Measures of multi-dimensional poverty Measures using aggregate data: The Human Poverty Index (HPI)

The Human Poverty Index (HPI), the fourth companion to the HDI, was introduced in the 1997 Human Development Report. Although very limited, it is a measure of multi-dimensional poverty that can be calculated in the absence of disaggregated data. The HPI looks at the same three dimensions as the HDI: health, education and standard of living. It differs from the HDI in that it focuses on deprivations, not achievements. There are two versions of the HPI, one for developing countries (HPI-1), the other for developed countries (HPI-2).

The indicators used for the HPI-1 are, the probability at birth of not surviving to the age of 40, the adult literacy rate, and the equally weighted average of (1) the percentage of population without access to an improved water source and (2) the percentage of children under weight for their age. The

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three indicators are deprivation indicators, name them DH, DE and DY (as in the HDI, the indicators are normalized to range between 0 and 1). Analogously to the case of the HDI, if dh, de and dy represent the distribution of deprivation among the population in each of the three dimensions, the indicators can be seen as the average deprivation across the population: DH =

(9) HPI – 1 = [(DH 3 + DE 3 + DY 3 )/3] 1/3

Because the elements of the index are deprivation indicators rather than across dimensions giving a higher weight to the dimensions that present higher

levels of deprivations. More demanding thresholds are used in HPI-2, and the HPI-2 includes a fourth dimension, ‘social exclusion’. The indicators of deprivation are: the probability at birth of not living to the age of 60, the percentage of adults (between 15 and 65 years) lacking functional literacy skills, the percentage of the population below the income poverty line (defined as 50 per cent of the median household income) and, for the social exclusion dimension, the rate of long-term unemployment (12 months or more). These indicators also correspond to the mean deprivation in

order 3. Both versions of the HPI provide summary information on aggregate deprivation, which is mainly helpful for constructing rankings between countries and regions. However, when designing policies, one may wish to identify the groups to target more specifically as well as the dimensions with the highest deprivation incidences and depths. For this reason one much newer and less established measure is introduced, because it is user friendly, robust and can be implemented with existing data.

Measures using disaggregate data: Alkire and Foster (2007) In this sub-section we present one family of measures of multi-dimensional poverty developed at the Oxford Poverty and Human Development Initiative (OPHI) by Alkire and Foster (2007) (AF measure). It looks at all the deprivations that batter a person or household together. The family of measures is a multi-dimensional extension of the uni-dimensional FGT class of poverty indices, introduced above. It is important to remark that other measures of multi-dimensional poverty have been proposed in the literature as by Tsui (2002), Bourguignon and Chakravarty (2003 – this is also a multi- dimensional extension of the FGT class) and Maasoumi and Lugo (2008), and counting-based measures are widely used in policy. We have chosen to present

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the AF measure because (1) it can be used with ordinal data, and (2) it uses a new identification criterion for the multi-dimensionally poor (most of the others use the union approach to identification), and (3) it can be decomposed by group and broken down by dimension, which brings a great deal of insight to policy.

As in the unidimensional case, there is a set of desirable properties for a multi-dimensional poverty measure. Most of them are natural extensions from the uni-dimensional case, which were introduced in the second section. For simplicity, we avoid its enumeration here, but mention them when describing the different members of the family of measures presented here. 32

The raw information for this family of measures is a matrix which gives the achievements of each person i, with i = 1,...,n, in each dimension j = 1,...,d. For simplicity, assume for the moment, that each dimension has one indicator, that each dimension is equally weighted and that we have data for individuals. Name such matrix X:

Achievement Matrix X =

Poverty lines z = [z 1 z 2 …z d ]

Each entry of the matrix x i indicates the achievement of person i in dimension j. Each row of the matrix indicates the achievements of individual i in the d dimensions, whereas each column indicates the distribution of achievements in

a specific dimension across the population. For each dimension, a poverty line is defined. The poverty lines can be presented in vector z = (z 1 ,z 2 ,...,z d ), where z j is the poverty line of dimension j. Similarly to the procedure followed in the uni-dimensional case (described in the third section), one can define a normalized gap for each individual in each dimension. For achievements that are below the poverty line, this gap is given by the distance between the poverty line in that dimension and the achievement, measured in poverty line units. If the person’s achievement is at or above the corresponding poverty line, the normalized gap is zero.

