# Variable Population Poverty Comparisons

Virtually all extant measures of poverty emphasize a *headcount ratio,* rather than an *aggregate headcount,* view of poverty. This, in turn, is because virtually all extant measures of poverty either explicitly or implicitly endorse the Replication Invariance Axiom or deny the Population Focus Axiom, even as they accept the Income Focus Axiom. To see the relationship between the headcount ratio and Replication Invariance, and the relationship between the aggregate headcount and Population Focus, note first that, under any k-fold replication of an income distribution, the headcount ratio will remain unaffected, while the aggregate headcount will register a k-fold increase; and, second, with an addition to the non-poor population, the aggregate headcount will remain unaffected, while the headcount ratio will register a decline. It would appear to be inconsistent to find merit in the Income Focus Axiom and none in the Population Focus Axiom; when this inconsistency is sought to be rectified by requiring poverty indices to also satisfy Population Focus, then we find—unsurprisingly perhaps, but also disquietingly—that Population Focus in conjunction with other axioms which traditionally emphasize a headcount ratio view of poverty leads to incoherence and impossibility. This section, which relies heavily on Subramanian (2002b, 2011b), Hassoun (2010) and Hassoun and Subramanian (2011), presents a small set of very elementary impossibility theorems which point to the difficulties inherent in variable population poverty comparisons.

Proposition 5.1 *There exists no anonymous poverty measure P:* X x *S ! R which satisfies Replication Invariance (Axiom RI), Weak Poverty Growth (Axiom WPG), and Weak Population-Focus (Axiom WPF).*

*Proof* Let the poverty line be z, and let *x* and *y* be two levels of income such that *x* > z

y) . By Axiom WPG, *P* (b; z) < P(c; z), and by Axiom RI, P(c; z) = *P* (a; z), whence P(b; z) < P(a; z)—which, however, is contradicted by P(a; z) < P(b; z), as dictated by Axiom WPF.

Proposition 5.2 *There exists no anonymous poverty measure P:* X x *S ! R which satis*fi*es Maximality (Axiom MX), Weak Poverty Growth (Axiom WPG), and Weak Population Focus (Axiom WPF)*.

*Proof* Let the poverty line be *z*, and let *x* be a level of income satisfying

*x* > z. Consider the income distributions a = (0, ....., 0), b = (a, x) and c = (b, 0).

We now have: P(c; z) > *P* (b; z) by Axiom WPG, and *P* (b; z) > P(a; z) by

Axiom WPF, whence P(c; z) > P(a; z)—which, however, is contradicted by P(a;

z) > P(c; z), as dictated by Axiom MX.

Proposition 5.3 *There exists no anonymous poverty measure P:* X x *S ! R which satis*fi*es Monotonicity (Axiom M), Replication Invariance (RI), and Population Focus (Axiom PF).*

*Proof* Let the poverty line be *z*, and let *x* and *y* be two levels of income such that 0 < x*P* (b; z).

Proposition 5.4 *(Corollary to Proposition 5.3). There exists no anonymous poverty measure P:* X x *S* ! *R which satisfies Transfer (Axiom T), Replication Invariance (RI), and Population Focus (Axiom PF)*.

*Proof* . The proof follows, given Proposition 5.3, from the fact that Axioms T and PF together imply Axiom M. To see this, imagine a situation in which *z* is the poverty line, *n* is a positive integer, A is a positive scalar, and x, y, u and v are four income vectors satisfying *x* = (xj, ...., *x _{n}); y* = (y

_{b}....,

*y*with

_{n}),*yi = xi8i = j*for some

*j*2 Q(x; z) and

*yj*=

*xj*+ A; u = (x, z + A); and v = (y, z). By Axiom PF, P(v; z) = P(y; z) and by Axiom T, P(u; z) > P(v; z), whence P(u; z) > P(y; z), which, together with P(u; z) = P(x; z) as implied by Axiom PF, leads to P(x; z) > P(y; z)— which, precisely, is what is dictated by Axiom M. We have shown that Axioms T and PF in conjunction imply Axiom M; from Proposition 5.3, we know that there exists no anonymous poverty measure P: X x

*S ! R*which simultaneously satisfies Axioms M, RI and PF; it follows that there exists no anonymous poverty measure

*P*: X x S ! R which simultaneously satisfies Axioms T, RI and PF.^

Propositions 5.1 and 5.2 are based on results available in Subramanian (2002b) and Subramanian (2011b) respectively, while Propositions 5.3 and 5.4 are available in Hassoun (2010) and Hassoun and Subramanian (2011). The impossibility results stated and proved above are fairly straightforward ones, and require little in the way of complicated reasoning to comprehend. The implications of these results, however, are of some significance for the measurement of poverty. In particular—and as argued in Hassoun (2010) and in Subramanian (2011b)—it would appear that there are at least two possible views one may take of what one calls ‘a measure of poverty’. Under the first view, one measures ‘how poor a society is’; under the second view, one measures ‘how much poverty there is in a society’. The latter view would deem all information relating to the status of the nonpoor population as being irrelevant for a measure of poverty, but not so the former. The latter view, that is, would defer to a Focus Axiom or what, in more general terms, Broome (1996) refers to as a ‘Constituency Principle’ of population ethics: the principle that, in comparing the ‘goodness’ of alternative states of the world, one takes account only of how good the states are for the *relevant constituency of individuals,* namely those individuals only—such as those that exist in both the states under review—whose preferences and interests can be validly seem to matter for the comparison.

