Journal of Economic Literature 2018, 56(2), 657–672
https://doi.org/10.1257/jel.20161454
On Measuring Multidimensional
Deprivation†
Prasanta K. Pattanaik and Yongsheng Xu*
This essay presents a critical review of the recent book by Alkire et al. entitled
Multidimensional Poverty Measurement and Analysis, and, in the course of doing so,
it also discusses some general issues that come up in this context. We outline the basic
structure of the problem of measuring multidimensional deprivation and critically
evaluate the methodology adopted by Alkire et al. (2015). In particular, we discuss
some problems associated with the methods used by them to identify the deprived and
to aggregate individual deprivations so as to derive an index of social deprivation. We
examine the interpretation in terms of unfreedoms of individuals, which Alkire et al.
put on one of their measures of social deprivation. We also suggest a variant of their
methodology for measuring multidimensional deprivation.( JEL C38, E02, I32, Z13)
1. Introduction
Measurement and Analysis by Alkire et al.
(2015) fills this gap admirably. We believe
that the publication of this very lucidly written volume is an important event in the literature on multidimensional deprivation.
The book can be roughly divided into
three parts. The first part, which consists of
chapters 1, 2, and 3, presents a background
of the problem of measurement of multidimensional deprivation and an overview of the
history of and motivation for research in this
area. It also provides a systematic discussion
of the various axioms used in the analytical
literature on multidimensional deprivation.
The second part of the book, consisting of
chapters 4, 5, and 6, focuses on the “counting approaches,” especially the methodology developed by Alkire and Foster (2011),
which has found fairly wide a­cceptance in
recent years. In many ways, this part, which
deals with the Alkire–Foster methodology
T
he analytical, as well as empirical, literature on the measurement of multidimensional deprivation1 has expanded
very rapidly over the last two decades or
so. What, however, was lacking was a booklength treatment of the subject, providing a
systematic account of the literature, a careful
examination of the formal assumptions figuring in various contributions and their intuitive content, and an assessment of the major
contributions. Multidimensional Poverty
* Pattanaik: Department of Economics, University of
California. Xu: Department of Economics, Andrew Young
School of Policy Studies, Georgia State University.
†
Go to https://doi.org/10.1257/jel.20161454 to visit the
article page and view author disclosure statement(s).
1 We use the terms “poverty” and “poor” in the context
of income poverty, while we use the terms “deprivation”
and “deprived” when talking about deficiencies in terms of
“real” attributes, such as nutrition, health, etc.
657
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Journal of Economic Literature, Vol. LVI (June 2018)
(the AF methodology) and several important conceptual issues relating to the measurements of multidimensional deprivation,
constitutes the heart of the volume. This
review essay will focus on this part. The third
part consists of the last four chapters. These
chapters consider how the analytical framework introduced earlier can be applied in
various settings and discuss many statistical
issues that come up in the course of such
application.
This review essay is organized as follows.
In section 2, we lay down some essential
notation and outline the basic structure of
the problem of measuring multidimensional
deprivation; in this section, we also briefly
present Alkire and Foster’s approach to the
problem, which occupies a central place
in Alkire et al. (2015). In section 3, we discuss certain difficulties with the approach of
Alkire and Foster and, in section 4, we consider a variant of the AF methodology that
avoids many of the difficulties discussed in
section 3. Section 5 discusses the interpretation, which Alkire et al. seek to give to one of
their measures of overall social deprivation,
in terms of individual unfreedoms. Section 6
deals with three general issues, namely, the
exclusion of the non-deprived individuals’
dimensional deprivations from the measure
of social deprivation, the treatment of an
individual’s achievements over and above
the deprivation benchmarks in some dimensions, and interpersonal comparisons of
overall deprivations of individuals. We conclude in section 7.
2.
Notation, Some Basic Concepts, and an
Outline of the AF Methodology
There are n
​ ≥ 2​individuals in the society,
to be indexed by ​1, … , n​, and ​m ≥ 2​ dimensions or attributes, to be indexed by ​1, … , m​,
in terms of which of an individual’s achievements are to be measured. Let ​N​and ​M​
denote the sets {1, 2, … , n} and {1, 2, … , m},
respectively. In this essay, we shall use the
terms “dimensions” and “attributes” interchangeably. Let the set of ​
m​ dimensions
​  ​​}​. For each
be denoted by F
​ = { ​f1​  ​​, … , ​fm
​​ j​​, of valdimension ​​fj​  ​​  ∈ F​, there is a set, X​​
ues or levels that dimension ​​f​ j​​​ can take. Some
dimensions, such as health status, may be only
ordinally measurable so that we can compare
different levels of health, such as “excellent
health,” “good health,” “poor health,” but
no meaning can be attached to statements
like “the difference between being in good
health and being in poor health is greater
than the difference between being in excellent health and being in good health.” When
an attribute ​​f​ j​​​ is ordinally measurable, ​​X​​  j​​ can
be a discrete set containing several real numbers. Sometimes, some dimensions such as
schooling may be cardinally measurable so
that comparing the difference between, say,
nine-year schooling and twelve-year schooling with the difference between ten-year
schooling and fourteen-year schooling makes
sense. When a dimension f​ ​​ j​​​ is cardinally measurable, we assume that X​​
​​ j​​is a nondegenerate real interval. Conventionally, for any
dimension ​​fj​  ​​  ∈ F​, a higher real number in​​
X​​  j​​indicates a higher level of achievement in
dimension ​​fj​  ​​​.
For each individual i​
∈ N​, let
​​x​ i​​  = (​x​ i1​​, … , ​x​ im​​)​ be ​i​’s achievement vector
in which x​ ​​ ij​​​ is ​i​’s achievement in dimension
​​fj​  ​​​. The achievements of all individuals in the
society can thus be summarized by an n
​ × m​
​ ​​​, where the i​ ​th row represents
matrix, ​​(x​ ​ ij​​)n×m
individual ​i​’s achievement vector ​​x​ i​​​ and the
​j​th column represents the achievements in
dimension ​​fj​  ​​​by all the individuals.
For each ​j ∈ M​, let ​​​ x ​​  j​​​be the cutoff level
¯
for dimension f​ ​​j​​​such that, for any individual ​i ∈ N​, ​i​is considered to be deprived
in dimension ​​
f​ j​​​ if and only if​ i​’s achievement, ​​x​ ij​​,​ in the dimension ​​fj​  ​​​, falls below ​​​ x ​​  j​​​.
