurement. Capability deprivation incorporates the true concept of poverty experienced by people in everyday life (Sen, 1999, p. 87). The deprived individuals are declined access to many primary social and political rights. Poverty can be conceived as an overall construct involving different aspects of living conditions.

In the direct method, each person possesses a vector of quantities of needs that serve as different dimensions of human well-being and a direct procedure of pinpointing the poor substantiates whether the person has “minimally acceptable levels” (Sen, 1992, p. 139) of achievements in these dimensions. These “minimally acceptable” quantities appear as the threshold limits or cutoff points for different dimensions. These demarcation lines are assumed to be exogenously given; they do not depend on the distributions of achievements in different dimensions. They are determined independently of the achievement distributions. A person whose achievements in different dimensions do not fall below the corresponding threshold limits is treated as nonpoor. On the other hand, if a person's achievement in a dimension is lower than the associated threshold limit, then he is counted as deprived in the dimension. Given a well-defined rule of diagnosing the set of poor persons, the construction of a poverty index involves aggregation of shortfalls of dimensional achievements from respective thresholds, in other words, of the dimensionwise deprivations. More generally, in the direct method, joint distributions of disadvantages and interconnections between disadvantages are taken into account.

It will now be worthwhile to make a systematic comparison between the income method and the direct method. In the income method, under aggregation, low achievements in some dimensions may be counterbalanced by high achievements in some other dimensions. However, dimensions demonstrating low achievements may be of policy relevance. For instance, in India, aggregate consumption is on the rise and the head-count ratio, the proportion of population living in poverty, is reducing, but still many children remain undernourished (see Foster et al., 2013b). In contrast, in the direct method, dimensionwise achievements are integrated, and hence, the possibility of offsetting higher deprivations by lower ones is ruled out.¹

Poverty in a society can be regarded as withdrawal of human rights from the affected persons. It is not an implication of the option “not to do.” To understand this in greater detail, recall the example on distinction between starvation and fasting, considered in Chapter 1. An individual who is not deprived in any dimension of well-being may be fasting to pay respect to some religious custom. Consider also a second individual who is starving because he is poor. Thus, for the first individual, fasting is not a consequence of the inability to possess food. In contrast, for the second individual, it arises out of enforced inability to achieve minimal level of food items resulting from deprivation. Such a person is unable to maintain a decent standard of living.

Since poverty alleviation is one of the economic policies of high concern in many countries of the world, it becomes worthwhile to identify the causal factors of poverty. People may get into poverty for many reasons including unemployment, family breakdown, death of a major earning member of the family, low earning as a consequence of low education, bad health status, and so on. People may earn low incomes because of existing minimum wage laws of the country. These may force some people to fall into poverty. Breakdown of the poverty index by dimensions, say, by health condition, educational qualification, income, and so on, will empower us to locate which of the dimensions considered contribute more to overall poverty. All poverty indices obeying this breakdown characteristic are known as factor decomposable.

Often, a subgroup of a population with a low fraction of population turns out to be highly poverty stricken, and its dissemination to overall poverty is also quite high. Elimination or reduction of poverty in such subgroups will certainly reduce national poverty to a large extent. This alternative notion of policy can be implemented if the underlying poverty index is subgroup decomposable; for any separation of the population into subgroups with respect to an attribute, say, age, sex, race, region, and so on, overall poverty is the population share weighted average of subgroup poverty values.² For instance, when we split up population with respect to sex, poverty reduction may be possible by pursuing policies that equalize opportunities between men and women.

The objective of this chapter is to present an analytical discussion on the axiomatic approach to the measurement of multidimensional poverty.³ The next section is concerned with a brief review of income poverty analysis. The background materials of the multidimensional analysis are introduced in Section 3.3. The problem of identification of the poor is discussed meticulously in Section 3.4. An investigation of the postulates for a multidimensional poverty index is carried out methodically in Section 3.5. Some implications of these postulates are also studied. In Section 3.6, a rigorous evaluation of multidimensional poverty indices is offered. The explored functional forms can be subdivided into three categories: dashboard-reliant composite indices, directly formulated indices, and the ones adopting inclusive measure of well-being approach. It often becomes worthwhile to investigate if a particular poverty index can rank two different distribution matrices in the same way for a range of poverty thresholds. This problem of multidimensional poverty ordering is studied in Section 3.7. We have noted that several dimensions of well-being possess ordinal characteristics. Poverty measurement in such a case requires a different treatment. We explore this matter in Section 3.8. Section 3.9 deals with ordering of distribution matrices using individual deprivation counts, the numbers of dimensions in which different individuals are deprived. The subject of Section 3.10 is aggregation of individual deprivation counts when the dimensions are of materialistic nature. Finally, Section 3.11 concludes the discussion.

3.2 A Brief Review of One-Dimensional Analysis

Since the one-dimensional analysis of poverty measurement has close relations with its multidimensional counterpart, in this section, we present a brief discussion on the former. It is assumed throughout the section that income is the only dimension of human well-being.

Let , the nonnegative orthant of the n-dimensional Euclidean space , stand for the set of income distributions in the n person society, we consider for our purpose, where is arbitrary and is the set of positive integers. Sometimes, it will be necessary to choose , with the origin deleted, or , the positive part of , as the income domain in an n-person society. The associated sets for all possible population sizes are indicated by , , and , respectively. We will write , where represents person i's income, for an arbitrary income distribution in an n person society.

Often, an absolute threshold income, called the poverty line, is used for poverty assessments of income distributions. An absolute poverty line is an exogenously given cutoff limit and is not affected by any change in the income distribution whose poverty is to be assessed. Given the income distribution, a person is identified as poor if his income falls below the threshold limit. In many developing countries of the world, individual absolute poverty lines are employed for poverty evaluations in the respective countries. Following a suggestion, put forward by Ravallion et al. (1991), a $1 per day poverty line was used by the World Bank for the developing world, and in a later contribution by Ravallion et al. (2009), this threshold was revised to $1.25 a day at the 2005 purchasing power parity (PPP). Deaton (2010) mentioned several problems associated with the construction of a global poverty line and the use of PPP exchange rates to correct international price differences. More recently, a new international poverty line of $1.90 was adopted by the World Bank (see Ferreira et al., 2015).

In contrast to the absolute poverty line, the relative poverty line depends on the income distribution and hence is responsive to alterations in the distributions. Examples of such poverty lines are 50% of the median (Fuchs, 1969) and 50% of the mean (O'Higgins and Jenkins, 1990). In Atkinson and Bourguignon (2001), a relative poverty line equal to the 0.37 times the mean income (or expenditure) was considered. Chen and Ravallion (2001), on the other hand, used 0.33 instead of 0.37 as the multiplicative coefficient. The EU standard posited the poverty line as 60% of the median. To the contrary, in the United States, since the 1960s, official poverty rates were determined using an individual's or family's pretax income and an absolute cutoff limit (Orshansky, 1965). In recent periods, a new supplemental poverty measure, which uses more general definitions and incorporates additional items such as tax payments and work experiences in the estimation of family's resources, has been proposed since 2011. In India, separate absolute thresholds are used for the rural and urban sectors (Subramanian, 2011).⁴

Throughout the section, we assume the existence of an absolute threshold limit and denote it by . Assume further that , where is the nonnegative part of the real line . In words, z presupposes values in a finite nondegenerate positive interval of . Now, person i is regarded as poor in , if . Equivalently, we can say that person has a feeling of deprivation since his income falls below the threshold point z. The person is nondeprived in income when is at least z, which means that . Thus, z is the level of income that a person requires minimally to be nondeprived in income.

Evidently, the concept of deprivation here arises from the comparison of an individual's income with that of another whose income meets the subsistence limit z. Equivalently, all those persons whose incomes fall below z regard z as their reference limit, and all those whose incomes are at least z may be assumed to constitute the reference group.^5,6

For any , we write for the set of poor persons in u. Analytically, . Then the number of persons in , , is the total number of poor in u. If nobody is poor in , then is an empty set. In contrast, if all the n persons in are poor, .

In his pioneering contribution, Sen (1976) argued that measurement of poverty involves two main steps: identification and aggregation. The concern of the issue of identification is the isolation of the set of poor. Consequently, given , identification requires determination of the set . The problem of aggregation is to summarize poverty information on the individuals in the society. More precisely, for any , the issue is to prepare a summary statistic of the extent of poverty suffered by the individuals in .

The most widely used summary statistic of poverty is the so-called head count, the proportion of poor people in the population. A second commonly used measure of income poverty is the poverty gap measure, the average of the proportionate income gaps of the poor from the poverty line multiplied by the head-count ratio. As Sen (1976) pointed out, these measures are violators of a basic postulate, which a useful poverty index should verify. This property, the transfer axiom, suggested by Sen himself, requires poverty to increase unambiguously under a transfer of income from a poorer poor to a richer poor such that the recipient does not become rich because of the transfer. Another basic poverty postulate suggested by Sen is the monotonicity axiom, of which the head-count ratio is a violator but not the poverty gap measure. This property demands poverty to increase whenever the income of a poor goes down. Sen also suggested a more sophisticated index that fulfills these desirable properties.⁷

Although each of the head-count ratio and poverty gap measure is a violator of at least one of the two basic postulates suggested by Sen, they have one attractive property, subgroup decomposability. According to subgroup decomposability, for any partitioning of the population into two or more population subgroups, the overall poverty is the population share weighted average of subgroup poverty levels.⁸ (See Chapter 2, for a formal discussion on subgroup partitioning of a population.) The contribution of a subgroup to overall poverty can be defined as the subgroup's poverty multiplied by the population share of the subgroup divided by the overall poverty value. Since the multidimensional analog of this property, including its policy implications, will be investigated extensively in a later section, we do not elaborate it further here.

