On Some Problems of Variable Population Poverty Comparisons

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Carnegie Mellon University Research CMU Department of hilosophy Dietrich College of Humanities and Social Sciences 2010 On Some roblems of Variable opulation overty Comparisons Nicole Hassoun
Carnegie Mellon University Research CMU Department of hilosophy Dietrich College of Humanities and Social Sciences 2010 On Some roblems of Variable opulation overty Comparisons Nicole Hassoun Carnegie Mellon University S. Subramanian Madras Institute of Development Studies Follow this and additional works at: art of the hilosophy Commons This Working aper is brought to you for free and open access by the Dietrich College of Humanities and Social Sciences at Research CMU. It has been accepted for inclusion in Department of hilosophy by an authoried administrator of Research CMU. For more information, please contact Working aper No. 2010/71 On Some roblems of Variable opulation overty Comparisons Nicole Hassoun 1 and S. Subramanian 2 June 2010 Abstract This paper demonstrates that the property of replication invariance, generally considered to be an innocuous requirement for the extension of fixed-population poverty comparisons to variable-population contexts, is incompatible with other plausible variable- and fixed-population axioms. Keywords: variable populations, fixed populations, Replication invariance, population focus, income focus, impossibility result JEL classification: I32, D63, D30 Copyright UNU-WIDER Carnegie Mellon University, hilosophy Department, ittsburgh, 2 Madras Institute of Development Studies, Chennai, This study has been prepared within the UNU-WIDER project on New Directions in Development Economics. UNU-WIDER gratefully acknowledges the financial contributions to its research programme by the governments of Denmark (Royal Ministry of Foreign Affairs), Finland (Ministry for Foreign Affairs), Sweden (Swedish International Development Cooperation Agency Sida) and the United Kingdom (Department for International Development DFID). ISSN ISBN Acknowledgements The authors would like to thank John Weymark, Teddy Seidenfeld and John Mumma for comments. The World Institute for Development Economics Research (WIDER) was established by the United Nations University (UNU) as its first research and training centre and started work in Helsinki, Finland in The Institute undertakes applied research and policy analysis on structural changes affecting the developing and transitional economies, provides a forum for the advocacy of policies leading to robust, equitable and environmentally sustainable growth, and promotes capacity strengthening and training in the field of economic and social policy making. Work is carried out by staff researchers and visiting scholars in Helsinki and through networks of collaborating scholars and institutions around the world. UNU World Institute for Development Economics Research (UNU-WIDER) Katajanokanlaituri 6 B, Helsinki, Finland Typescript prepared by Liisa Roponen at UNU-WIDER The views expressed in this publication are those of the author(s). ublication does not imply endorsement by the Institute or the United Nations University, nor by the programme/project sponsors, of any of the views expressed. 1 Introduction Much of the canonical work in poverty measurement considers desirable properties of poverty indices in the context of populations of fixed sie. Recently, however, several scholars, motivated by philosophical work on population ethics,1 have demonstrated that some of these properties have implausible consequences in variable population contexts (Kundu and Smith 1983; Subramanian 2002, 2005a, 2005b; axton 2003; Chakravarty, Kanbur and Mukherjee 2006; Kanbur and Mukherjee 2007; Hassoun 2009). It is a standard feature of the poverty measurement literature to invoke the replication invariance axiom, which allows poverty comparisons in fixed-population contexts to be extended to poverty comparisons in variable population contexts. Replication invariance is the purportedly unexceptionable requirement that the extent of poverty in a situation should remain the same if the population is replicated any number of times. Many of the common measures of poverty satisfy replication invariance because they incorporate within themselves the headcount ratio (or the proportion of the population in poverty).2 The axiom occupies a central place in all of distributional, welfare, and poverty analysis: phenomena such as Loren dominance, generalied Loren dominance, and Stochastic dominance which are essential ingredients of the sorts of assessments under review depend for their meaningfulness on acceptance of the replication invariance axiom. (With specific reference to the context of poverty comparisons, the test of Stochastic dominance is employed to verify that an unambiguous poverty ranking of distributions can