¯
More formally, we define the “deprivation
Pattanaik and Xu: On Measuring Multidimensional Deprivation
s­ tatus,” to be denoted by d​
​​ ij0​  ​​, of an individual
​i ∈ N​along a dimension ​​f​ j​​  ∈ F​ below:
1  if ​x​ ij​​ < ​​ x ​​  j​​
¯ ​​​.​
​​d​ ij0​  ​  = ​ ​
{0  if ​x​ ij​​  ≥ ​​ x ​​  j​​
¯
When a dimension ​​f​ j​​​ is cardinally measurable, for each individual i​ ∈ N​, the normalized deprivation gap of i​​in dimension f​ ​​j​​​, to
be denoted by ​​d​ ij​​​, is defined as follows:
​​ x ​​  j​​ − ​x​ ij​​
______
if ​x​ ij​​  < ​​ x ​​  j​​
​  ¯ ​​ x ​​  j ​
¯ ​​​.​
​  ¯ ​​ ​
​
​di​  j​​ = ​ ​
{0
if ​x​ ij​​ ≥ ​​ x ​​  j​​
¯
When all attributes are cardinally measurable, the state of the dimensional deprivations of all individuals in the society can
be summarized by a deprivation matrix,​
​ ​​​, where the ​ij​th entry, ​​c​ ij​​​, of ​C​ is​
C = ​(​c​ ij​​)n×m
i​’s normalized deprivation gap in dimension​​
f​ j​​​. In this case, the class, ​​, of all permissible
matrices is then the class of all ​n × m​ matrices ​C​such that each entry of C
​ ​belongs to
the interval [0, 1]. It is not obvious how one
should construct a deprivation matrix for the
society when attributes are only ordinally
measurable. When all attributes are only
ordinarily measurable, one possibility is to
assume that for each individual i​​and each
attribute ​​fj​  ​​​, the deprivation, ​​c​ ij​​,​ of ​i​in dimension ​j​can take exactly one of only two values,
namely, 0 (​i ​is not deprived in dimension ​​f​ j​​​)
and 1 ​(i​is deprived in dimension f​ ​​j​​​). In this
case, the deprivation of every individual in
every dimension coincides with her deprivation status in that dimension, and the class, ​​,
of all permissible matrices is then the class of
all n
​ × m​matrices ​C​such that each entry of​
C​is either 0 or 1. As we shall see, Alkire et
al. take this route when all attributes are only
ordinally measurable.
The problem of measuring social deprivation is viewed as finding a function
​P :  → [0, 1]​with the ­following interpretation: for every deprivation matrix ​C ∈ ​,​
659
P(C)​denotes the level of social deprivation
corresponding to ​C​with a higher level of ​P​
indicating greater social deprivation. Thus,
the measure of social deprivation ​P​ aggregates each deprivation matrix D
​ ∈ ​to a
real number indicating the level of deprivation in the society.
In the literature on measuring multidimensional deprivation, two types of procedures have emerged as possible ways of
aggregating deprivation matrices: row-first
procedures and column-first procedures.
Chapter 3 of Alkire et al. (2015) provides an
overview and critical evaluation of the major
existing methods in the literature on measuring multidimensional deprivation based on
the two procedures. A row-first procedure
first aggregates each individual’s deprivation
vector to an overall deprivation index for this
individual and then aggregates the individuals’ overall deprivations to a social deprivation. A column-first procedure, on the other
hand, first aggregates all the individuals’
deprivations in each dimension to a degree
of social deprivation in that dimension and
then aggregates these degrees of dimensional social deprivations to an index of
overall social deprivation. Column-first procedures are very useful and are the only viable methods to compute social deprivation if
there are only aggregate data (from different
sources) available. A well-known application
of such a procedure is the United Nations
Development
Programme’s
(UNDP’s)
Human Poverty Index. Row-first procedures
have a solid foundation in welfare economics
and demand micro-level data in which information on each dimension is available for
each individual (so that the data and information about joint distribution are known).
It may be remarked that, in general, the two
procedures can yield different rankings of
social deprivation (and therefore different
index numbers) for the society; see Dutta,
Pattanaik, and Xu (2003) for further discussion on this aspect of the two procedures.
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Journal of Economic Literature, Vol. LVI (June 2018)
Alkire et al. (2015) focus on row-first procedures in aggregating deprivation matrices.
We now outline the approach to multidimensional deprivation adopted in Alkire
et al. (2015). Let ​w = (​w​ 1​​, … , ​w​ m​​)​ be the
vector of the positive weights attached to
​​ ​w​
​ j​​  = 1​.
the different attributes, with ​​∑j∈M
For each individual i​​and given the vector,
​ )​
,
of
her
dimen​​di​  0​  ​  = (​di​  01 ​,​  … , ​di​  0m ​
sional deprivation statuses, her depriva ​
​​​w​ j​​ ​di​  0j​  ​​. Let ​k​
tion score is defined as ∑
​​ j∈M
(1 ≥ k > 0)​​be the cutoff value for the
deprivation score such that a person is
deprived overall if and only if her deprivation score is at least as great as the cutoff
​
​​ ​w​ j​​ ​dij​  0​  ​  ≥ k​. The set, ​​N​​  ∗​​,
value ​k​, i.e., ​​∑j∈M
of deprived individuals, is defined below:
{
}
​​N​​  ∗​  = ​ i ∈ N : ​  ∑ ​​ ​  ​w​ j​​​dij​  0​  ​ ≥ k ​​.
j∈M
It may be noted that this approach to the
identification problem in a multidimensional
deprivation framework is an extension of a
historically important measure for multidimensional deprivation—the “counting”
method—which has been developed and
used in empirical research in two regions,
Latin America and Europe, since the 1970s.
However, the motivations and applications
are different. In Latin America, it was developed in the context of basic-needs approach,
and in Europe, it was developed in the context of social exclusion (see chapter 4 of Alkire
et al. 2015 for more details). It can be easily
checked that
1
(1) if
​​w​ 1​​​  = ​​w​ 2​​​  = ⋯ = ​​w​ m​​  = ​ __
m ​,
then for every k (1 ≥ k > 0)​, there
exists a unique integer t​(k) ∈ M​,
such that i​ ∈ ​N​​  ∗​​if and only if i​​ is
deprived in at least t​ (k)​ dimensions.
Once the set of individuals who are
deprived is determined, the AF methodology aggregates the overall deprivations of
these individuals to reach an index of social
­deprivation. For any α
​ ≥ 0​and any deprivation matrix ​D ∈ ​, define the social deprivation, ​​Pα​  ​​ (D)​, associated with ​D​, as follows:
(2) if ​α > 0​, then
1  ​​​ ​∑
​​ ​  ​w​ j​​ ​(​dij​  ​​)​​  α​​
​​Pα​  ​​ (D) = ​ __
n ​ ​  ∑
∗
i∈​N​​  j∈M
​
and
(3) if ​α = 0​, then ​​P0​  ​​ (D)​, the “adjusted
head count ratio,” is given as follows:
1  ​​ ​ ​∑
​​P​ 0​​ (D) = ​ __
​​​w​
​  j​​ ​dij​  0​  ​​.
n ​ ​  ∑
∗
i∈​N​​  ​j∈M
Note that, given ​
k​and the weights,​
​ ​ 1​​, ​w​ 2​​, … , ​w​ m​​,​if every entry of ​D​is either 0
w
or 1, then, for every ​α > 0​, ​​Pα​  ​​ (D)​ = ​​P0​  ​​ (D)​.
