The Adjusted Headcount Ratio: A Measure of Capability Poverty?

Suppose that there are a set of dimensions, each of which represent functionings or capabilities that a person might or might not have—things like being well nourished, being able to read and write, being able to drink clean water, and not being the victim of violence.

How might such an M₀ reflect capability poverty? The key insight is this: in such a measure, a higher value of M₀ 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

⁴ Section 6.1 and Box 6.1 draw upon Alkire and Foster (2007).

⁵ We are especially conscious in this chapter of being unable to cite or engage all the many scholars who have creative insights on measures of well-being and poverty that draw on the capability approach. Their work deserves its own in-depth constructive survey, building on other such surveys that already exist, including Chiappero-Martinetti and Roche (2009), Clark (2008), and Robeyns (2006), as well as references in recent applied work such as Arndt and Volkert (2011) and Van Ootegem and Verhofstadt (2012). We would refer interested readers to the Human Development and Capability Association (HDCA) and the bibliographies on capabilities published annually in the Journal of Human Development and Capabilities.

⁶ This assumption can be relaxed to obtain more general results if required. would value being non-deprived in it, then we anticipate that deprivations among the poor could be interpreted as showing that poor people do not have the capability to achieve the associated functionings. Thus M₀ would be a (partial) measure of unfreedom, or capability poverty.

As noted above, such an interpretation of M₀ relies on assumptions regarding the parameters:

(a) indicators measure or proxy functionings or capabilities;

(b) people generally value attaining the deprivation cutoff level of each indicator;

(c) the weights reflect a defensible set or range of relative values on the deprivations;

(d) the cross-dimensional poverty cutoff reflects ‘who is capability poor’.

Such an interpretation implicitly also relies upon assumptions about data quality and accuracy, and empirical techniques (that measures are implemented accurately). It has quite a restricted and uniform approach to values: for example, using a non-union poverty cutoff to permit ‘some’ abstinence from functionings.^[178] But it might at least signal an avenue worth pondering.

In fact, as Box 6.1 elaborates more formally, under these conditions, our identification strategy and Adjusted Headcount Ratio can be related to Pattanaik and Xu’s signature work (1990), except that we now focus on unfreedoms rather than on freedoms. In their lucid and illuminating paper, Pattanaik and Xu elaborate on Sen’s claim that freedom has intrinsic value, thus that the extent of freedom in an opportunity set matters—independently of its relationship to preferences and utility. In developing this claim axiomatically they propose that the ranking of two opportunity sets in terms of freedom should depend only on the number of options present in each set.^[179]

Sen, responding to Pattanaik and Xu (1990), observed that not every additional option (singleton) would contribute to an expansion of freedom—only those options that a person values and has reason to value. ‘The evaluation of the freedom I enjoy from a certain menu must depend to a crucial extent on how I value the elements included in that menu’. For example, ‘if a set is enlarged by including an alternative which no one would choose in relevant circumstances (e.g., being beheaded at dawn), the addition of that alternative may not necessarily be seen as a strict enhancement of freedom.

(Sen 1991: 21 and 25). Nor would a deprivation in that negatively valued alternative be seen as impoverishing.

Our assumptions regarding the choice of parameters avoid Sen's critique if each dimension of poverty reflects something that people value and have reason to value. Further, we follow Anand and Sen (1997), who argued that it may be easier to obtain agreement on the value of a small set of unfreedoms than an ample set of freedoms.^[180] As Sen points out, ‘in the context of some types of welfare analysis, e.g.

Note that this capability interpretation of M₀ does not directly represent ‘unchosen' sets of capabilities in a counterfactual sense (Foster 2010). Nor does it necessarily incorporate agency (Alkire 2007). Rather, in a manner parallel to Pattanaik and Xu, it interprets the deprivations in at least a minimum set of widely valued, achieved functionings as unfreedom, or capability poverty (Box 6.1).

Naturally, capability poverty measures that have different specifications and reflect different purposes could be constructed for the same society. There might be a child poverty measure or a capability poverty measure reflecting the values of a specific cultural group such as nomadic populations, or there might be a national capability poverty measure that reflects important deprivations about which there is widespread agreement across social groups. Thus the decision to measure capability poverty does not generate one unique measure; decisions as to the scope and purpose of the measure and the data sources guide measurement design even if the choice of space has been settled.

We also hasten to point out that many legitimate and tremendously useful measures could be constructed using M₀ but located in a different space or in a mixture of spaces. These would not be measures of capability poverty but could be powerful tools for reducing capability poverty. For example, the dimensions might be resources such as service delivery (hopefully identifying whether marginalized groups have real access and clarifying the quality of the services). The point is that our measurement framework can be used for different purposes including those unrelated to capabilities.