We will wind up this section with a brief discussion on three well-known subgroup-decomposable indices whose multidimensional extensions will be analyzed in one of the following sections. Consequently, this brief introduction serves as a background material.

For this purpose, we need some more preliminaries. Let be the censored income related to . Denote the censored income distribution associated with by . In a censored income distribution, each income is replaced by poverty line if the income value does not fall below the poverty line. Otherwise, the income quantity remains unchanged (see Takayama, 1979). In our presentation of subgroup-decomposable indices, we will restrict attention on censored income distributions. Since a poverty index should focus on the incomes of the poor, this is a sensible assumption (Sen, 1981). This, however, does not rule out the possibility of dependence of the index on the nonpoor population size.

The poverty shortfall of person i may be measured using the absolute difference . This shortfall is 0 if person i is nonpoor and positive otherwise. When expressed in relative or proportionate term, it is given by ; . By definition, each is homogeneous of degree 0, that is, invariant under equiproportionate changes in and the threshold limit . It is continuous and decreasing in the censored income level . We refer to as the deprivation index of person i. It is a measure of the extent of deprivation suffered by person i in the profile .

The entire family of subgroup-decomposable income poverty indices can be written as , where is arbitrary and satisfying is a transformed deprivation indicator. It is also assumed to be decreasing, continuous, and strictly convex in incomes of the poor. As we derive from person i's deprivation by applying the transformation , we can ascribe as his transformed income deprivation. Since , in the amalgamated formula , only deprivations of the poor are taken into account. Decreasingness and strict convexity of are necessary to ensure that the poverty index fulfills respectively the monotonicity and transfer axioms. For and , where and are parameters, the resulting indices turn out respectively to be

3.1

and

3.2

The measuring devices and are respectively the Chakravarty (1983) and Foster et al. (1984) income poverty indices. For and , they coincide with the poverty gap measure. For , reduces to the head-count ratio. While is defined directly on the deprivations of the poor, is a normalized version of the utilitarian gap , where U is a continuous, increasing, and strictly concave identical utility function. The constant elasticity of the underlying marginal utility function is given by . Both e and α capture different aspects of income poverty. For all and , under a transfer of income from a poorer poor to a richer poor, increase in poverty becomes higher, the poorer the donor is.⁹

Finally, consider the transformation , where , which means that in this case, the domain of is . Then the underlying measuring instrument becomes the Watts (1968) index of income poverty:

3.3

One common feature of all the three indices is that they assume the minimum value 0 when nobody is deprived in the society ( for all i). Both and reach the upper bound 1 when everybody in the society is maximally deprived ( for all i). However, is undefined in this situation. This index is unambiguously more sensitive to income transfers lower down the income scale.¹⁰

3.3 Preliminaries for Multidimensional Poverty Analysis

To formulate and discuss the preliminaries rigorously, we follow the notation adopted in Chapters 1 and 2. As before, the number of dimensions of well-being in an -person society is given by , where N is the set of positive integers. Each dimension may be regarded as representing some basic need of human well-being. In this n-person society with d dimensions of well-being, a typical distribution matrix or an achievement matrix, which we also refer to as a social matrix or social distribution, is denoted by . The quantity , the (i, j)th entry of X, indicates the achievement of person i in dimension j, where . The matrix X is assumed be an element of , that is, . The set is the restriction of the general set M when the population size is fixed n, where . The necessity of the general set M arises whenever poverty comparisons involve differing population sizes. The general set M can be anyone of the three matrices in the set , where , , and are the same as in Section 1.2. The column vector of X represents the distribution of achievements in dimension j among n-persons. On the other hand, person i's achievement profile, the achievements of person i in d dimensions, is denoted by the row vector , the ith row of .

In the multidimensional framework under consideration for each dimension , there is a unique (exogenously given) poverty threshold , which represents means of supporting a person's level of living at a minimal level with respect to dimension _. Often, we will refer to as dimension reference limit or cutoff limit. Since represents the least possible quantity of achievement in dimension for a subsistence level of living with respect to the dimension, we can say that a person whose achievement falls below this barely adequate quantity will regard as his targeted achievement in the dimension. Consequently, person feels deprived in dimension , or the dimension is meager for the person if his achievement falls below the corresponding threshold , that is, if . The person is nondeprived in the dimension in the case when his achievement is at least , which means that . Thus, is the level of achievement that a person requires minimally to be nondeprived in dimension . Person i's deprivation in dimension is maximal if .

Evidently, relativity in the concept of deprivation in a dimension arises from the comparison of an individual's achievement in the dimension with that of another whose achievement meets at least the subsistence limit. Equivalently, all those persons whose achievements in dimension fall below regard as their dimensional reference limit, and all those whose achievements are at least may be assumed to constitute the dimensional reference group.

The threshold limits taken together for all the dimensions constitute the vector , which is assumed to be an element of , a finite subset of , the strictly positive part the d-dimensional Euclidean space. More precisely, .

The aforementioned method of determination of multidimensional poverty in terms of dimensional deprivations using the vector of dimensional reference limits is known as the direct method (Sen, 1981). In contrast, in the income method (Sen, 1981), the poverty line incorporates the monetary values of some necessaries required to maintain a minimal standard of living (see Townsend, 1954). (See also Booth, 1894, 1903; Rowntree, 1901; Bowley and Burnett-Hurst, 1915 and Gillie, 1996.) Evidently, “a commodity focussed concept of basic needs underlay the income method…, as the poverty line indicated the minimum amount of resources to cover such needs” (Alkire et al., 2015, p. 125).

For any person and any dimension , let stand for the censored quantity of achievement in dimension j enjoyed by person i. Then the deprivation indicator of person i in dimension j is specified by . We will also refer to as the proportionate deprivation, the proportionate shortfall of the achievement in the dimension from the dimensional cutoff limit. Hence, if individual i is deprived in dimension j, then he encounters a positive deprivation; otherwise, his deprivation is zero.

3.4 Identification of the Poor and Deprivation Counting

Identification of the poor in a multidimensional framework is not as simple as in the one-dimensional case. In the one-dimensional situation, given the unique income poverty line and a person's income, it can be unambiguously claimed that the person is poor (respectively, nonpoor) if his income falls below (respectively, not below) the poverty line. Since in a multivariate setup, the number of dimensions is at least 2, the selection of the number of deprived dimensions for judging whether a person is in poverty or not becomes an issue of natural discussion.

There can be two extreme positions in this context, each one possessing arguments for and against it. The first is the union approach that counts a person as poor if he is deprived in at least one dimension. On the other hand, we have the intersection criterion that regards a person as poor if all his achievements are simultaneously below the corresponding reference limits (see Tsui, 2002, Atkinson, 2003 and Bourguignon and Chakravarty, 2003). Finally, there is an intermediate way that can be reckoned as a compromise between the two extremes. In this case, the number of dimensions for which individual achievements have to be compared with the corresponding dimensional reference limits can vary from the minimum to the maximum (see Mack and Lansley, 1985; Nolan and Whelan, 1996; Gordon et al., 2003 and Alkire and Foster, 2011a,b).

To analyze the recommendations formally, following Bourguignon and Chakravarty (2003), let us define a deprivation identification function for each of them. The formulation will be presented in terms of an arbitrary element M of the set . For the union mode, the identification function can be rigorously defined as follows. For all ,

3.4

In other words, takes on the value 1 if person i is deprived in at least one dimension; otherwise, its value is 0. We designate as a deprivation identification function since it clearly indicates whether the person is deprived in some dimension. It is definitely not necessary to check whether deprivation occurs in all the dimensions. One goes on searching for a dimension in which deprivation prevails, and once this is established, the search stops. The person is then regarded as poor by the union criterion. Accordingly, the number of persons who are counted as poor in X by the union criterion is given by the sum .

The set of persons who are regarded as poor by the union criterion, the set of union poor, in is given by . Consequently, the number of union poor persons in X can alternatively be expressed as , the number of persons in .

Consider a person who is nondeprived in all the dimensions except one with a low shortfall from the reference limit will be counted as poor by the union rule. In the aggregation, his contribution to total poverty will be quite low. This can be checked easily if the poverty index satisfies subgroup decomposability, an axiom that has attractive policy appeals. Isolation of such minor contributions is not a problem under subgroup decomposability. If the population size is large, the set of union poor may also be large. But identification of the set of poor with low deprivations, under subgroup decomposability, is an extremely easy task.

In the intersection mode, the poor are identified as those persons who experience deprivation in each dimension. As a result, the corresponding deprivation identification function can be formally defined as follows. For all ,

3.5

In consequence, the set of intersection poor in X becomes , and the number of persons suffering from deprivation in all the dimensions in the matrix X is .

In this alternative identification arrangement, a person is poor if he is deprived in all the dimensions, and this leads us to identify the number of poor as the total number of persons who are deprived in all the dimensions. A person deprived in one dimension but nondeprived in another is not designated as poor by the intersection method. But trade-off between achievements in the two dimensions may not be possible. Shortage of essential durables cannot be neutralized by housing. Consider a person with high deprivations in all dimensions except one, and in the nondeprived dimension, his achievement is at the corresponding reference point. Although this person is assigned a nonpoverty status by the intersection criterion, from antipoverty policy perspective, this may be difficult to accept. An old street beggar who does not even earn hand-to-mouth daily by begging is judged as nonpoor by the intersection system because of high life expectancy. But overall, he maintains a very low level of living and is definitely a union poor. Consequently, treating a person as poor if he is deprived in at least one dimension appears to be more sensible.