3. An Evaluation of the AF Methodology
In evaluating the AF methodology for
measuring multidimensional deprivation, it
is important to distinguish two alternative
frameworks based on different assumptions
about the measurability of individual deprivations in terms of the different attributes.
Consider the following two alternative
assumptions.
ASSUMPTION 1 (binarily ordinal measurement of normalized individual deprivation in
terms of every attribute): The deprivation of
every individual in terms of every attribute
is either 0 (the individual is not deprived in
terms of the attribute) or 1 (the individual is
deprived in terms of the attribute) so that the
class ​​of permissible deprivation matrices
is the class, ​​​​  1​​, of all ​n × m​ matrices​ C​ such
that each element of C
​ ​is either 0 or 1.
ASSUMPTION 2 (cardinal measurement of
normalized individual deprivation in terms
of each attribute): The normalized deprivation gap of an individual in terms of every
Pattanaik and Xu: On Measuring Multidimensional Deprivation
a­ ttribute is cardinally measurable along the
[0, 1] interval, so that the class 
​ ​of permissible deprivation matrices is the class, ​​​​  2​​, of all​
n × m​matrices​C​such that each element of C
​ ​
is a number in the interval [0, 1].
Clearly, these two assumptions refer to
polar cases. One can think of cases where an
individual’s deprivation in a particular dimension is ordinally measurable but can have more
than two levels (e.g., no deprivation, moderate
deprivation, serious deprivation, and extreme
deprivation) rather than just two levels,
namely, 0 and 1 (see Pattanaik and Xu 2018
for an exploration of the more general case).
Also, an individual’s deprivations in terms of
different attributes may have different types
of measurability. Much of the literature, however, deals with these two extreme cases and
we still do not have any satisfactory way of
handling more realistic but complex analytical structures where different attributes may
be measurable in different ways. The choice
between a framework based on assumption 1
and a framework based on assumption 2 will,
of course, depend on the type of information
about dimensional deprivations, which may be
available.
3.1. The AF Method for Identifying the
Deprived
An important contribution of the AF
“counting method” for identifying the
deprived is to draw attention to the limitations of the criteria for identifying the
deprived in what have been called the “union
approach” and the “intersection approach” to
the problem of identifying the deprived (see
Atkinson 2003). Under the union approach,
a necessary and sufficient condition for a
person to be deprived overall is that she be
deprived in terms of at least one attribute.
While the condition is plausible as a necessary condition for a person to be deprived
overall, it lacks plausibility as a sufficient
condition: in a framework, where we have
661
ten different attributes that are all c­ ardinally
measurable, we may not be willing to say
that a person is deprived overall if she is not
deprived in terms of any attribute other than
calorie consumption and her normalized calorie deprivation is only 0.005. Even when we
are working with assumption 1, we may not
be prepared to say that a person is deprived
overall if she is deprived in terms of only one
of fifteen different attributes, all of which we
consider equally important. Under the intersection approach, a person is deprived overall if and only if she is deprived in terms of
all attributes. While the condition may have
appeal as a sufficient condition for a person
to be deprived overall, it lacks plausibility as
a necessary condition to be deprived. Given
assumption 2, if there are ten attributes
and a person is severely deprived in terms
of nine out of ten attributes, all of which
are considered to be equally important, but
is not deprived in terms of the remaining
attribute, one can hardly regard the person
as ­non-deprived overall. A similar objection
can be raised against the intersection criterion for identifying the deprived, when one
is working with assumption 1.
The AF method for identifying the deprived
provides a generalization of both the union
and intersection approaches by permitting
the possibility that a person may be deprived
overall even when she is not deprived in all
dimensions, as well as the possibility that a
person may be non-deprived overall even
when she is deprived in some dimensions.
But the exact way in which this generalization
is achieved causes some intuitive problems in
the case where the (normalized) deprivations
of an individual in terms of the different attributes are cardinally measurable.
The root of the problem is this: when the
dimensional deprivations happen to be cardinally measurable and the relevant information regarding such cardinal measurement is
available to us, our intuition about whether
someone is deprived overall often depends
662
Journal of Economic Literature, Vol. LVI (June 2018)
on the depths of the ­individual’s d
­ eprivation
along different dimensions. The AF method
for identifying the deprived, however, completely ignores such information about
the depths of dimensional deprivations.
Consider an example, where ​​​​  2​​ is the class
of all permissible matrices and we have
five attributes, all of which are cardinally
measurable and considered to be equally
important, so that the weights attached to
the attributes in the AF counting criterion
for identifying the deprived are given by
w​  5​​  = 1/5.​ Suppose
​​w​ 1​​  = ​w​ 2​​  = ⋯ = ​
the cutoff value k​​is 3/5 so that a person
is deprived if and only if she is deprived in
at least three out of five dimensions. Now
consider two individuals, 1 and 2, whose
normalized deprivation vectors are, respectively, ​​d1​  ​​  = (ϵ, ϵ, ϵ, 0, 0)​ with ​ϵ > 0​ and
​​d2​  ​​  = (0, 0, 0, 1, 1)​. Then under the AF counting method, given our simplifying assumptions, individual 1 will be considered deprived
overall while 2 will not be considered
deprived. This will be so no matter how small​
ϵ​is so long as ​ϵ​ is positive. We find this counterintuitive. It seems to us that, if assumption 2 is satisfied and we have information
about the depths of dimensional deprivations, then in assessing whether a person is
deprived, not only do we take into account
the list of the dimensions in which the person
is deprived, but we also consider the extent to
which the person is deprived in terms of each
attribute in that list. The AF criterion for
identifying the deprived when assumption 2
holds is not cognizant of the information that
may be available about the depths of an individual’s dimensional deprivations.
We now discuss a second feature of the
AF method of identifying the deprived in
the case where assumption 2 holds. We shall
illustrate our point with reference to the
class of multidimensional social deprivation
​ > 0,​
measures ​​Pα​  ​​​, such that, for some α
​  n1 ​​​​  ∑ ​​​​  ∑ ​​​  ​w​ij​​(​dij​ ​)​​ α​
​​Pα​ ​​(D) =​ ​__
i∈​N​​ ∗​j∈M
for all D
​ ∈ ​​​ 2​.
Recall that, in the above expression, ​​N​​  ∗​​ is
the set of all deprived individuals defined
in section 2 and ​​w​ 1​​, … , ​w​ m​​​ are the positive
weights attached to the different attributes,
​
​​​w​ j​​  = 1.​Given ​i ∈ ​N​​  ∗​,​ it seems
with ​​∑j∈M
plausible to interpret ∑
​​ j∈M
​
​​ ​w​ j​​​(​di​  j​​)​​  α​​as a measure of individual ​i​’s “real deprivation.” Now
consider an individual ​i′ ∈ (N − ​N​​  ∗​)​. Note
that since social deprivation is being visualized as an aggregate measure of the overall
deprivations of all deprived individuals in the
society ,2 and since the deprivation score of i​′​
is not large enough for ​i′​to be considered a
deprived individual, the overall deprivation
of i​′​is excluded from the measure of social
deprivation given above. But that does not
mean that ​i′​does not have any dimensional
deprivations; nor does it mean that, if one
asks the question, “How should one aggregate the dimensional deprivations of ​i′ ,”​ then
one will be asking a meaningless question.