BOX 6.1 UNFREEDOMSAND M

Let M be a poverty methodology satisfying decomposability, weak monotonicity, non-triviality, and ordinality The first three properties are satisfied by all members of methodology M; however, M₀ = (ρ_k,M₀) is the only adjusted Foster-Greer-Thorbecke (FGT) measure that satisfies ordinality, and it is this property that ensures that its poverty levels and comparisons are meaningful when the dimensional variables are ordinal.

By decomposability, the structure of M depends entirely on the way that M measures poverty over singleton subgroups; and by dichotomization, this individual poverty measure can be expressed as a function p(v) of the individual's deprivation vector v (which is any row g_i. of deprivation matrix g⁰). In the case of (ρ_k,M₀), we have p(v) = μ(v(k)), where v(k) is the censored distribution defined as v(k) = v if ₁ Vj ≥ k and v(k) = 0

if jj_{s 1} vj < k. We will now explore the possible forms that p can take for dichotomized measures. Note that while the definition of M₀ is based in part on the dimensional cutoff k, we have not specified the identification method employed by the general index M. Hence a second question of interest is what forms of identification might be consistent with various properties satisfied by M₀.

The individual poverty function p for M₀ has two additional properties of interest. First, it satisfies anonymity or the requirement that p(v) = p(v∏), where ∏ is any d ? d permutation matrix. This property implies that all dimensions are treated symmetrically by the poverty measure. Second, it satisfies semi-independence, which states that if vj = vj = 1, and p(v) ≥ p(v'), then p (v - j ≥ p(v' - e_j∙).¹° Under this assumption, removing the same dimensional deprivation from two deprivation vectors should preserve the (weak) ordering of the two.

Let p be the individual poverty function associated with a dichotomized poverty measure. p satisfies anonymity and semi-independence if and only if there exists some k = 1,...,d such that for any deprivation vectors v and v' we have: p(v') ≥ p(v) if and only if μ(v'(k)) ≥ μ(v(k)).

In other words, p ranks individual deprivation vectors in precisely the same way that (ρ_k,M₀) does for some k = 1,...,d. This result is especially powerful since it simultaneously determines both the individual poverty index (p) associated with the Adjusted Headcount Ratio (M₀) and the identification method (based on a dimensional cutoff k) consistent with the assumed properties. To establish the result we extend the generalization of Pattanaik and Xu given in Foster (2010). In particular, if full independence were required, so that the conditional in semi-independence were converted to full equivalence, then a direct analogue of the Pattanaik and Xu result would obtain, namely, p(v^l) ≥p(v) if and only if μ(v^l) ≥ μ(v). In this specification, p would make comparisons of individual poverty the same way that the union-identified M₀ does—by counting all deprivations.

While our result uniquely identifies the individual poverty ranking, it leaves open a multitude of possibilities for the overall index P—one for each specific functional form taken by p. For example, the individual poverty function p(v) = [μ(v(k))]², when averaged across the entire population to obtain P, would place greater emphasis on persons with many deprivations. It would be interesting to explore alternative forms for p and the properties of the associated index P. Note that because of dichotomization, each of these measures would provide a way of evaluating multidimensional poverty when the underlying variables are ordinal.

Given the arguments in Foster (2010), it is straightforward to establish the above result. In particular, let C = [v ∈ R^d : vj = 0 or vj = 1] for all j be the set of all individual deprivation vectors, and let p : C → R be an individual poverty function associated with a standard dichotomized poverty measure such that p satisfies

¹⁰ The symbol βj refers to the j^th usual d-dimensional basis vector whose j^th entry is equal to 1 and the rest of the elements are equal to 0. Note that semi-independence is a weakening of the property of‘independence’ found in Pattanaik and Xu (1990).

BOX 6.1 (cont.)

anonymity and semi-independence. By anonymity, all vectors v,v ∈ C with ₁ v∣ = ∑f= ₁ vj must satisfy p(v) = p(v'). In other words, the value of p(v) depends entirely on the number of deprivations in v. Weak monotonicity implies that p(v) ≤ p(v') for ₁ v∣ ≤∑j ₁ vj, and so the value of p(v) is weakly increasing in the number of deprivations in v. By non-triviality and decomposability, it follows that p(v) > p(0) for Óó ₁ v∣ = d.¹¹ Let k be the lowest deprivation count for which p(v) is strictly above p(0); in other words, p(v) = p(0) for

₁ v∣ < k, and p(v) > p(0) for ∑j= ₁ v∣ ≥ k. Semi-independence ensures that p must be increasing in the deprivation count above k. For suppose that p(v) = p(v') for v, v ∈ C with k ≤∑d= ₁ ^vj critical ethical objectives. Most empirical outworkings of the capability approach have used drastic simplifications, and these can often be cheered as true advances, even while their limitations are borne in mind. ‘In all these exercises, clarity of theory has to be combined with the practical need to make do with whatever information we can feasibly obtain for our actual empirical analyses. The Scylla of empirical overambitiousness threatens us as much as the Charybdis of misdirected theory' (Sen 1985: 49). In this sense, our methodology may be a step forward in operationalizing the measurement of capabilities.

6.2