In between the union and intersection procedures, there lies the intermediate identification system, which treats a person as poor if he has a minimum deprivation score of . For any , the deprivation score of person is defined as , where is the deprivation indicator of person i in dimension j, is the weight assigned to dimension j, and . Note that if person i is deprived in dimension j; otherwise, it takes on the value 0. Hence, , where the lower and upper bounds are attained respectively in the extreme cases when the person is nondeprived and deprived in all the dimensions. Following Duclos and Tiberti (2016), we refer to as the weighted proportion of deprived dimensions for person i. If all the dimensions are equally weighted, that is, for all , then becomes the person's deprivation count, the number of dimensions in which the person is deprived, divided by the maximum attainable value of the deprivation count.

Alkire and Foster (2011a,b) defined person i as multidimensionally poor if the inequality holds, where the poverty cutoff is a threshold parameter. Formally, the intermediate deprivation identification function can be formally defined as follows. For all ,

3.6

Hence, the set of persons in X who are regarded as poor by the intermediate system of identification is given by , and the number of such persons in the matrix is . This intermediate identification criterion coincides with the union mode if the poverty cutoff does not exceed the minimum of the weights: . On the other hand, the intersection mode drops out as a particular case of the intermediate norm if . For , this intermediate identification criterion coincides with the union or the intersection rule depending on whether or . For the intermediate mode to be different from the union and intersection methods, the number of dimensions should be at least 3. The choice of and the weights are matters of value judgment. (See Duclos and Tiberti, 2016, for further discussions on this criterion of identification and its implications.)

Under the equal weighting scheme in the intermediate technique, we assign equal importance to each dimension by treating them symmetrically. Under symmetry, it becomes necessary to look at a person's deprivation count. As soon the count leads us to , the person is labeled as poor by the intermediate process. (More extensive applications of deprivation counts are available in Sections 3.8–3.10.)¹¹

We may now illustrate the union, intersection, and intermediate practices of identification graphically. In Figure 3.1, the horizontal and vertical axes represent the achievements in dimensions 1 and 2, respectively. Their upper bounds are denoted respectively by and . The positions of the respective cutoff limits are indicated by the horizontal and vertical lines z₁ and z₂. All those individuals whose achievements are in the two-dimensional poverty space-, exhibited by the region , are enumerated as poor by the intersection criterion. Since the union rule recognizes someone as poor if he is deprived in either of the two dimensions, all individuals with achievements below the line or to the left of the line become poor by this criterion. Finally, the curve illustrates an intermediate approach.

The two one-dimensional poverty spaces and are displayed by the regions and , respectively. While in the former region, a person is deprived in dimension 2, in the latter, deprivation arises from low achievement in dimension 1. The intersection of the two one-dimensional spaces generates the two dimensional space . However, by summing up the numbers of individuals whose achievements lie inside the space, () will overestimate the number of poor. The reason behind is that the number of persons whose achievements lie inside is counted twice. Suppose that a person's achievements in the first dimensions are zero and the achievement in the remaining dimension coincides with its threshold limit. This person is not counted as poor by the intersection method, however large or small the value of d may be. Thus, although the person is maximally deprived in all the dimensions except one and he is nondeprived in only one dimension, he is intersection nonpoor. This clearly indicates a shortcoming of the intersection rule as an identification criterion. Nevertheless, the union method identifies this person as poor.

In the following discussion, we follow the union criterion of identification. This mechanism has installed itself as a notable identification criterion because of its easy implementation and long usage. For any , , we denote the set of all poor persons in X by . For any given , we write the number of poor persons in by . The head-count ratio, the proportion of persons in poverty, is .

3.5 Axioms for a Multidimensional Poverty Metric

A multidimensional poverty index P is required to aggregate the deprivations of the individuals in a society along different dimensions of well-being in terms of a nonnegative scalar. For any vector of exogenously given threshold limits and an achievement matrix, this scalar signifies the extent of poverty related to the matrix. More precisely, it is a summary statistic of the distributions of individual deprivations in different dimensions of well-being. Unless specified explicitly, we assume that the set of all distribution matrices is given by . Then . For any , the scalar is the level of poverty associated with social matrix , where is arbitrary.

The structure of the current section, whose subject is the analysis of the properties for a multidimensional poverty index, parallels that of Part 2.3.1.1 of Section 2.3.1. Most of the postulates discussed next for an arbitrary P are generalizations of different postulates suggested for an income poverty index. However, any axiom that necessitates existence of at least two dimensions does not have any unidimensional sister.

3.6 Multidimensional Poverty Measurement

In this section, we analyze some functional forms for multidimensional poverty indices. Each of these functional specifications is a summary statistic of deprivations of the poor along different dimensions of well-being in terms of a real number. We begin by presenting dashboard-based composite indices. Next, we discuss the direct approach, the main method adopted in the literature, in which axioms and poverty measures are defined directly on multidimensional matrices (see also Duclos and Tiberti, 2016). We then turn to the indices that can be interpreted from the inclusive measure of well-being perspective.

3.6.1 The Dashboard Approach

The example we consider in this subsection is based on a dashboard of dimensional indices. For any , , signifies the level of poverty associated with , the distribution of achievements in dimension j. (Unless stated, the assumption will be maintained throughout the section.) The dimensional poverty dashboard is given by . For illustrative purpose, assume that the dimensional metrics are given by , where is arbitrary and the transformation satisfies . Since indicates person i's deprivation in dimension j, is the corresponding transformed deprivation indicator. It is also assumed to be decreasing, continuous, and strictly convex in the concerned dimensional achievements of the poor. Each of the d dimensional metrics is the arithmetic average of transformed deprivations in the dimension. The resulting dashboard then becomes . Evidently, if nobody is deprived in dimension j.

Decreasingness of ensures that a reduction in the achievement of a poor person in any deprived dimension increases . Strict convexity of guarantees that a transfer of achievement from a poor person to a richer poor person such that the poverty status of the recipient does not change increases , where the deprived dimension is arbitrary. This implies that if each component of the dashboard is increasingly related to the composite poverty index to be designed from the dashboard, then the underlying composite index satisfies the Bourguignon and Chakravarty (2003) one-dimensional transfer principle. The one-dimensional transfer principle necessitates a raise of multidimensional poverty under a regressive transfer of achievement in a deprived dimension from a poorer poor to a richer poor, given that the poverty status of the richer person, achievements of the two affected persons in all other dimensions, and achievements of all the remaining persons are kept unchanged.

Sequential applications of one-dimensional transfers miss the essence of multidimensionality and hence are not suitable in the current context. The appropriate notion of redistribution is the multidimensional transfer axiom.

An extremely simple way to arrive at a composite index is to take a weighted average of dimensional indices incorporated into the dashboard. Formally, the formula for the composite index turns out to be

3.7

which can alternatively be expressed as

3.7a

where is the weight assigned to poverty assessed in dimension j, . Chakravarty et al. (1998) suggested (3.7a) assuming subgroup decomposability and factor decomposability at the outset. But it emerges that in (3.7a) can also be accommodated within the dashboard framework.

By assigning positive weights to individual dimensional poverty amounts in the dashboard, we implicitly assume that all the dimensions are important in global poverty evaluation. However, any choice of the weights will reflect a particular value judgment. (See Section 1.5 for a discussion. See also Aaberge and Brandolini, 2015.) For instance, if we choose , then all the dimensions are marked as equally important.

By construction, the symmetric, population replication invariant, dashboard-based family satisfies monotonicity. It as well satisfies monotonicity sensitivity and dimensional monotonicity axioms. It expands if the threshold limit of a dimension in which there is at least one deprived person goes up. Although it verifies the multidimensional transfer principle, it does not change under a correlation-increasing switch and hence treats the dimensions as “independents.”

Several members of the family (3.7) are worthy of investigation. For the transformation , where , the underlying measuring instrument becomes the Watts (1968) index of multidimensional poverty:

3.8

In order to develop a characterization of this index, Tsui (2002) assumed in the first place that a poverty index is symmetric, population replication invariant, strongly focussed, monotonic, and continuous in achievements. He then demonstrated that axioms of strong ratio-scale invariance, decreasingness under a uniform majorization operation among the poor, nondecreasingness with respect to a correlation-increasing switch among the poor, subgroup consistency, and poverty criterion invariance hold together if and only if is one of the members of a two-member family. Subgroup consistency requires that for any partitioning of the population into two or more subgroups, if poverty in one subgroup goes down and stays fixed in the other subgroups, then global poverty should reduce as well (see Foster and Shorrocks, 1991). Poverty criterion invariance demands that there should be no change in poverty ranking if one or more threshold limits change, keeping the number of poor unaltered.

Chakravarty and Silber (2008) characterized this index from a welfare theoretic perspective. Poverty index is defined as welfare loss generated as a consequence of shortfall of dimensional achievements of the poor from the respective cutoff limits. It was then demonstrated that the only strongly focused poverty index satisfying subgroup decomposability, strong ratio-scale invariance, montonicity, and continuity in achievements is .