Further, if the overall deprivation of every​
, is given by ​​
∑j∈M
​​ ​w​
​ j​​​(​di​  j​​)​​  α​​, then it
i ∈ ​N​​  ∗​​
is not clear why the overall deprivation of
any individual ​i′​who is deprived in terms of
some attributes, but who is not deprived in
terms of a sufficient number of attributes to
be considered to be deprived overall, should
not be measured in an analogous fashion. In
saying this, we are not suggesting that, if the
overall deprivation of every individual ​i​ in
​​​w​
​ j​​​(​dij​  ​​)​​  α​​, then it logically
​​N​​  ∗​​is given by ​​∑j∈M
follows that, for every ​i′​in ​N − ​N​​  ∗​, such that
i′​is deprived in terms of at least one attribute,
the overall deprivation of i​′​must be given by​​
∑j∈M
​
​​​w​ j​​​(​di′j​  ​​)​​  α​​. All that we are ­suggesting is
that there is no intuitive reason why the overall deprivation of any such ​i′​must be computed differently from the way in which the
overall deprivation of every i​​ in ​​N​​  ∗​​ is computed, though the dimensional deprivations
2 See
section 6.1.
Pattanaik and Xu: On Measuring Multidimensional Deprivation
of ​i′​may not be sufficiently widespread for
her to be regarded as a deprived ­individual.
If this intuitive position is accepted, then
the AF methodology runs into a problem.
Consider again our earlier example, where
we have five attributes with equal weights;
the cutoff value ​k​is 3/5 (so that, given the
assumption of equal weights for all attributes,
an individual is deprived if and only if she is
deprived in at least three dimensions); and
there are two individuals, 1 and 2, such that
1’s deprivation vector is ​​d​ 1​​  = (ϵ, ϵ, ϵ, 0, 0)​,
with ​
ϵ > 0​
, and 2’s deprivation vector is
​​d2​  ​​  = (0, 0, 0, 1, 1)​. Then calculating the two
individual’s overall deprivations as suggested
above, it can be easily seen that the overall deprivation of 1, who is deprived under
the AF rule for identifying the deprived, is
given by ​(3/5) ​(ϵ)​​  α​​, while the overall deprivation of 2, who is not deprived under the
AF criterion, is given by 2​ /5.​If ϵ​ ​ (​ϵ > 0) ​is
sufficiently small, then the overall deprivation of 2, who is not deprived overall under
the AF criterion, is higher than the overall deprivation of 1, who is deprived overall
under that criterion. This would seem to be
intuitively odd. At the risk of emphasizing
the obvious, we would like to clarify again
that, when we use the term “odd” here, we
are not referring to any logical contradiction.
Clearly, there is nothing logically contradictory if: (i) one agrees to use a criterion, γ
​ ​, to
identify the deprived individuals in a society;
(ii) one agrees to the use of a principle, ​δ​, in
assessing the overall deprivation of only those
individuals who have already been identified as being deprived under criterion γ
​ ​; and
(iii) it turns out that, had one used the same
principle ​δ​ to assess the overall deprivation of
some individual ​i′​, who is not deprived under
criterion ​γ​, then the deprivation of ​i′​ under
principle ​δ​ would have been greater than the
deprivation, under principle ​δ​, of some individual ​i​whom criterion ​γ​ has declared to be
deprived. We do, however, believe that the
conjunction of (i), (ii), and (iii) indicates an
663
intuitive tension between criterion ​γ​ and
principle ​δ​.
What happens when assumption 1 holds
and there are only two levels of an individual’s deprivation in terms of any attribute,
namely, 1 (the individual is deprived in terms
of the attribute under consideration) and 0
(the individual is not deprived in terms of the
attribute)? Given our simplifying assumption that all attributes are given the same
weight, in this case of 0–1 measurement,
the AF procedure would specify an integer
​t ∈ {1, 2, … , m}​, such that an individual
is regarded as being deprived overall if and
only if she is deprived in at least t​ ​ dimensions
and, for every D
​ ∈ ​​​  1​​, the society’s overall
1  ​
1
__
deprivation is given by ​​
​
​​ ​__
N​​  ∗​​​ ​∑ j∈M
n ​ ​∑ i∈​
m ​ ​ d​ ij​​​,
∗
where ​​N​​  ​​is the set of all individuals who are
judged to be deprived overall.
It is clear that, in this case of 0–1 measurement of dimensional deprivations, there
cannot be any counterpart of the first difficulty that we discussed when the attributes
are cardinally measurable. Similarly, in the
case of 0–1 measurement of an individual’s
deprivation in terms of every attribute, there
cannot be any counterpart of the second
intuitive difficulty discussed above.
3.2. Aggregation of Individual Deprivations
in the AF Methodology
Under the AF methodology, the measure of social deprivation takes the form of​​
1  ​
​​ j∈M
​
​​​w​ j​​​(​dij​  ​​)​​  α​​for some ​α ≥ 0​.
P​ α​​  = ​__
N​​  ∗​​∑
n ​ ​∑ i∈​
It may be noted that, for each individual i​​
​​​w​
​ j​​​(​di​  j​​)​​  α​​
who is identified as deprived, ​​∑j∈M
can be regarded as ​i​’s “real overall deprivation.” Consequently, the AF measure of
social deprivation can be regarded as summing up the “real overall deprivations” of
deprived individuals and then dividing this
sum by the population size n
​ ​. Function ​​Pα​  ​​​
has several attractive properties as discussed
in chapter 5 of the book. For example, P​
​​ α​​​
is separable across dimensions and across
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Journal of Economic Literature, Vol. LVI (June 2018)
deprived individuals, making it easy and simple to be implemented empirically. On the
other hand, ­separability across dimensions
comes at a “cost,” as it has some undesirable
implications illustrated below. For brevity
and simplicity, we focus on 0–1 deprivation
levels, though our discussion can easily be
formulated for cardinally measurable deprivation levels in the different dimensions.
Also, we assume that: (i) there are exactly
five dimensions; (ii) all dimensions have the
same weight, namely, 1/5; and (iii) the cutoff value ​k​is 2/5.
We believe that most people will agree
with Stiglitz, Sen, and Fitoussi’s Commission
on the Measurement of Economic
Performance and Social Progress (2009)
report when it says, “the consequences for
quality of life having multiple disadvantages far exceed the sum of their individual
effects” (emphasis added). In fact, Alkire
et al. also seem to agree with this statement,
which they cite (see Alkire et al. 2015, p. 21).
