Distinctive Characteristics of the Adjusted Headcount Ratio
The M0 measure described in section 5.3 has several characteristics that merit special attention. First, it can be implemented with indicators of ordinal scale that commonly arise in multidimensional settings.
In formal terms, M0 satisfies the ordinality property introduced in section 2.5. The ordinality property states that whenever variables (and thus their corresponding deprivation cutoffs) are modified in such a way that their scale is preserved—what has been defined in section 2.3 as an admissible transformation—the poverty value should not change.[167]The satisfaction of this property is a consequence of the combination of the identification method and the aggregation method. Because identification is performed with the counting approach, which dichotomizes achievements into deprived and non-deprived, equivalent transformations of the scales of the variables will not affect the set of people who are identified as poor. Note that the weights attached to deprivations are independent of the indicators' scale and implemented after the deprivation status has been determined. This is clearly relevant for consistent targeting within policies or programmes using ordinal indicators. In turn, aggregation to obtain the M0 measure is performed using the censored deprivation matrix, which represents the deprivation status of each poor person in every dimension and also uses the 0-1 dichotomy. In the aggregation procedure, the deprivations of the poor are weighted, but, again, the weights are independent of the indicators' scale and implemented after the deprivation status of the poor has been determined. Thus, equivalent transformations of the scales of the variables will not affect the aggregation of the poor and thus will not affect the overall poverty value.
The fact that M0 satisfies the ordinality property is especially relevant when poverty is viewed from the capability perspective, since many key functionings are commonly measured using ordinal (or ordered categorical) variables.
Virtually every other multidimensional methodology defined in the literature (including M1, M2, and, in general, the Mα measures with a > 0, which are defined in section 5.7) do not satisfy the ordinality property. In the case of the Mα measures with α > 0, while the set of people identified as poor does not change under equivalent representations of the variables, the aggregation procedure will be affected as it is no longer based on the censored deprivation matrix but on a matrix that considers the depth of deprivation in each dimension. In other measures, the violation of ordinality occurs at the identification step. Moreover, for most measures, the underlying ordering is not even preserved, i.e. an increase in poverty may become a decrease just by changing representations. Special care must be taken not to use measures whose poverty judgements are meaningless (i.e. reversible under equivalent representations) when variables are ordinal.Consider a methodology that combines the identification method used in the AF measures ρk with the headcount ratio as the aggregate measure: M(ρk,H). This methodology which was used in previous counting measures surveyed in Chapter 4, satisfies the ordinality property. But it does so at the cost of violating dimensional monotonicity, among other properties. In contrast, the methodology that combines a counting approach to identification and M0 as the aggregate measure, M (ρk,M0), provides both meaningful comparisons and favourable axiomatic properties and is arguably a better choice when data are ordinal.
Second, while other measures have aggregate values whose meaning can only be found relative to other values, M0 conveys tangible information on the deprivations of the poor in a transparent way. As stated in section 5.3, it can either be interpreted as the incidence of poverty ‘adjusted' by poverty intensity or as the aggregate deprivations experienced by the poor as a share of the maximum possible range of deprivations that would occur if all members of society were deprived in all dimensions.
As we shall see in section 5.5.3, the additive structure of the M0 measure permits it to be broken down across dimensions and across population subgroups to obtain additional valuable information, especially for policy purposes.Third, the adjusted headcount methodology is fundamentally related to the axiomatic literature on freedom. In a key paper, Pattanaik and Xu (1990) explore a counting approach to measuring freedom that ranks opportunity sets according to the number of (equally weighted) options they contain. Let us suppose that the achievement matrix X has been normatively constructed so that each dimension represents an equally valued functioning. Then deprivation in a given dimension is suggestive of capability deprivation, and since M0 counts these deprivations, it can be viewed as a measure of ‘unfreedom' analogous to Pattanaik and Xu. Indeed, the link between M(ρk, M0) and unfreedom can be made precise, yielding a result that simultaneously characterizes ρk and M0 using axioms adapted from Pattanaik and Xu.[168] [169] This general approach also has an appealing practicality: as suggested by Anand and Sen (1997), it maybe more feasible to monitor a small set of deprivations than a large set of attainments. 5.5