An attractive feature of the Watts index is that we can split up the change in poverty between two periods into its multidimensional growth and multidimensional redistribution components (see Chakravarty et al., 2008). Inquiry concerning relationships among poverty, economic growth, and inequality is highly motivated by policy-related issues including “trickle-down” eventualities of economic growth and impact of structural adjustment plans for stabilizing the economy (See, e.g., Ravallion and Huppi, 1991; Datt and Ravallion, 1992; Kakwani 1993 and Lipton and Ravallion, 1995.) To diagnose the growth and redistribution ingredients explicitly, suppose that and stand for the achievement matrices of a society in periods 1 and 2, and let the corresponding population sizes be and , respectively. Then assuming that the threshold limits stay put, the poverty change between the two periods happens to be

3.9

where stands for the mean of the jth dimensional achievements of the poor in period t (t = 1, 2). The first third bracketed term on the right-hand side of (3.10) represents the redistribution factor showing the change in poverty resulting from a change in inequality among the poor, given that the means of their dimensional achievements stay put. The inequality index that appears here is a member of the multidimensional generalized entropy family (see Equation (2.9c)). It is a multidimensional extension of Theil's mean logarithmic deviation (see Equation (2.8)). The second third bracketed term of the right-hand expression is the growth ingredient of the decomposition. It signifies the change in the poverty that follows from changes in means of achievements of the poor under the a priori assumption that inequality stays fixed (see also Chakravarty, 2009). It shows the effectiveness of economic growth on poverty change.¹³

A multidimensional generalization of the one-dimensional Chakravarty (1983) index was suggested by Chakravarty et al. (1998) entailing additivity over transformed individual dimensional deprivations. It corresponds to the transformation , where is a constant. The resulting summary measure becomes

3.10

For any value of , the index satisfies the monotonicity sensitivity property. For , overall poverty assumes the form of weighted average of dimensional deprivations of an average person. Although the transfer principle is violated in this case, it may be worthwhile to note that for computational purpose, we only need information on dimensionwise deprivations of different individuals and the dimensional weights.

Finally, consider the specification , where is a parameter. The consequential multidimensional index comes out to be

3.11

This is a simple multidimensional extension of the Foster–Greer–Thorbecke unidimensional index (see Foster et al., 1984). It satisfies the monotonicity sensitivity axiom for all . The values and coincide when .

3.6.2 The Direct Approach

The first example we provide in this subsection is the second member of the Tsui (2002) family, defined as

3.12

where is arbitrary and the parameters are positive for all . The subscript indicates that this strongly relative (more generally, strongly ratio-scale invariant) index was proposed by Tsui, and stands for the d dimensional vector of parameters . This quantifier of poverty is an alternative multivariate extension of the one-dimensional Chakravarty (1983) index. (See Eq. (3.1)). While the first member of the Tsui family involves additive aggregation both across dimensions and across individuals, in (3.12), a multiplicative aggregation across dimensions is employed. The parameters have to obey some inequality restrictions for the fulfillment of different axioms. For , verification of the transfer axiom is ensured by the restrictions and . It is unambiguously increasing with reference to a correlation-increasing switch.

The next proposal we wish to analyze is the Bourguignon and Chakravarty (2003) index , one of the most widespread indices of multidimensional poverty:

3.13

where the parameters α and θ are positive, and is the weight assigned to the transformed deprivation indicator of dimension j, . Accordingly, the individual poverty function in (3.13) is obtained by raising the weighted average of such indicators across dimensions to the power . Multidimensional poverty is then defined as the simple arithmetic average of individual poverty functions. As a consequence, by construction, the index comes to be subgroup decomposable (hence symmetric and population replication invariant). Further, it is continuous, bounded, strongly ratio-scale invariant, and dimensionally monotonic. It also gladdens the monotonicity sensitivity (hence monotonicity) and threshold limit sensitivity axioms. The transfer axiom is satisfied when θ is greater than 1. For a given θ, as the value of α increases, sensitivity of dimensionwise regressive transfers at lower down the distributions of achievements increases. It may be viewed as a simple parametric generalization of the Foster et al. (1984) one-dimensional index to the multidimensional framework.

Its value increases or decreases under a correlation-increasing switch depending on whether α is greater or less than θ. As a result, while under the restriction , the dimensions affected by the switch are treated as substitutes, they are counted as complements whenever the reverse restriction holds. That is why, is referred to as the “Bourguignon–Chakravarty substitutes” index for , and it is known as the “Bourguignon–Chakravarty complements” index for (Vélez and Robles, 2008, p. 217).

For , coincides with in (3.11) and hence gets converted into a composite index, which shows insensitivity to a correlation-increasing switch. Bourguignon and Chakravarty (2003) proposed and analyzed this index extensively by assuming that the deprivation indicator of person i in dimension j is given by , where for all , and they need not be the same across dimensions. The resulting index, which Alkire et al. (2015) denoted as , comes to be . We can as well obtain it by assuming first parametric variability in . More precisely, first α is replaced by α_j and θ is replaced by where is arbitrary, and variability of these 2d such parameters across dimensions is allowed. At the second step, we assume that for all and deduce . Alkire et al. (2015) refer to , which is known in the literature as the Bourguignon–Chakravarty index, as . For α = 1, θ = 1, the value of matches with those of and when , indicating its insensitivity to a correlation-increasing swap as well as transfers.

The parameter θ is directly related to the curvature of an isopoverty contour, a locus of achievements in that generate the same value of the individual poverty function. For any , an isopoverty contour becomes strictly convex to the origin. As the value of θ over the interval rises, convexity of the contour increases under ceteris paribus assumptions. Because of this, the marginal rate of substitution, the trade-off between the proportionate deprivations in the given dimensions, becomes increasingly difficult with an increase in the value of θ.

The constant elasticity of substitution between proportionate shortfalls associated with the two dimensions is . For θ = 1, there is perfectly elastic trade-off between the proportionate deprivations; the isopoverty contour becomes a straight line in this polar situation. At the other extreme, as , we get the rectangular isopoverty contour.

Given α, in the limit as , approaches

3.14

As the two-dimensional individual isopoverty curves corresponding to the formula (3.14) are of rectangular shape, substitution between the two relative shortfalls is ruled out. The information required for this representation of multidimensional poverty is rather basic; knowledge of the proportionate shortfalls is sufficient to perform the underlying aggregation. (See Bourguignon and Chakravarty, 1999, for further discussion.) In consideration of the fact that in (3.14) employs a Leontief-type aggregation, Vélez and Robles (2008, p. 217) refer to it as the “Bourguignon–Chakravarty–Leontief” index. The second expression of (3.14) clearly demonstrates that in this limiting case, is the average of the relative maximin individual poverty functions.

Bourguignon and Chakravarty (2003) considered extensions of in (3.13) by allowing variation of the elasticity of substitution between dimensional proportionate shortfalls with the level of poverty. For instance, if a person is highly deprived in one dimension but is characterized with a very low level of deprivation in a second dimension, then elasticity of substitution may be of minor significance. This is because deprivation in the first dimension becomes a major determinant of overall poverty. But if the extents of deprivations are low in both the dimensions, then the roles of deprivations in both the dimensions are important. One may assume the opposite as well; scope of substitution decreases with the level of poverty. These two positions can be incorporated into the general Bourguignon–Chakravarty index (3.13) by assuming that the parameter θ depends on the level of poverty. More precisely, we replace θ in (3.13) by , a positive-valued function of the extent of poverty P. The consequential variant of in (3.13) that emerges under this assumption about the value of θ is

3.15

Two simple forms of that will ensure decreasingness and increasingness of the elasticity of substitution as poverty level increases are, respectively, and . (P here stands for and hence is bounded so that the examples of we have considered are well defined.) Under these specifications of , we can solve equation (3.15) numerically. The variant (3.15) of (3.13) inherits all postulates of the original poverty representation (3.13), except unambiguous change under a correlation-increasing swap between two poor persons. The swap may increase or decrease overall poverty depending on whether the affected persons are very poor or moderately poor.

In their excellent contributions, Lasso de la Vega and Urrutia (2011, 2009) characterized axiomatically as a multidimensional deprivation index by employing several intuitively understandable axioms defined on the set of deprivations of the individuals in different dimensions of well-being. The set of axioms invoked in the characterization includes continuity, monotonicity, weak dimension separability, homotheticity, subgroup decomposability, and normalization. The first axiom, continuity, stresses continuous variation in the index with respect to its arguments. The next axiom, monotonicity, says that overall deprivation is increasing with respect to dimensionwise deprivations. The third axiom, weak dimension separability, requires that for any separation of the set of dimensions Q into two or more subgroups, if deprivation for one subgroup goes up, while deprivations in the other subgroups stay put, then global deprivation should go up as well. According to homotheticity, multiplication of all dimensionwise deprivations in two different societies, by the same positive scalar, will not reverse the original deprivation ranking of the societies. The normalization axiom parallels the boundedness axiom of multidimensional poverty. Since when coincides with for , the framework also includes a member of the Chakravarty et al. (1998) family. A highly novel characteristic of their axiomatization is its generality; many perceptions of deprivation, including multidimensional poverty, can be accommodated within their general framework.

For any , , and the poverty threshold denote the set of persons who are treated as poor by the intermediate identification rule by . Then the Alkire and Foster (2011a,b) multidimensional poverty index is the simple average of the deprivation scores of the individuals in a population. Formally,

3.16

where and

3.17

In other words, the deprivation scores that are not lower than the poverty threshold are included in the aggregation involved in (3.16). This strongly focused subgroup-decomposable index is strong ratio-scale invariant, normalized, continuous, symmetric, population replication invariant, insensitive to a correlation-increasing switch, and monotonic in dimensions and achievements of the poor. It meets transfer postulate for and increases if the cutoff limit of a deprived dimension of a poor goes up. For , simplifies to the adjusted head-count ratio, the product of the head-count ratio , and the average deprivation score of the poor (see Alkire et al., 2015). Since an increase in the number of deprived dimensions of a poor increases the average deprivation score, this product functional representation endorses the dimensional monotonicity property, although it is a violator of the monotonicity and the transfer postulates. For , coincides with the adjusted poverty gap, the product of the head-count ratio, and the average deprivation of the poor.

Evidently, under the union rule of spotting the set of poor persons, coincides with when . A similar observation reveals that equates with if the union criterion of isolating the set of poor persons is adopted.