Consider, however, the implications of the
measure of a deprived individual’s overall
“real deprivation” embedded in an AF measure of social deprivation. Given a deprivation vector d​
​​ i​​​ of individual ​i,​ let the “real
overall deprivation” of i​​be denoted by ​h(​di​  ​​)​.
Consider the AF measure of multidimensional deprivation ​​P​ α​​​ where ​α > 0​. Implicit
in this measure is the measure of a deprived
individual’s real overall deprivation given
∑j∈M
​
​​​w​ j​​ ​(​dij​  ​​)​​  α​​, where ​​w​ j​​​ is the
by h
​(​di​  ​​) ≡ ​
weight attached to attribute ​​f​ j​​​. In the case,
where we have five attributes with equal
weights and k​ = 2/5​
, take five vectors,
(1, 1, 0, 0, 0),
(1, 1, 1, 0, 0),
(1, 1, 0, 1, 0),
(1, 1, 0, 0, 1), and (1, 1, 1, 1, 1). Given the
separability of ​h​across dimensional deprivations, it is clear that the addition to ​
i​’s
real overall deprivation when her deprivation vector changes from (1, 1, 0, 0, 0) to
(1, 1, 1, 1, 1) is exactly equal to the sum of:
(i) the addition to ​i’​s real overall deprivation
when her deprivation vector changes from
(1, 1, 0, 0, 0) to (1, 1, 1, 0, 0), (ii) the addition
to i​ ​’s real overall deprivation when her deprivation vector changes from (1, 1, 0, 0, 0) to
(1, 1, 0, 1, 0), and (iii) the addition to ​
i​’s
real overall deprivation when her deprivation vector changes from (1, 1, 0, 0, 0) to
(1, 1, 0, 0, 1). This goes counter to the intuition underlying Stiglitz, Sen, and Fitoussi’s
(2009) statement.
4.
A Variant of the AF Methodology
It seems to us that a variant of the AF
methodology will avoid many of the difficulties that we discussed in section 3. Assume
that the class, , of permissible deprivation matrices is either ​​​​  1​​ or ​​​​  2​​ and, for
every ​k (1 ≥ k > 0)​and for every α
​ > 1​,
define a class of social deprivation measures​​
Q​ α​​​ as follows:
(4) for
all
​
D ∈ ,​
​​Qα​  ​​(D)​
α
∗∗​ ∑  ​​​w​
​​​(​
​
​​​ d​
)
​​
​​
,
​
​
where​​
= ​​∑i∈​
N​​  ​
j∈M j ij
​​ ​w​
​ j​​
w​ 1​​  > 0, … , ​w​ m​​  > 0​ with ​​∑j∈M
= 1​are the weights attached
to the different attributes and​​
​​​w​
​ j​​ ​ d​ ij​​  ≥ k}​ is
N​​  ∗∗​  ≡ { i ∈ N : ​∑j∈M
the set of deprived individuals.
It may be checked that, given ​α > 1​, this
simple variant of the AF measures avoids
the problems of identification, as well as the
problem of aggregation that we discussed
in sections 3.1 and 3.2. In fact, Alkire et al.
(2015) very briefly mention this measure
(when​ α = 2​ and ​​​​  1​​is the set of permissible deprivation matrices) in the penultimate paragraph on page 191 of their book.3
the case of ­
cardinally
When ​  = ​​​  2​(​
3 See also Dhongde et al. (2016) and Pattanaik and Xu
(2018) for an axiomatic analysis and discussion of a broader
class of measures that subsume the aggregation rule (4) as
a specific example.
Pattanaik and Xu: On Measuring Multidimensional Deprivation
measurable normalized dimensional deprivations), the rule for identification of the
deprived embedded in (4) is a little more
complex than its AF counterpart, since it
takes into account the depths of dimensional
­deprivations. However, if we are to avoid the
­intuitive problems d
­ iscussed in section 3.1,
then there does not seem to be any way of
avoiding the necessity of taking into account
the depths of dimensional deprivations when
information about the depths of dimensional
deprivations are available.4 The aggregation
rule implicit in (4) is also fairly straightfor ​​​w​
​ j​​ ​dij​  ​​)​​  α​​
ward. But since the expression (​
​​ ∑j∈M
is not separable across the attributes, we cannot any longer calculate how much the individuals’ deprivations in a specific dimension
contribute to the total overall deprivation of
the society, as we can do under the AF methodology. But separability of an individual’s
overall deprivation across the different attributes is precisely what contradicts Stiglitz,
Sen, and Fitoussi’s (2009) intuition, which
we have discussed in section 3.2 and which
we find rather compelling.
5. The Adjusted Head-Count Ratio and
Individual Freedom
In a framework where assumption 1 holds
so that every individual’s deprivation in each
dimension takes exactly two values, 0 (the
individual is not deprived in terms of the
attribute) and 1 (the individual is deprived
in terms of the attribute), Alkire et al. (2015)
have sought to link their adjusted headcount ratio ​​(P​ 0​​​) to the literature on the measurement of freedom in welfare economics.
There are two distinct levels at which these
links are established. First, at a purely formal level, they demonstrate that, given 0–1
measurement of dimensional deprivations
4 It may be noted that when ​  = ​
​​  1​​, for each given
vector ​
w = (​w​ 1​​, …, ​w​ m​​
)​and cutoff value ​
k,​ we have
​​N​​  ∗​  = ​N​​  ∗∗​​.
665
of individuals, an extension of the axiomatic
analysis in Pattanaik and Xu’s (1990) paper
on the measurement of freedom yields the
results that: there exists t​ ∈ {1, 2, … , m}​,
such that a person is deprived if and only if
she is deprived in at least t​​dimensions (this
is exactly the criterion the AF methodology
uses to identify the deprived when assumption 1 is satisfied and the same weight is
attached to each attribute); and for any two
deprived persons, ​i​and ​i′​, with deprivation
vectors ​​di​  ​​​ and ​​di′​  ​​​, respectively, ​i​is at least as
deprived as i​′​if and only if the number of
dimensional deprivations in ​​d​ i ​​​(i.e., the number of 1s in the vector ​​d​ i​​​) is at least as great
as the number of dimensional deprivations in​​
d​ i′​​​. Second, at an intuitive level, Alkire et al.
claim that the numbers of deprivations figuring in the deprivation vectors of individuals
give us a certain type of information about
these individuals’ “unfreedoms.”
The demonstration by Alkire et al. (2015)
that a modified version of the set of axioms
in Pattanaik and Xu’s (1990) contribution on
the measurement of freedom can be used
to highlight the formal structure of the AF
methodology is ingenious and elegant. This
demonstration of similarity between the
mathematical structures of the two very different problems provides valuable insights
but by itself, it does not establish any direct
intuitive connection between the measure
of an individual’s overall deprivation in their
analysis and the intuitive conception of an
individual’s freedom or unfreedom. Alkire
et al. (2015) seek to establish this intuitive
connection through a different route. We
consider their argument below. For the sake
of simplicity, in the rest of this section (i.e.,
section 5), we assume that all the attributes
are considered to be of equal importance so
that the weights attached to them in calculating the deprivation score of an individual
are the same.