In the context of poverty measurement, it is arguable that the poverty ranking of alternative distributions must depend solely on the interests and preferences of the *poor* constituency of the population. What is important to note is that if such a view is to be defended, it must be defended in its entirety, that is to say, one must defer to what in Sect. 5.2 has been labelled a *Comprehensive Focus Axiom,* one which respects both Income Focus *and* Population Focus. Alternatively, one may reject both the Income Focus and the Population Focus Axioms. An index that satisfies Comprehensive Focus is any standard measure of poverty which incorporates the headcount ratio, such as the Sen Index, multiplied by the total population: the headcount ratio in the expression for the Sen Index would then be replaced by the aggregate headcount (call it *A*), and the resulting measure (call it *S' = A [I* + (1 - *I)G ^{P}])* would defer to both Income Focus and Population Focus.

^{[1]}An example of a measure which

*violates*both Income Focus and Population Focus is Anand’s (1977) modification of the Sen Index, given, for ‘large’ numbers of the poor, by the expression

*S''*= H[I* + (1 - I*)G

^{P}] where I* is a modified income-gap ratio which measures the shortfall of the average income of the poor from the poverty line as a proportion of the average income of the entire population rather than of the poverty line

*(I**= 1 -

*yf/p,*and i is the average income of the entire population). Without entering into the substantive merits of a Constituency Principle, one may still pronounce on a matter of consistency, as such: namely, that it would be consistent to violate both Income and Population Focus, or to respect Comprehensive Focus, but inconsistent to defer to one of the Focus Axioms while violating the other. In this sense, the measure

*S'*is a consistent measure (in that it satisfies both Income and Population Focus), just as the measure S" is also a consistent measure (in that it violates both Income and Population Focus), whereas, unfortunately, most extant measures of poverty are inconsistent, in that they tend to insist on the sanctity of Income Focus, while apparently seeing no case for Population Focus. It is this inconsistency which is at the heart of the impossibility results subsumed in Propositions 5.1-5.4: Replication Invariance and Maximality are properties of a poverty measure which uphold a ‘how poor a society is’ view of poverty, while Population Focus is a property that upholds a ‘how much poverty there is in a society’ view of poverty. Combining these conflicting views of poverty inevitably leads to incoherence.

An issue that is directly precipitated by the above considerations has to do with the rival claims of the headcount ratio (H) and the aggregate headcount (A) as the appropriate indicator of the prevalence of poverty. This problem has been considered in Subramanian (2005a, b), and in Chakravarty et al. (2006). Perhaps one of the earliest efforts at dealing with the problem from a conceptual perspective is to be found in related work done by Arriaga (1970) on the measurement of urbanization. As pointed out in Subramanian (2005a, b), the headcount ratio violates, and the aggregate headcount satisfies, the Constituency Principle; on the other hand, the headcount ratio satisfies, and the aggregate headcount violates, what one may call a ‘Likelihood Principle’, which is the principle that an assessment of the extent of poverty in a population should carry some indication of the probability of encountering a poor person in that population. Thus, arguably, each of *H* and A has something to commend it, but each also has something to detract from it. Under the circumstances, it may always be best, in empirical work dealing with the prevalence of poverty, to report on both the headcount ratio *and* the aggregate headcount. This is not a particularly common practice, but two notable exceptions are reflected in the work of Sundaram and Tendulkar (2003) and Reddy and Miniou (2007).

An alternative to providing a disaggregated picture of the headcount ratio and the aggregate headcount is to *combine* the two indices in a composite headcount indicator of poverty. Examples of this approach are available in Arriaga (1970), Chakravarty et al. (2006), and Subramanian (2005a). The last-cited work advances an axiom of *‘Flexible Replication Responsiveness’* (Axiom FRR), in terms of which a k-fold replication of an income distribution induces a k^{p}—fold increase in the extent of measured poverty, where b is a parameter in the interval (0, 1): the closer b is to zero, the closer the FRR Axiom is to Replication Invariance; and the closer b is to unity, the closer the FRR Axiom is to Replication Scaling. If we pitch p at the mid-point (1/2) of the unit interval, then a ‘compromise headcount index’ which combines the headcount ratio and the aggregate headcount in a ‘mixed’ measure is given—under some reasonably undemanding axiomatic restrictions—by the quantity *M* = a2(1 + H), a measure which has been advanced and discussed in Subramanian (2005a). A possibly useful feature of the measure M is that when two income distributions are indistinguishable in terms of the headcount ratio, M ranks the distributions according to the aggregate headcount; and when two distributions are indistinguishable in terms of the aggregate headcount, M ranks the distributions according to the headcount ratio.

The competing appeals of H and *A* are, in the end, only a specific manifestation of the more general conceptual difficulties that preside over an appropriate interpretation of what it means to measure ‘the extent of poverty’ in situations—which are the rule rather than the exception—wherein poverty comparisons have to be effected across populations of variable size. This section has provided a summary of some of these difficulties relating to poverty measurement and population ethics. A similar exercise is undertaken, in the following section, on problems relating to inequality measurement and population ethics.

- [1] This, obviously, is also true for the well-known P„>0 family of poverty measures due to Fosteret al. (1984). A distinguished member of this family is the P2 index, given, for all x 2 X and z 2 S,as we have seen earlier (in Sect. 5.3), by: P2(x; z) = H(x; z)[I2(x; z) + (1 — I(x; z))2CP(x; z)].This is a measure of ‘how poor’ a society is. A corresponding measure of the ‘quantity ofpoverty’ in a society would be given by: P2'(x;z) = A(x;z)[I2(x;z) + (1 - I(x;z))2CP(x;z)]: allone has to do to derive P2' from P2 is to replace the headcount ratio by the aggregate headcount.