The strong ratio-scale invariance axiom is a representative of some particular concept related to poverty evaluation. An alternative reflection of poverty assessment arises through gratification of strong translation-scale invariance, a postulate that requires invariance of poverty under equal absolute changes in achievements and threshold limits, where the absolute amounts need not be the same across dimensions. Thus, while the strong ratio-scale invariance postulate summarizes poverty with respect to proportionate shortfalls, the strong translation invariance axiom treats poverty in terms of absolute shortfalls .

Tsui (2002) characterized the following class of strongly translation-scale invariant multidimensional absolute indices:

3.18

where , are arbitrary, and for all . The positive parameters are also assumed to obey the condition that is strictly convex. While the first functional form in (3.18) is a multidimensional extension of the Zheng (2000a,b) income poverty index, the second expression is a multidimensional generalization of the absolute poverty gap.

Often, for many policy purposes, we need to assess poverty using unit-consistent indices. The following family of strongly unit-consistent multidimensional poverty indices, fulfilling subgroup decomposability and continuity, has been characterized axiomatically independently by Diez et al. (2008) and Chakravarty and D'Ambrosio (2013):

3.19

where , are arbitrary; t is a real number; and choices of the parameters c and are dictated by the constraints for all . It increases (respectively, decreases) under a correlation-increasing exchange between two poor persons if and only if , where .

Given , the necessary and sufficient condition for to fulfill the strong ratio-scale invariance axiom is . Since strong ratio-scale invariance implies strong unit consistency, for any real t, strong unit consistency is fulfilled. Consequently, the family of strongly unit-consistent multidimensional poverty indices given by (3.19) is rather large. Nevertheless, no suitable choices of c and will empower to fulfill strong translation-scale invariance, where is arbitrary.

If there are only two dimensions, the transfer axiom holds if and only if and . If decreases under a correlation-increasing swap (the two dimensions are regarded as complements), for expositional ease, the value of c may be chosen as 1. Then satisfaction of all the conditions stipulated earlier are ensured for all choices of . The index increases by a larger quantity under a bundle of regressive transfers, the higher is the value of over the interval However, if increases under the swap (the two dimensions are treated as substitutes), a suitable choice of c may be . In such a situation, if , then all the conditions are satisfied. To establish this formally, observe that for , the necessary and sufficient condition for to increase with respect to the swap is . The necessary and sufficient conditions for the transfer postulate to hold get modified as and . These three inequalities hold at the same time for any .

If there is only one dimension, may be treated as a multidimensional extension of the Zheng (2007b) unit-consistent income poverty index. A worthwhile observation here is that the Zheng index itself is a two-parameter generalization of the Chakravarty and the Clark et al. (1981) ratio-scale invariant income poverty indices.

3.6.3 The Inclusive Measure of Well-being Approach

A natural question that arises is whether it is possible to interpret multidimensional poverty indices from the inclusive measure of well-being perspective. The objective of this subsection is to explore this possibility. It is necessary to check whether a standard multidimensional poverty quantifier relying on the direct method corresponds to a one-dimensional poverty measure computed using a vector of individual well-being numbers. (See also Ravallion, 1994, for a discussion.)

We can illustrate the issue using some examples. The first example we consider is based on the welfare theoretic characterization of the Watts multidimensional poverty index. Chakravarty and Silber (2008) argued that the index can be deduced as a normalized value of the welfare difference , where U is a metric of individual well-being and is the dimensional profile of censored achievements of person i. But use of the symmetric utilitarian well-being function for poverty assessment here is not logically acceptable when one adopts the inclusive measure disposition. The reason behind this is that it does not demonstrate uniform sensitivity to a person's dimensional achievements below and above the corresponding cutoff limits. All dimensional achievements above appropriate cutoff limits are replaced by the cutoff points themselves, and this is done irrespective of how large or small they are. Consequently, any change in such achievements provided that they never fall below the respective cutoff points is completely ignored. Consequently, the Watts index cannot be treated as an inclusive measure of well-being approach-based index.

As a second example, we may consider Tsui's (2002) generalization of the Chakravarty index (Chakravarty, 1983). From the functional form, it appears that at the initial stage for each person, a product-type well-being function is employed to cluster allocation of the d dimensions into a metric of personal well-being, and then at the second stage, transformed values of individual well-being levels are clumped by using simple arithmetic averaging rule. But the same problem, as we have encountered in the context of the Watts index, arises here as well. The same remark can be made for the Chakravarty et al. (1998), the Bourguignon and Chakravarty (2003), and the Alkire and Foster (2011a, 2011b) indices as well.

Maasoumi and Lugo (2008) suggested a family of multidimensional poverty measures by employing the Bourguignon and Chakravarty (2003) index within an information theory framework. Subgroup decomposability of the family is assumed in the first place. The individual well-being function, which requires aggregation across achievements of a person, was derived by using the same aggregation technique considered by Maasoumi (1986) (see Equation (2.29)). The same transformations that were employed for accumulating dimensional achievements were used to pile the threshold limits as well. The individual poverty function was then defined as the relative gap between one and the individual well-being function and the compounded poverty limit. The overall index, which was assumed to be increasingly related to the individual indices, was deduced by combining these individual components using a Foster et al. (1984) type transformation.

Formally, the family is given by

3.20

where , are positive weights that add up to 1, , , and are parameters. The denominator of each of the two members of the family can be regarded as the comprehensive poverty line. This is essentially a one-dimensional threshold limit.

The first bracketed term in the second member of the family is the shortfall of the ratio between a Cobb–Douglas function of censored achievements and the composite poverty line from unity. Hence, this notion of averaging is different from the usual ones where dimensionwise deprivations are calculated for each person, which are then amalgamated across dimensions and persons. An analogous sequential compounding process is performed for the first functional form as well. As a result, under this notion of composition, some degree of substitution between dimensional achievements is permitted before related deprivations are deduced from them. From poverty assessment perspective, this requires careful analysis. For , the index value becomes 1 for all distribution matrices. For all values of , it is sensitive to dimensional achievements.

By construction, an IMWB-based index is a violator of the strong focus axiom. One way to resolve this issue is to adopt the suggestion put forward by Decancq et al. (2015). In a different approach, these authors proposed to use heterogamous preferences in poverty evaluation analysis. Since their framework does not start with existence of dimensionwise threshold limits, it deviates substantially from the direct methodology. Instead, it is assumed that there exists a common cutoff bundle with regard to minimum degree of satisfaction of individual preferences.

They have demonstrated that the only way of measuring poverty is by imposing a minimum consumption bundle, which is the same for all. Consequently, if a person's consumption bundle turns out to be worse than the common minimum bundle , then he is regarded as poor; otherwise, he is counted as nonpoor. Their preference-based axiomatic study imposes a strong decomposition restriction on the poverty standard. The central idea is to look at individual preferences in order to identify the poor and aggregate dimensional achievements. Consequently, this contribution offers a two-fold suggestion: endogenizing the poverty thresholds and using individual preferences in the context of identification of the poor.

Under these authors' proposition, the strong Pareto principle is satisfied among the poor. Furthermore, the assessment of complementarity or substitutability between dimensions is left to the individuals themselves. This contrasts with the direct approach where the complementarity–substitutability issue is resolved by imposing parametric restrictions in the aggregated standard, which may or may not respect individual preferences (see Bourguignon and Chakravarty, 2003).

We denote the preference relation of person i, defined on the bundles of d-dimensional private consumption goods, by . For any two bundles and , indicates that person prefers to . The matrix of allocation of private consumption goods among n individuals in the society is denoted by x. We write for the vector of preferences of the persons. In this model, an economic situation is represented by . Let stand for the indifference component of . The only preference relation identified by their axiomatic characterization is the Leontief preference relation, and the resulting index emerges as

3.21

where is continuous and increasing, is continuous, decreasing, and convex and if and only if . Here the individual poverty function can be any decreasing, convex function worked out from the proportion of z to which a person shows indifference.¹⁴ By accommodating preference relations in the formulation, the model subsumes high degree of flexibility in the context of the choice of the minimal consumption vector. (Maniquet, 2016, provides an excellent discussion on this scheme.)

3.7 Multidimensional Poverty Orderings

As we have noted, given the threshold limits, there may exist several poverty indices that fulfill a set of desirable axioms. However, often subjective opinions may dictate the selection of such reference limits. (See the discussion presented in Section 3.2.) As a result, involvement of arbitrariness in the choice of the lines of demarcation may lead to unreliable conclusions based on that index. For instance, it may happen that a poverty index ranks two achievement matrices in different directions for two different threshold limit vectors. In consequence, it becomes worthwhile to verify if two achievement matrices may be ordered commonly by the same poverty index when the threshold limit vectors are allowed to vary within some reasonable limits. We refer to this notion of ordering as poverty-line limits ordering.

An alternative concept of ordering is to rank two achievement matrices unanimously by a family of poverty indices satisfying a set of plausible postulates, when the threshold limits are assumed to be fixed. This natural line of investigation is a consequence of existence of too many poverty indices satisfying such postulates, and emergence of opposite directional ordering of two different matrices by two alternative members of the family is quite likely. We term this concept of ordering as poverty-measure ordering¹⁵ (see Zheng, 2000a,b).

In this section of the chapter, we address the former notion of ordering. We begin by presenting a brief review of one-dimensional orderings along this line. This serves as the background material of the analysis carried out for multidimensional orderings.

3.7.1 A Brief Outline of One-Dimensional Orderings

In the one-dimensional framework, the problem is to judge whether one distribution of income does not have lower poverty than the other for all poverty lines. (For simplicity of exposition, we use income as the dimension of well-being. The analysis applies equally to any other dimension.) In such a case, we say that the former distribution poverty dominates the latter or the latter is poverty dominated by the former.