Alkire et al. (pp. 188–89) write, “A higher
value of [the adjusted head-count ratio]
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Journal of Economic Literature, Vol. LVI (June 2018)
represents more unfreedom, and a lower
value, less. Given that the set of indicators
will be unlikely to represent everything that
­constitutes poverty, if each element is widely
valued, and if people who are poor and are
deprived in a dimension would value being
non-deprived in it, then we anticipate that
deprivation among the poor could be interpreted as showing that poor people do not
have the capability to achieve the associated
functionings.” Suppose we have seven attributes, all of which are considered equally
valuable by everybody (so that each attribute
is given the weight of 1/7 in the adjusted
head-count ratio). Suppose ​k = 3/7​so that
an individual is deprived if and only if she
is deprived in at least three dimensions (see
(1)). Assume that the observed deprivation
vector of an individual ​i​is (0, 0, 0, 0, 1, 1, 1).
In the context of this specific example, how
do we interpret the statement that “deprivation among the poor could be interpreted
as showing that poor people do not have the
capability to achieve the associated functionings”? We consider some possible interpretations and their plausibility.
First, can our observation that (0, 0, 0, 0,
1, 1, 1) is the actual deprivation vector of
individual ​i​justify the inference that ​i​ could
not possibly have attained a deprivation vector where she would not have been deprived
in any of the attributes, ​​
f​ 5​​, ​ f6​  ​​,​ and ​​f7​  ​​​?
It is difficult to think of a set of reasonable
assumptions under which the observation of
the deprivation vector (0, 0, 0, 0, 1, 1, 1) can
lead us to conclude that the individual did
not have the freedom to achieve a deprivation vector,5 say, (1, 1, 1, 0, 0, 0, 0), where
she would have been non-deprived in terms
5 The terminology of “achieving” or “choosing” a deprivation vector is slightly awkward, and it would be more
natural to talk about choosing a vector of attribute levels,
which implies the deprivation vector under consideration.
But, for the sake of brevity, we shall continue to use the
terminology of achieving or choosing a particular deprivation vector.
of each of the attributes ​​f​ 5​​, ​f6​  ​​, ​and ​​f7​  ​​​. There
is no compelling reason why we would never
see the individual choosing to attain the
deprivation vector (0, 0, 0, 0, 1, 1, 1), when
(1, 1, 1, 0, 0, 0, 0) is available to her even if
we are prepared to make the following rather
strong assumptions:
(i) the set of attributes under consideration in our analysis coincides with
the set of all attributes that i​ ​values,
(ii) the individual’s values attach the same
weight to all attributes,
and
(iii) the individual never chooses to have
deprivation in terms of a larger number of attributes if she has the option
of being deprived in terms of a smaller
number of attributes.
Given the above assumptions, and given
the observed deprivation vector (0, 0, 0, 0, 1,
1, 1), it is, however, reasonable to infer that,
when ​i​achieved the deprivation vector (0, 0,
0, 0, 1, 1, 1), no achievement vector involving deprivation in less than three dimensions
was available to her. In that sense, one can
say that the observation of the deprivation
vector (0, 0, 0, 0, 1, 1, 1) of ​i​gives us some
specific type of ​i​’s unfreedom, namely, the
information that the opportunities available
to her did not include the possibility of having deprivation in less than three attributes.
The number, 3, then becomes a measure
of the extent to which i​​falls short of complete freedom from deprivations. What the
adjusted head-count ratio does is to sum up
these numbers across all deprived individuals identified by the “counting criterion” and
to take the average for the entire society. But
it is important to note that such inference is
crucially dependent on assumptions (i) and
(iii), which, as empirical assumptions, are
Pattanaik and Xu: On Measuring Multidimensional Deprivation
far from being innocuous. First, consider
(i). The list of all attributes that an individual belonging to a society values may be very
different from the list of a­ttributes, which
the social scientist/­philosopher ­studying the
overall deprivation of that society6 considers
to be reasonable. On the other hand, if the
list of attributes figuring in a study is the list
that has emerged from democratic deliberation and decisions in the society under
consideration (cf. Sen 1999), again there is
no reason to expect that the list will coincide with the list of all attributes that some
specific individual belonging to the society
considers valuable7. It is obvious that if, for
whatever reason, (i) does not hold, then very
little can be inferred about an individual’s
unfreedoms (in the sense explained above)
from that individual’s deprivation vector as
observed by the social scientist.
Assumption (iii) is also suspicious as an
empirical assumption. It is worth recalling
that the functioning and capability approach
to living standards and deprivation, which
has provided considerable inspiration for the
measurement of multidimensional deprivation in recent decades, often emphasizes
that there may be large gaps between what
people value and what they prefer or choose
(see Sen 1985, 1987). Thus, from what nutritionists tell us, it seems that a significant
number of people in many societies suffer
from serious health problems (e.g., obesity,
high cholesterol, and high blood pressure)
6 In particular, the set of attributes chosen by the social
scientist/philosopher is likely to miss out many of the vast
number of attributes that an individual may consider to be
valuable in her life.
7 The difference between the list of all attributes that
the individual values and the list of all relevant attributes
that emerges from democratic deliberation and voting is
somewhat analogous to the intuitive difference, discussed
in welfare economics, between an individual’s value judgment about social states and the ranking of social states
emerging from a social decision process (see, among others, Little 1952, Bergson 1954, Arrow 1963, Sen 1970, and
Pattanaik 2005).
667
because of lifestyles (e.g., heavy dependence
on unhealthy fast food, lack of exercise, etc.),
which they can easily avoid but which they
adopt despite knowing that those lifestyles are
harmful for their health. If it is accepted that,
as an empirical assumption, (iii) is of doubtful
validity and that often there may not be any
tight link between what people value in their
lives and what they choose to have, observed
deprivation vectors of people can hardly bear
the burden of the interpretation in terms of
unfreedoms of the type discussed above.
Alkire et al. (2015) are aware of problems
involved in trying to infer people’s unfreedoms from the observation of their actual
deprivation vectors. Referring to cases where
people may choose not to achieve the minimum satisfactory level of an attribute even if
they could do so and, presumably, having in
mind Sen’s (1999) well-known example of a
person who chooses to fast though she can
afford to eat, Alkire et al. write:
. . . suppose that we can anticipate what percentage of people would refrain from such
achievements in certain functionings—those
who might be fasting to the point of malnutrition, at any given time, for example. . . . Assume
that in identifying who is poor, the calibration
of poverty cut off ​k​reflects these predictions
of voluntary abstinence, as well as anticipated
data inaccuracies. . . . Applying such a poverty
cut off reduces errors in identification—for
example, by permitting people who would voluntarily abstain not to be identified as poor. . . .