(10) Normalized gap for each person

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Now, we can create matrix G , which contains each normalized gap raised to

At this point one needs to define an identification criterion of the multi- dimensionally poor. The authors propose the following. Consider the matrix G 0 , in which all entries are 1 if the individual is deprived in that dimension, or 0 if she is not. With such matrix one can ‘count’ the number of deprivations that

each individual in the distribution has, and construct a vector c = (c 1 ,...,c n ) l where each element c i indicates on how many dimensions person i is deprived. Then, it is a matter of deciding on how many dimensions should someone be deprived so as to be considered multi-dimensionally poor. That is equivalent to setting a second poverty line, which the authors call the ‘dimension cutoff’, and name it k. A person i is considered multi-dimensionally poor if c i ≥ k.

This is why the approach is said to use a dual cutoff, first the within dimension cutoffs zj and second, the across dimensions cutoff k. The decision

on the value of k is left to the researcher, and several different values can be tested. One could set k = 1, requiring an individual to be deprived in at least one dimension to be considered multi-dimensionally poor, and this would correspond to the union approach. Others would set k = d, requiring an individual to be deprived in all the considered dimensions so as to be multi- dimensionally poor, the intersection approach. Clearly, other intermediate values for k may be more appropriate.

Once the multi-dimensionally poor have been identified, one wants to focus only on them. To do that, the normalized gaps of people who have not been identified as multi-dimensionally poor are replaced by zero. Name this censored normalized gaps as g ij (k). These are given by:

With these censored gaps, one can construct the censored matrix :

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The multi-dimensional poverty measure of AF is simply obtained as the mean of this matrix:

(k))

Example: An example will help to clarify the computation of the measures and some of the salient members of this family. Suppose that there are four people and four considered dimensions, say, consumption, years of education, empowerment and access to health care. For the purpose of a numerical example the dimensions itself could be any. The matrix of achievements and the vector of poverty lines are given by:

z = ( 13

12 3 1) Then, the G 0 matrix, and the vector of deprivation counts c, are given by:

Suppose that a cutoff of k = 2 is selected. The second and third person will be considered multi-dimensionally poor, but not the fourth person. The G (k) matrix is therefore given by:

The Multidimensional Headcount Ratio H multi-dimensional headcount ratio. That is, the fraction of the population that

is multi-dimensionally poor H = q/n, where q is the number of multi- dimensionally poor. In this example, H = 1/2. In the uni-dimensional case, we noted that the headcount ratio is insensitive to the depth and distribution of

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poverty, and that remains true in the multi-dimensional case. Moreover, the multi-dimensional headcount ratio is not sensitive to the number of deprivations the poor experience. For example, if the second person becomes deprived in three dimensions rather than two, the multi-dimensional

headcount ratio would not change. 33 For this reason, the multi-dimensional H is said not to satisfy dimensional monotonicity.

The Adjusted Headcount Ratio M 0

0 , obtained as the mean of the G 0 (k) matrix. In this case, M 0 = HA = (0.5)(0.75) = 0.375. Note that M 0 is lower than H, and that is precisely why it is called the ‘adjusted’ headcount ratio: it is the multi-dimensional headcount ratio (H) multiplied by the average deprivation share among the poor (A). The average deprivation share among the poor can be obtained adding the number of deprivations of the multi-dimensionally poor and dividing them by the total number of deprivations and poor people. In this case: A = (2 + 4)/(4*2) = 3/4.