Let u^I and u^II be two nondecreasingly ordered distributions of income over a given population size. These distributions can as well be represented by the respective cumulative distribution functions G^I and G^II (distribution functions, for short). Then the value of at the income level stands for the cumulative proportion of persons with income not exceeding , is nondecreasing, and , where is the maximum value that incomes can assume, where .

Assume that the poverty lines can vary over the interval . For any and a poverty index P, we write for the poverty level associated with the distribution function . Then we say that G^II poverty dominates G^I with respect to the poverty index P if and only if for all .

Suppose that poverty evaluation is done using the head-count ratio. In terms of the distribution functions, this means that . In other words, the value of the head-count ratio for when the poverty line is z is simply the value of at z. Accordingly, poverty dominance of G^II over G^I with respect to the head-count ratio holds if and only if for all . As Foster and Shorrocks (1988) noted, this is the same as the condition that G^I with first order stochastic dominates G^II. Atkinson (1987) demonstrated that of two income distributions, if one first order stochastic dominates the other, then the former cannot have higher poverty than the latter for all subgroup-decomposable poverty indices that are nonincreasing in incomes of the poor. Graphically, first order stochastic dominance means that the graph of G^II never lies below that of G^I. In other words, we need pointwise dominance of the curve of G^II over that of G^I (see Figure 3.2).

Next, assume that poverty assessment is performed with the poverty gap ratio. Foster and Shorrocks (1988) demonstrated that dominance of G^II over G^I with respect to the poverty gap ratio is equivalent to the requirement that G^I second order stochastic dominates G^II. Graphically, at any income, the area under the curve of G^I never exceeds the corresponding area under the curve of G^II (see Figure 3.3). Equivalently, we say that areawise dominance of the graph of G^II over that of G^I has to hold.¹⁶ Second order stochastic dominance of G^I over G^II is also equivalent to the condition that G^I is not generalized Lorenz inferior to G^II; the generalized Lorenz curve of G^I does not lie below that of G^II.

From Figure 3.3, it follows clearly that up to any level of income z, the area under the curve of G^II is not lower than the corresponding area under the curve of G^I. Hence, second order stochastic dominance of G^I over G^II holds. From the figures, it also follows that first order dominance implies the second order dominance, but the converse is not true. In other words, first order dominance is a sufficient but not a necessary condition for second order dominance. This shows that as we move from the first order condition to the next higher order condition, the ranking power becomes thinner. The reason behind this is that in the case of the latter, we need areawise dominance starting from the initial income level. As a result, one or more intersections between the curves of the distribution functions are permissible; the only restriction we need to follow is that area under G^I does not exceed the area under G^II at any income level. But for the first order dominance to hold, we need pointwise dominance, and as a result, no intersection between the curves is allowed.¹⁷ (Formal treatments of stochastic dominance relations are available in Hadar and Russell, 1969; Whitmore, 1970; Levy, 2006; Shaked and Shanthikumar, 2006 and Chakravarty, 2013.)

3.7.2 Multidimensional Orderings

Duclos et al. (2006a) and Bourguignon and Chakravarty (2009) developed conditions for poverty ranking of alternative multivariate distributions of achievements. Recall that each column of an achievement matrix represents the marginal distribution of a particular dimension (see Section 3.3). Ranking of dimension-wise marginal distributions in two distribution matrices ignores association among dimensions, an important characteristic in multidimensional well-being analysis. Duclos et al. (2006a) compared child poverty in two dimensions between Madagascar and Cameroon and noted statistically significant dominations for each of the two marginal distributions. Nevertheless, no unambiguous domination was observed for the joint distributions. This clearly confirms that simple comparisons of marginal distributions across social matrices do not give us a comprehensive picture of the issue. In consequence, the ranking analysis should explicitly involve the matrices as a whole.

Gravel and Mukhopadhyay (2010) employed generalizations, to three dimensions or more, of the Atkinson and Bourguignon (1982) first and second order dominance criteria to provide a normative assessment of growth in India over the 1987–2002 period. The four dimensions, considered in their analysis, are real consumption (measured at the individual level), literacy rate, under-5 mortality, and violent crime rates (all measured at the district levels). Their findings indicate that India was better off in 2002 than in 1995 or 1987 by the second order dominance rule. However, if violent crime rates are excluded from the list of dimensions, the dominance was found to be steady over the entire period.¹⁸

As we have observed, in a multidimensional setup, identification of the poor is an important ingredient of poverty evaluation. A “poverty frontier,” defined as alternative combinations of all dimensional achievements that produce an overall achievement, which equals an aggregate or comprehensive poverty line, is taken as the identification indicator. The comprehensive poverty line is assumed to render a subsistence standard of living. It is gathered by aggregating the poverty cutoff limits across all dimensions, where the underlying aggregation function is the same as that employed to deduce an overall achievement. Subsequently, a person is identified as poor if his achievements in all the dimensions generate a lower level of aggregated achievement than the comprehensive poverty line. The frontier, thus, separates the poor from the nonpoor.

Given a particular poverty frontier, of two social distributions , X is said to be poverty dominated by Y in the Duclos–Sahn–Younger sense if and only if for all . We denote this relation by .

Duclos et al. (2006a) developed stochastic dominance-based poverty ordering results for achievement matrices with two dimensions of well-being. To analyze these results, let us denote the achievements in the two dimensions by x₁ and x_2, respectively, assuming that the dimensions are substitutes. Let stand for the joint distribution of x₁ and x₂. Formally, . The joint distribution shows simultaneously the proportion of population enjoying achievement levels x₁ and x₂ not exceeding q₁ and q_2, respectively. The marginal distribution of x_i is expressed by , where .

Duclos et al. (2006a) showed that under substitutability assumption between the dimensions, if for all , then holds for all two dimensional subgroup-decomposable poverty indices that are nonincreasing in dimensional achievements, where is the joint cumulative distribution function for . The inequality means that the joint cumulative distribution function of Y does not lie below that of X cumulated up to . The underlying class of indices includes the family of indices suggested by Chakravarty et al. (1998), Tsui (2002), Chakravarty et al. (2008), Alkire and Foster (2011a,b) and some members of the family proposed by Bourguignon and Chakravarty (2003) (see also Bresson and Duclos, 2015).

The poverty frontier is implicitly defined by a locus , where the aggregator is continuous and nondecreasing in its arguments. Bidimensional poverty is now defined by the combinations for which , and the poverty domain is given by .

In Figure 3.4, the poverty frontier is represented by the curve A. Since signifies the proportion of population with achievements simultaneously less than or equal to q₁ in dimension 1 and q₂ in dimension 2, under , the inequality is an intersection type condition. If is not lower than for all rectangles B, C, and so on that fit within , then the dominance follows. Hence, an attractive feature of this dominance condition is that we only need to check intersection-type conditions.

Bourguignon and Chakravarty (2009) developed related first order bidimensional dominance conditions involving the joint distributions and the marginal distributions. In their highly innovative contribution, Aaberge and Brandolini (2015) provided an excellent overview of the Bourguignon–Chakravarty orderings. Here we furnish a sketch of their main findings. They demonstrated that poverty dominance for two bidimensional achievement matrices for all poverty indices that are strongly focussed, symmetric, subgroup decomposable, population replication invariant, nonincreasing in achievements of the poor, and nondecreasing under a correlation-increasing switch (substitutability) holds if and only if there is unambiguous dominance for each marginal distribution and the intersection area. Graphically, the areas of all intersection rectangles for the dominant matrix should not be less than the corresponding areas of the dominated ones, and the marginal distribution function for each dimension in the dominated matrix cannot lie above that of the dominant one. (See Figure 3.4.)

In a subsequent result, Bourguignon and Chakravarty (2009) retained all the poverty postulates stated earlier except nondecreasingness under a correlation-increasing switch, which has been replaced by nonincreasingness under the switch (complementarity). They then established that necessary and sufficient conditions in such a situation are dominance for each marginal distribution and dominance with respect to joint distribution in union area (see Figure 3.1).

3.8 Dimensions of Well-Being with Ordinal Significance and Multidimensional Poverty

It has already been observed in Chapter 1 that values of ordinally measurable dimensions are only indicators of order. Specific numerical values assigned to represent the ordering are insignificant. Recall that each notion of measurability allows some admissible transformation that can be applied to dimensional achievements so that no loss of information occurs. In other words, the value of a quantifier that relies on dimensional achievements remains unaltered when we transform the achievements by applying some admissible transformations.

We write for the set of all dimensions of well-being that are measurable on ordinal scale. The corresponding set of dimensional thresholds is denoted by . Let (say) be the number of elements in . This is the exogenously given number of dimensions of well-being with ordinal significance. For any , let stand for the set of all dimensional matrices of achievements in different dimensions. The set of all such matrices is given by , where . A multidimensional poverty index in an ordinal framework is a real-valued function defined on . More precisely, . In other words, for any , , is the level of poverty that exists in the society given that the dimensions of well-being are ordinally significant.

The information invariance assumption can now be analytically stated as follows. For any , , we say that is derived from using a permissible transformation if for each , there exists an increasing transformation such that and , where is the threshold point of . In other words, is the jth coordinate of the vector . We do not assume that the transformations are the same across dimensions. The ordinality property, which we also refer to as ordinal information invariance assumption, then requires that .