We are puzzled by the method suggested
by Alkire et al.(2015) to handle the problem
of “voluntary abstinence” by adjusting (presumably upwards) the deprivation cutoff level​
k > 0​. As we have noted earlier (see (1)),
given any given cutoff value ​k​for the deprivation score, there exists a unique integer t​ (k) ∈ ​
{1, 2, … , m}​​such that the set of all i​ ∈ N​, for
​  ​​  > k​holds, coincides with
whom ​​∑m
j=1​​​ ​w​  j​​ ​dij
the set of all i​ ∈ N,​such that i​​is deprived in
at least ​t(k​) dimensions. So, instead of defining deprived ­individuals in terms of k​ ​, we can
668
Journal of Economic Literature, Vol. LVI (June 2018)
equivalently define deprived individuals in
terms of ​t(k)​. Suppose we have seven attributes and a person is considered deprived if
and only if she is deprived in at least three
dimensions. Assume that (i) 30 percent of
the population have deprivation in at least
three dimensions; (ii) 5 percent of the population are actually deprived in exactly three
dimensions, though they had the opportunity of being deprived in exactly two dimensions (these individuals being the “voluntary
abstainers”); (iii) 7 percent of the population
are deprived in exactly three dimensions and
did not have the opportunity of having deprivations in fewer than three dimensions; and
(iv) 18 percent of the population are deprived
in four attributes and did not have the option
of being deprived in fewer than four attributes. Given this configuration, the only way
of eliminating the “voluntary abstainers” from
the group of the deprived is to raise the cutoff number of dimensional deprivations from
three to at least four. But, if we do this, not
only will the voluntary abstainers be eliminated from the newly specified deprived
group, but 7 percent of the population, who
have exactly three deprivations and did not
have any opportunity to have fewer than three
deprivations, will also be thrown out from the
set of deprived individuals specified under the
new, stricter criterion. Thus, it may not always
be possible to adjust upward the deprivation
benchmark ​k​so as to throw out the “voluntary
abstainers” from the set of deprived individuals without simultaneously throwing out from
the set of deprived people a chunk of the population who are deprived overall not because
of voluntary abstinence, but because the individuals concerned did not have the option of
having fewer deprivations than they actually
have in their observed deprivation vectors.
6.
Some Broader Issues
In this section, we comment on three
general issues, namely, the exclusion of the
non-deprived individuals’ dimensional deprivations from the measure of the society’s
overall deprivation, the relation between an
individual’s extra achievements beyond some
dimensional deprivation benchmarks, and
interpersonal comparisons of overall deprivations of individuals.
6.1. The Dimensional Deprivations of
Individuals Who Are Not Deprived
Overall
The AF approach makes a very useful distinction between a person being
deprived and a person having some dimensional deprivations. When we have only
one dimension or attribute, say, income,
it seems natural to say that a person is
deprived if and only if her achievement falls
short of the deprivation benchmark in that
single dimension, no matter how small that
shortfall may be. In contrast, when there
are multiple dimensions, a person may have
mild deprivations in a few of these dimensions, but we may not still consider such
dimensional deprivations, together, to be
serious enough to regard the person as a
deprived individual or, if we like, as an individual who is deprived overall, just as, in our
ordinary language, we may not be prepared
to say that a person is a bad person simply because she has a few minor character
flaws. In view of this distinction between an
individual who has some dimensional deprivations and an individual who is deprived
overall, does it necessarily follow that, in
measuring multidimensional social deprivation, we should take into account only the
dimensional deprivations of people who
are considered deprived overall and ignore
the dimensional deprivations of other individuals in the society? It seems to us that
the answer will depend on one’s motivation. Every dimensional deprivation of an
individual damages her quality of life even
if the damage caused to her quality of life
by her different dimensional deprivations,
Pattanaik and Xu: On Measuring Multidimensional Deprivation
taken together, may not be large enough
to regard her as being deprived overall.
Given this, there may be two different types
of concerns in seeking to measure overall
deprivation at the social level. One may
be exclusively ­concerned with the urgent
problem of measuring (and relieving) the
deprivation of people whom one considers to be deprived overall; in that case, it
would be justifiable to ignore the deprivation of people who have some dimensional
deprivations, but whose dimensional deprivations are not extensive or deep enough to
regard them as being deprived overall. On
the other hand, there is no reason why one
may not be interested in devising an aggregate measure of the damages that dimensional deprivations cause to the qualities of
life of all individuals in a group, irrespective
of whether the damage to an individual’s
quality of life, caused collectively by her
dimensional deprivations, is large enough to
regard her as being deprived overall.8 When
Alkire et al. (2015) base their measure of
social deprivation on the dimensional deprivations of only those individuals in the group
who are classified as being deprived overall,
they are presumably guided by the former
motivation.9 But, even if one accepts that an
individual having some dimensional deprivations may not necessarily be deprived, in
measuring a group’s deprivation one may
still decide to take into account the dimensional deprivations of all individuals in the
group if one is guided by the latter concern
mentioned above.
8 It is clear that when there is only one dimension,
say, income, this distinction between the two motivations
becomes irrelevant.
9 The distinction between the two motivations disappears if one takes the position that a person is deprived
overall if and only if she is deprived in at least one
dimension.
669
6.2. The Role of an Individual’s Achievements
above Deprivation Benchmarks for Some
Dimensions in the Determination of Her
Overall Deprivation
Most measures of social deprivation
incorporate explicitly or implicitly some
­conception of a person’s overall deprivation.
In assessing a person’s overall deprivation, it
is a common practice to ignore her achievements, if any, over and above the respective
deprivation benchmarks in some dimensions. (For the sake of convenience, we shall
call an individual’s extra achievement beyond
the deprivation benchmark for a dimension
simply her extra achievement in that dimension.) Like most people, including us, who
have worked in this area, Alkire et al. (2015)
have followed this convention. As far as we
can see, there are at least two different ways
in which one can justify this convention
intuitively. First, one may take the position
that the damage to a person’s quality of life,
caused by even a very small shortfall from
the deprivation benchmark in one dimension cannot be mitigated to any extent by the
beneficial contributions to the individual’s
quality of life made by extra achievements,
however large, in other dimensions. It is not
clear how plausible this position is, especially
when one is simultaneously prepared to allow
trade-offs between deprivations in different
dimensions. The second justification may be
as follows. A social scientist or philosopher
may admit the possibility that the damage to
an individual’s quality of life, which is caused
by some dimensional deprivations, can be
mitigated, at least partially, by the contributions to the individual’s quality of life of extra
achievements in other dimensions. But, having admitted this possibility, the social scientist or philosopher may also argue that: for
the moment, she (i.e., the social scientist/
philosopher) is interested in the exercise of
assessing just the damage to an individual’s quality of life made by the dimensional
670
Journal of Economic Literature, Vol. LVI (June 2018)
deprivations that the individual may have;
and if anyone is interested in finding out
to what extent such damage is compensated or overcompensated by the individual’s extra achievements in other dimensions,
that is a matter of another distinct exercise, namely, the exercise of ­assessing the
­individual’s well-being/overall quality of
life/living standards. This can be a reasonable defense of the convention of ignoring
an individual’s dimensional extra achievements in the assessment of the individual’s
overall deprivation if one assumes that the
damage caused to an individual’s quality of
life by deprivations in certain dimensions is
independent of the extra achievements of
the individual in other dimensions. But, if
one takes this line of defense, then, in the
absence of further assumptions, one may
have to accept, at least in principle, the possibility that a policy maker may be justified
in devoting resources to increase people’s
dimensional extra achievements, instead
of devoting the same resources to reducing their dimensional deprivations, since
that may be the most cost-effective way of
increasing their well-being/overall quality of
life/living standards.