Now it is clear that M 0 = HA = (0.5)(0.75) = 0.375 – which is, again, the mean of the matrix. Note that if the second person became deprived in the first dimension – consumption, the M 0 measure would increase to M 0 = 7/16 = 0.437, because the average deprivation share among the poor would increase, satisfying dimensional monotonicity. However, if any of the multi-

dimensionally poor became more deprived in one dimension, M 0 would not change, violating monotonicity. (For example, if the third person had 9 years

of education rather than 10, M 0 would be the same).

The Adjusted Poverty Gap M 1 Consider now the actual value of each normalized gap – just as we did for the

the mean of the matrix is M 1 = 3.3/16 = 0.206. But what does that mean? This measure is the product of three informative partial indices. In addition to HA, it considers the average poverty gap G across all instances in which poor persons are deprived. Here, G = (0.04 + 0.42 + 0.17 + 0.67 + 1 +1)/6 = 0.55.

So the formula is M 1 = HAG = (0.5) (0.75) (0.55) = 0.206. It is worth noting that, because M1 is based on the poverty gaps that individuals experience, if a poor person becomes more deprived in one dimension, the M 1 measure will

> 0). (If the third person had 9 years of education rather than 10, M 1 = 3.38/16 = 0.211). However, as in the poverty gap, an increase in deprivation counts the same for the barely poor and for the extreme poor. To favour the poorest of

the poor we turn to the M 2 measure.

The Adjusted Squared Poverty Gap M 2 Now we consider the matrix of squared gaps, just like we considered the

is now M 2 = 2.66/16 = 0.166. As with the previous members of the family, this

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measure is also the product of other informative partial indices. We replace the average gap G, by the average squared gap, which represents the severity of deprivations S across all instances in which poor persons are deprived. Here S

= (0.04 2 + 0.42 2 + 0.17 2 + 0.67 2 +1 2 +1 2 )/6 = 0.44. So the formula for M 2 = HAS = (0.5)(0.75)(0.44) = 0.166. This measure satisfies monotonicity and, moreover, favours the extreme poor. For example, if the third person had 9

years of education rather than 10, then M 2 would increase to M 2 = 0.168, whereas if the second person had 6 years of education rather than 7, this would increase M 2 to M 2 = 0.170. This measure (and all members of the family

designing policy: they are decomposable in subgroups of people, and – after the identification step has been completed – they can also be broken down by dimensions. In this way, it is possible to identify the percentage contribution of

a particular group to overall multi-dimensional poverty, as well as the percentage contribution of deprivation in each particular dimension to overall multi-dimensional poverty.

simplicity. However, any weighting structure can be used. Let w = (w 1 ,w 2 ,...,w d )

be a d dimensional row vector, where each element w j is the weight associated with dimension j. The weights must add to the total number of dimensions

( d j=l w j = d). One can define the matrix G , where the typical element is the weighted normalized gap g j=l =w j ((z j –x ij )/z j ) , when x ij <z j , while g ij =0 otherwise. As before, one can obtain a vector of deprivation counts c i , which

indicates for each individual the weighted number of dimensions in which she is deprived (for example if an individual is deprived in consumption and health, and consumption has a weight of 2, while health has a weight of 0.5, then c i = 2.5 and not 2, as it would be with equal weights). The dimensional cutoff for the identification step of the multi-dimensionally poor k, now ranges between the minimum weight (k = min{w j }, which corresponds to the union approach) and the total number of dimensions (k = d, which corresponds to the intersection approach). As before, once the multi-dimensionally poor have been identified, the censored matrix G calculated as the mean of such matrix. 34

As a last comment on these measures, it should be noted that they treat dimensions as independent rather than assuming all are substitutes or all are complements. The issue on how to account for interactions across dimensions (either substitutability or complementarity) is a much discussed topic in multi- dimensional poverty measurement and although some proposals have been made (such as Bourguignon and Chakravarty’s (2003) measures – which however cannot be broken down by dimension), no consensus has been achieved yet.

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Measures of multi-dimensional poverty Measures using aggregate data: The Human Poverty Index (HPI)