Essentials to the construction of a multidimensional poverty index when dimensions have ordinal significance are deprivation counts of the individuals. The deprivation count of a person relies on a dichotomous or binary variable. A variable may be binary by definition: for instance, whether a person is a citizen of a given country or not. But often it may be useful to derive binary variables from nonbinary variables. For instance, weights of individuals in a region can assume more than two values. However, we can segregate these persons into two subgroups: those who weigh less than 60 kilograms and those weighing at least 60 kilograms. This is a simple creation of a binary variable from a nonbinary variable.

In the current context, we dichotomize each ordinal dimension by comparing a person's achievement in the dimension with its norm, the threshold limit. If somebody's achievement in the dimension is associated with a category that has a lower rank than the category to which the threshold limit corresponds, then he is deprived in the dimension, given that the categories are ranked in increasing order of preferences, that is, from the worst to the best. Otherwise, the person is nondeprived in the dimension. We assign the values 1 and 0, respectively, to indicate deprivation and nondeprivation of a person in a dimension possessing ordinal characteristic.

For any , , , let , (, for short) denote person i's deprivation metric for . More precisely, if person i is deprived in the ordinal dimension j and , otherwise. Then the deprivation profile of the person associated with the ordinal dimensions in the set is , where . Then the deprivation count of person i in ordinal dimensions is . Clearly, is a nonnegative integer bounded between 0 and k. The lower bound is achieved if the person is not deprived in ordinal dimensions, and the upper bound is attained when he has deprivations in all the dimensions. The distribution of deprivation counts in ordinally significant dimensions across individuals in the society is given by . Person i is regarded as union poor in this ordinal framework if . However, the intersection rule treats the person as poor if . (See Cappellari and Jenkins, 2007, for a discussion.)

As an illustrative example, assume that in a four-person society, there are only two dimensions of well-being: self-reported health status and literacy grade. In the dimension self-reported health condition, we assign the numbers 1, 4, 9, 16, 25, and 36 to the six health categories “very poor,” “poor,” “fair,” “good,” “very good,” and “excellent” to indicate that a higher number is assigned to a better category.¹⁹ Suppose that the category “good” represents the threshold category, and it is identified with the number 16. Assume that out of four individuals in the society, the first and third individuals have health conditions characterized by the categories “very poor,” “poor” so that they suffer from health deprivation. The heath conditions of the second and fourth persons belong to the categories “excellent” and “good,” respectively. Consequently, they are not health-deprived. For the second dimension, literacy level, the numbers assigned to the categories “illiteracy,” “knowledge just to read and write,” “elementary school graduation,” “high school graduation,” and “college graduation” are respectively 0, 0.1, 0.2, 0.3, and 0.4. The threshold limit of this dimension is 0.1, that is, the ability just to read and write only is taken as the threshold education category. Assume that the levels of education of the four persons are respectively “illiteracy,” “high school graduation,” “knowledge just to read and write,” and “college graduation.” Then the threshold education level is identified by the number 0.1. If we denote the two dimensions by 1 and 2, respectively, and the distribution matrix by , then and .

In , while person 1 is deprived in both the dimensions, person 3 is deprived only in dimension 1. On the other hand, none of persons 2 and 4 is deprived in any dimension. Consequently, the deprivation profiles of these four persons are given, respectively, by , , , and . This in turn shows that the deprivation counts of the four persons, the numbers of dimensions in which they are deprived are given respectively by 2, 0, 1, and 0.

Now, let us assign the numbers 1, 2, 3, 4, 5, and 6 to the self-reported health categories by taking square roots of the previously assigned numbers. Similarly, we assign the numbers 0.01, 0.04, 0.09, and 0.16 by simply squaring the numbers allotted earlier to the grades of literacy. Formally, we transform the dimensional achievements in these two dimensions by applying the increasing transformations and , where is the increasing transformation applied to the achievements in dimension and t denotes a dimensional achievement. The threshold limit vector gets converted into and the transformed achievement matrix becomes . It is easy to check that the individual deprivation counts remain unchanged under these ordinal transformations of dimensional achievements.

Hence, union poverty head-count ratio turns out to be , whereas according to the intersection rule, this ratio becomes . The ordinal information invariance assumption demands that .

It is then easy to check that the values of the union and the intersection head-count ratios are the same for and , given that the corresponding threshold limit vectors are and , respectively. In other words, the head-count ratio remains unaltered under allowable transformations in the ordinal framework. In consequence, the head-count ratio measure of multidimensional poverty has an advantage over many other indices of multidimensional poverty indices in the sense of its fulfillment of the ordinality property. This holds irrespective of whether the union, intersection, or the intermediate identification rule is employed to determine the set of poor persons. A second index that fulfils this property is the Alkire and Foster (2011a,b) adjusted head-count ratio. This index satisfies dimensional monotonicity property as well, whereas the head-count ratio is a violator of the postulate.

3.9 Orderings Based on Deprivations Counts

Chakravarty and Zoli (2012) used an axiomatic framework to derive some integer majorization results that can be easily transformed into deprivation profile ranking structure. These results apply to both ordinal and nonordinal dimensions of well-being. Consequently, the set of dimensions can include pension income, interest income, public goods, environmental conditions, and so on. Given that there are such dimensions of well-being, denote the functioning score of person by , where is arbitrary. In other words, is the number of dimensions in which person i is not deprived. Evidently, . The lower bound 0 of arises when the person is deprived in all the dimensions. On the other hand, the upper bound is achieved whenever the individual becomes nondeprived in all the dimensions. Let be a profile of functioning scores of the persons in the society. The functioning score distribution is an element of , the n-fold Cartesian product of . Clearly, the deprivation count of person is given by . That is, given any profile of functioning scores, we can write its deprivation counterpart as , where 1ⁿ is the n-coordinated vector of 1s. It then follows that the profile of deprivation counts is also an element of .

The nonincreasingly ordered permutation of τ is denoted by , that is, and . Similarly, we write for the nondecreasingly ordered permutation of τ, that is, . Similarly ordered permutations of are denoted by and , respectively.

Chakravarty and Zoli (2012) considered distributions of functioning scores across persons and established the equivalence between five dominance conditions. Of particular interest is their demonstration that if one vector of functioning scores generalized Lorenz dominates that of another, then the former is preferred to the latter by the generalized Gini welfare function. The converse is true as well.

As a background material of deprivation count ordering, we now analyze the Chakravarty–Zoli dominance results. Of two profiles of functioning scores , we say that τ is obtained from by a favorable composite change if there exists individuals such that and and for all . In , person i's functioning score exceeds that of person j by at least 2, and is transformed into τ by reducing i's functioning score by 1 and increasing that of j by 1. The transformation decreases the variance of the distribution of functioning scores without changing the mean. Consequently, it leads to an improvement in well-being of the society. The restriction ensures that , that is, the relative position of the affected individuals remains the same. (See Chakravarty and D'Ambrosio, 2006 and Chakravarty and Zoli, 2012.)

We say that τ is obtained from by a simple increment if for some such that , and for all . In other words, we deduce τ is from by increasing person j's functioning score by 1 and keeping all other persons' scores unchanged.

Now, consider a functioning evaluation standard , a nonnegative real-valued function defined on . Formally, . For any , indicates the extent of well-being enjoyed by the individuals in the society corresponding to the distribution τ of functioning scores across the individuals. A standard is said to be increasing if whenever τ is obtained from by a simple increment. It satisfies the principle of favorable composite change if its value increases under a favorable composite change.

The following dominance theorem of Chakravarty and Zoli (2012) can now be stated.

In condition (i), decreasingness of the population-size-dependent positive sequence ensures that the value of generalized Gini functioning evaluation is higher for the profile τ compared to that for the profile (see Weymark, 1981 and Yaari, 1987). According to condition (ii), of two distributions τ and of functioning scores, the former integer generalized Lorenz dominates that of the latter, that is, the generalized Lorenz curve of the former lies nowhere below that of the latter and lies above in at least some places. Condition (iii) says that the profile τ is regarded as socially better than the profile by all evaluation standards that are increasing, symmetric, and increasing under favorable composite changes. Conditions (iv) and (v) are self-explanatory. Conditions (ii)–(v) can be regarded as integer counterparts of some well-known majorization results existing in the literature on univariate inequality measurement (see Marshall et al., 2011, p. 14 and p. 87).²⁰

To illustrate condition (v), consider the functioning score profile . The profile is deduced from by increasing person 1's score by 1 accompanied by a rank-preserving favorable composite change involving persons 3 and 4.

Recall that for all , . In consequence, we can easily convert the dominance conditions of vectors of functioning scores into ordering of deprivation count distributions. Note that for any , . Let and , respectively, be the deprivation counterparts of the profiles τ and of functioning scores. If is obtained from by a simple increment, an increment in functioning score of a person, we can equivalently say that is deduced from by an increment in his deprivation count. Similarly, if is deduced from by a favorable composite change affecting functioning scores of two individuals, we can as well say that is derived from ξ by an unfavorable composite change that affects deprivation counts of the concerned persons. In the example, and assume that . We can then say that is transformed into by increasing person 1's deprivation count by 1 and an unfavorable composite change entailing persons 4 and 3. (We increase the deprivation count of person 4, the most deprived person in terms of counting, by 1 and reduce that of person 3, who is less deprived than person 4, by 1.)

Then condition (ii) of Theorem 3.1 can be restated in terms of dominance between the integer inverse generalized Lorenz curves of the distribution of deprivation counts.²¹ Formally, this condition becomes (ii) for all , with > for at least one j. This is the same as the condition that the integer inverse generalized Lorenz curve of dominates that of . Condition (i) of the theorem can be rewritten as . In the aggregation, higher weight is assigned higher deprivation counts. This inequality means that the value of the generalized Gini index of deprivation count distribution is higher than that of the distribution .