Our discussion in the preceding paragraph
brings us to another possible approach to the
notion of an individual’s overall deprivation,
namely, the approach through the notion of
an individual’s well-being defined with reference to the same space of attributes or
indicators, which is considered relevant for
the measurement of the individual’s overall
deprivation. Suppose there is an agreedupon “well-being function” or “quality of life
function,” W
​ ​, to evaluate an individual’s overall well-being or quality of life on the basis of
her dimensional achievements, and there is a
given well-being threshold, ​​ W ​​, below which
¯considered to
the individual’s well-being is
be “unacceptably low.” Then, one can say
that an individual, i​,​with the achievement
​ (​x​ i​​)
vector ​​x​ i​​​, is deprived if and only if W
< ​ W ​​(see Bourguignon and Chakravarty
¯ and Tsui 2002). The magnitude of ​i​’s
2003
shortfall from the deprivation benchmark can then be taken to be 0 if
​W(​x​ i​​) ≥ ​ W ​​and it can be taken to be
W(​x¯
​ i​​)​if ​
W(​x​ i​​) < ​ W ​​
. The specifica​​ W ​​– ​
¯
tion of the function W
​ ​is ¯
the analytical core
of this approach. Once the function ​W​ is
­specified and the level of deprivation of each
individual is assessed in this fashion, the
remaining exercise of aggregating the individual deprivations in terms of well-being
to reach an assessment of social deprivation
becomes exactly analogous to the aggregation problem in the case of unidimensional
deprivation. This approach, discussed by
Bourguignon and Chakravarty (2003) and
Tsui (2002), has the advantage of bringing
together the analytical framework for measuring well-being or living standards and that
for measuring deprivation. It may be thought
that this approach will push the issue of
dimensional deprivations of individuals to
the background, but it is not clear why this
should be so. Dimensional deprivations of an
individual are important because of the damage they do to the individual’s well-being or
quality of life. If the well-being function for
an individual is defined with respect to the
same space of attributes used in the analysis of multidimensional deprivation, then all
our concerns relating to dimensional deprivations of an individual would come to the
surface when we seek to determine what
the individual’s well-being function should
be and how it should take into account the
individual’s dimensional deprivations. In
general, it seems to us that the approach
to the problem of multidimensional deprivation through the notion of an individual’s
­well-being function deserves more attention
than it has received so far.
6.3. Intergroup Comparisons of Deprivation
w​ m​​​,
Consider the weights ​​
w​ 1​​, ​w​ 2​​, … , ​
which are attached to different dimensional
Pattanaik and Xu: On Measuring Multidimensional Deprivation
deprivations in the AF methodology (see
section 2). These play a crucial role in the
AF method for identifying the deprived and
their method for aggregating the individual
deprivations to assess the magnitude of social
deprivation. Even in a given s­ ociety, however,
there may be different distinct ­
culturally
identifiable groups, which may have different sets of weights for the dimensions; this
is true especially in the case of large and very
diverse societies, such as in India. How does
one determine the set of weights to be used
to assess the deprivation status (deprived
versus non-deprived) of an individual and
the magnitude of the overall deprivation of
an individual identified as being deprived?
Following Sen (see, for instance, Sen 2009,
241–43), it is often argued that the set of
weights to be attached to different dimensions should be determined through “public
reasoning,” which takes the form of democratic deliberations and debates. While the
importance of democratic deliberations and
debates in the evolution of values can hardly
be doubted, such deliberations and debates
do not necessarily lead to unanimity of values. If after extensive public discussion, there
still remain differences between the sets of
weights that different cultural groups attach
to different dimensions, the question arises
of what should be done about the choice of
weights to be used to determine whether
an individual is deprived and to measure
the extent of her overall deprivation if she is
judged to be deprived. Should one use the
same set of weights, possibly reached through
some form of democratic decision procedure,
for every individual in the society irrespective of what cultural group she belongs to?
If one feels, as we do, somewhat uncomfortable with such “absolutism,” which does not
make any allowance for cultural differences
of different groups, and if one allows for
some difference between the sets of weights
to be used for computing the overall deprivations of individuals coming from different
671
cultural groups, then can one do so without
running into problems with some very plausible principles of interpersonal comparisons
of overall deprivation? In particular, can
one allow the set of weights to vary between
individuals while adhering to the principle
that, if, for every dimension, the ­deprivation
of a deprived individual, i​​
, is at least as
high as that of another deprived individual,
​i′, ​and, if, for some dimension, the deprivation of ​i​is strictly greater than the deprivation
of ​i′​, then the overall deprivation of i​​ must
be at least as high as that of ​i′​ ? The discussion of the exact counterpart of this issue in
the context of the measurement of standard
of living when the attributes are cardinally
measurable (see, among others, Pattanaik
and Xu 2007, 2012 and Fleurbaey 2007) has
turned up some rather depressing negative
results, and it is not difficult to prove the
counterpart of those negative results for the
problem of measuring overall deprivation of
individuals when the attributes are cardinally
measurable.
7.
Concluding Remarks
In this essay, we have tried to provide
a critical evaluation of Alkire et al. (2015)
and, in the course of doing so, we have also
commented on certain general issues in the
literature on the measurement of multidimensional deprivation. While we have some
reservations about the AF method for measuring multidimensional deprivation for a
society—especially when individual dimensional deprivations are cardinally measurable—and about the interpretation by Alkire
et al. (2015) of certain measures of deprivation in terms of freedom/unfreedom, such
reservations hardly detract from our admiration for the volume as a comprehensive,
meticulous, and authoritative treatment of
the problem of measuring multidimensional
deprivation. The volume will be cherished by
all who are interested in the measurement of
672
Journal of Economic Literature, Vol. LVI (June 2018)
multidimensional deprivation, irrespective
of whether they are seasoned researchers
or beginning students looking for a highly
readable but sophisticated account of the
issue of measuring multidimensional deprivation, and it should stimulate further future
research in this important area.
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