In general, a deprivation count assessment metric is a nonnegative real-valued function defined on the set of profiles of deprivation count. Formally, . Such an index provides a social assessment of deprivation counts existing at the individual levels.

Conditions (iii)–(v) of the Theorem 3.1 can be modified analogously. Hence, the following theorem, which may be regarded as the deprivation count twin of Theorem 3.1, can now be stated.

The most attractive feature of Theorem 3.2 is equivalence of integer inverse generalized Lorenz ordering with the remaining four conditions each of which has its own merit. Once the dominance condition between integer inverse generalized Lorenz curves of two social matrices hold, we can be sure about their ranking by the generalized Gini deprivation index, where the positive increasing weights assigned to nondecreasingly ordered deprivation counts are arbitrary. Similar remarks hold for other equivalent conditions (see Chakravarty and Zoli, 2012).

Lasso de La Vega (2010) examined dominance conditions to ensure unanimous poverty ranking in a counting framework. These orderings coincide with the integer first and second order stochastic dominance orderings. Theorem 3.2 of the current section is formally equivalent to the integer second order stochastic dominance condition. However, its equivalence with ranking in terms of the generalized Gini index of the count distribution has not been demonstrated rigorously earlier (see also Silber and Yalonetzky, 2014 and Yalonetzky, 2014). Aaberge and Peluso (2011) compared distributions of deprivation counts by employing summary measures of deprivation within the rank-dependent social evaluation framework (Sen, 1974; Yaari, 1987).

3.10 Multidimensional Material Deprivation

Material deprivation can be regarded as a situation of economic condition, which follows as a consequence imposed incapability of an individual to acquire achievements not below cutoff limits in different dimensions related to material living conditions. More precisely, multidimensional material deprivation deals with deprivations in dimensions of well-being regarding material conditions.²² We refer to such a dimension as a material dimension. Consequently, qualitative dimensions such as communing with friends and political participation are excluded from the domain of the material deprivation measurement. As maintained by EU policy, indices of material deprivation are to be consolidated with income-dependent poverty measures and quantifiers of low employment.

Examples of material deprivations in nonmonetary dimensions of well-being considered in the “The European Union Statistics on Income and Living Conditions (EU-SILC)” project are incapability to face unexpected required expenses, inability to bear expenses to have a meal with meat, chicken, fish (or vegetarian protein equivalent) every second day, lacking capacity to afford to pay for a one-week annual holiday from residence, not having a durable good such as color television, washing machine, telephone, or car, and so on.

Bossert, Chakravarty and D'Ambrosio (2013) developed an axiomatic characterization of a multidimensional index of material deprivation that relies on the union mode to the identification of the poor. The suggested index turns out to be a weighted sum of material deprivation counts of the individuals, where the material deprivation count of a person is defined as the number of relevant dimensions from which the person is deprived.²³

The framework considered deprivation counts because the dimensions were assumed to be discrete in nature. Consequently, it was not possible to determine deprivation in a dimension in terms of shortfall from its threshold limit, which is also nonquantifiable. For instance, possession of a durable good was taken as the threshold limit of the corresponding dimension, and its nonpossession was regarded as deprivation in the dimension. As a result, it was simply necessary to check whether the person has achievement in the dimension or not. The material deprivation count of a person is deduced by calculating the number of material dimensions in which he is deprived. The model rules out the possibility of using continuous dimensions so that an axiom such as ratio-scale invariance becomes inappropriate.

Since the union poverty rule has been employed, a person becomes materially deprived if his deprivation count in appropriate dimensions is at least 1. As a result, for a person to be materially nondeprived, it is necessary that he is not deprived in any accordant dimension, and in such a case, the person's deprivation count is 0. This does not rule the possibility that he is not multidimensionally poor; he may be deprived in one or more nonmaterial living conditions.

Of the three axioms that were employed in the characterization result, the first, zero normalization is a cardinality property. According to this postulate, the index takes on the value zero whenever in a one-person society, the person is not deprived in any dimension. The second axiom is the factor decomposability postulate introduced by Chakravarty et al. (1998). The third axiom is population subgroup decomposability.

To present the index analytically, we assume that there are material dimensions of well-being and denote the material deprivation count of person by . The superscript m signifies that our concern here is with material deprivation. Clearly, is a nonnegative integer bounded between 0 and , the total number of materialistic dimensions. The lower bound is achieved if the person is not materially deprived and the upper bound is attained when he has deprivations in all the dimensions. The material deprivation profile in an n-person society is denoted by . The vector is an element of the set , the n-fold Cartesian product of the set .

The Bossert et al. (2013) material deprivation index is defined as

3.22

where , are arbitrary, is the set of all persons who are materially deprived in , and is the weight assigned to the jth material dimension. It represents the importance assigned to dimension j in the aggregation. The index in (3.22) satisfies a monotonicity property; if a person who was deprived in a material dimension now becomes nondeprived in it because of, improvement in living conditions, say, then value of B_m reduces. The individuals are treated symmetrically in (3.22) in the sense that if two individuals are deprived in the same set of material dimensions, then their material deprivation counts are the same. If for all material dimensions j, then B_m is simply the head-count ratio of material deprivation, the proportion of persons who are materially deprived.

Earlier Chakravarty and D'Ambrosio (2006) characterized the following measure of social exclusion, which can be regarded as a material deprivation metric:

3.23

where , and f is increasing. Increasingness of f guarantees that the value of C_m increases if the material deprivation count of a person goes up.

A functional form of f under which increasingness of material deprivation index holds is , where is a parameter. The resulting functional form of C_m becomes . The parametric restriction ensures that is monotonic. (Jayaraj and Subramanian (2010) applied the index C_m to study multidimensional deprivation in India using data from National Family Health Surveys.)

An example of a nonadditive material deprivation index is the generalized Gini index defined by , where the decreasing, positive sequence depends on the population size affected by material deprivation, and is the nonincreasingly ordered permutation of , that is, for all . Assignment of higher weights to higher material deprivation counts in the generalized Gini index reflects the idea that the deprivation index has an increasing marginal.

To state the increasing marginal material deprivation property formally, consider the change in material deprivation when person i's materialistic deprivation count increases by 1, where is arbitrary. Analytically, given , this change is given by . Since this difference in the deprivation index arises from a marginal increase in person i's deprivation count in materialistic dimensions, we refer to it as marginal material deprivation. Of two arbitrary persons , if the material deprivation of each of them increases by 1, where i has higher number of deprivation count than j in the relevant dimensions, then increasingness of marginal material deprivation means that , which we can rewrite as . Under additivity of F_m, that is, if it is of the type , where , then increasingness of marginal evaluation requires the satisfaction of the inequality , where and are arbitrary.

We can now formally say that the index F_m satisfies the increasing marginal material deprivation property if for all , , where .

We note that the material deprivation count of each of the two persons i and j increases by 1 and j has lower count than i. Then increasingness of material deprivation demands that the increase in society material deprivation resulting from worsening in person i's deprivation count in material dimensions is higher than the corresponding increase that arises as a result of similar aggravation in person j's position. In other words, higher weight is assigned to increase in social assessment of material deprivations as the number of materially deprived dimensions of a person increases.²⁴ Recall that this has been done in the generalized Gini material deprivation index . For the additive index , the parametric restriction is necessary and sufficient for fulfillment of this postulate.²⁵

3.11 Concluding Remarks

Several methodologies have been proposed in the literature for the evaluation of poverty from multidimensional standpoints. Each of them has its own objectives and represents a particular mode of assessing poverty using specific means. None of them is designed to overthrow one or more of the others.

In the dashboard mechanism, valuation of poverty is carried out in a primitive framework; the dashboard comprises dimension-by-dimension poverty figures. In the axiomatic approach to the evaluation of multidimensional poverty, a set of desirable postulates for alternative poverty indices is proposed at the outset. Each of these indices or quantifiers is a summary measure of the levels of deprivations arising from insufficiency of achievements in different dimensions of well-being. Broadly speaking, the job then becomes scrutiny of a measure with respect to gratification of one or more of these desiderata. A related issue is to order two or more social matrices, which describe different individuals' positions in terms of their achievements in given dimensions of well-being, indicating society's preference for lower poverty. Several notions of ordering depend on a cumulative joint distribution function, a statistical concept, of a social matrix. Further, it also often becomes necessary to enquire whether poverty ranking by cumulative distribution functions agrees with that produced by axiomatic indices. Consequently, sometimes the ordering process may be interpreted as a mixture of axiomatic and statistical designs, where the latter incorporates many tools that can be invoked for the purpose of poverty evaluation. A second example of a statistical technique employed within the axiomatic framework is the correlation between dimensional achievements that judges interdimensional association.

Additional examples of statistical techniques include principle component analysis; factor analysis (see Chapter1); multiple correspondence analysis used to reveal veiled structures in a given data set; cluster analysis, concerned with grouping of individuals bearing similarity with respect to multidimensional deprivations and structural equations models that are capable of intimating relationships between latent and observed variables.²⁶

Another well-accepted process that exists in the literature is the fuzzy set approach that arises when information concerning the identification of the poor is incomplete (see Chapter 4). Multidimensional poverty analysis sometimes also makes use of information-theory-based technique that relies on information function and related entropies. (See Maasoumi and Lugo, 2008; Roelen et al., 2009; Alkire et al., 2015; Chakravarty and Lugo, 2016 and Section 3.5 of this chapter.) An alternative mention worthy technique employed in this context is the distance function that looks at the distance between achievements of an individual and a yardstick vector (see, e.g., Anderson et al., 2008 and Ramos, 2008). In the Venn diagram method, joint distributions of deprivations of the individuals are presented diagrammatically when the number of dimensions does not exceed 3 (see Alkire et al., 2015).²⁷

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