be effected one that does not, for instance, depend on what particular poverty line, or what particular poverty index, is used.) This paper argues, however, that the replication invariance axiom may be problematic. Others have made similar points (see the earlier-cited work by Subramanian 2002; axton 2003; Chakravarty, Kanbur and Mukherjee 2006; Hassoun 2009).3 Specifically, it turns out that, if one accepts certain other canonical axioms for poverty measurement, one would be compelled to choose between replication invariance and some sort of population focus axiom, for it is impossible to hold them all. The population focus axioms at issue suggest that the magnitude of poverty ought not to be sensitive (in various ways) to additions to the non-poor population of a society. Variations of this desideratum are reflected in Subramanian s (2002) strong focus axiom, in axton s (2003) poverty non-invariance axiom, and in Hassoun s (2009) no mere addition axiom. To see why one might be disposed favourably towards some population focus axiom, consider an argument in favour of the strongest version of this axiom on which changes in the non-poor population do not affect poverty in any way. One should take this 1 For a canonical text on population ethics, see arfit (1984). Two other important contributions are: Blackorby and Donaldson (1984) and Blackorby, Bossert and Donaldson (2005). 2 A prominent exception is the income-gap ratio, which measures the proportionate shortfall of the average income of the poor from the normative income level that separates the poor from the nonpoor. 3 The axiom also acquires salience in the context of welfare and inequality comparisons across populations of different sies (as discussed in, among other works, Blackorby and Donaldson (1984); Bossert (1990); Trannoy and Weymark (2009)). In this connection, see also Subramanian (2010). 1 population focus axiom seriously for the same reason, we suggest, that what is popularly known as the focus axiom has been traditionally taken so seriously in the poverty measurement literature. The focus axiom is really an income focus axiom, which demands (in a fixed-population context) that measured poverty ought not to be sensitive to increases in non-poor incomes. The rationale for this requirement is that poverty is a feature of the poor, and not of the general population: in making poverty comparisons, one ought, in this view, to focus attention only on the income distribution of the poor. If there is merit in this contention, its scope ought to extend also to a population-focus view of the matter. That is to say, if poverty is a characteristic of the poor, then additions to the non-poor population exactly like additions to non-poor incomes ought not to affect the magnitude of poverty. Both income-focus and population-focus requirements can then be seen as being allied to Broome s (1996) constituency principle in population ethics, a principle which, in the present context, would require assessments of the extent of poverty to be based exclusively on information regarding the constituency of the poor. If this argument is persuasive, and one accepts any of the canonical (fixed-population) axioms for poverty measurement relied upon in the elementary inconsistency proofs that follow, then there is reason to question any routine acceptance of replication invariance. 2 Concepts An income distribution is an n-dimensional vector x = ( x 1,..., x i,..., x n ), in which the typical element x i stands for the (non-negative) income of person i in a community of n individuals, n being a positive integer. The poverty line, designated by, is a positive level of income such that anybody with an income less than is labelled poor.4 The set of all individuals whose incomes are represented in the distribution x is N (x), while, given that the poverty line is, Q( x ; ) is the set of poor people in x, that is, Q( x ; ) { i N( x) xi }. For any x, and any 0, the vector of poor incomes is designated by x, while the vector of non-poor incomes is designated by R x. If X n is the set of all n-dimensional income vectors, then the set of all conceivable income vectors is given by X X. n= 1 n We shall let Π stand for a weak binary relation of poverty defined on X : specifically for all x, y X, we shall write xπ y to signify that there is no more poverty in x than in y. The asymmetric and symmetric factors of Π are represented by Π and Π respectively: x, y X : xπy [ xπy & ( yπx)] ; and x, y X : xπy [ xπy & yπx]. 4 This is what Donaldson and Weymark (1986) call the weak definition of the poor, which excludes an individual on the poverty line from the count of the poor. 2 That is, we shall write xπ y to signify that there is less poverty in x than in y, and xπ y to signify that there is exactly as much poverty in x as there is in y. It will be assumed that Π is reflexive ( x X : xπx ) and transitive ( x, y, X : xπy & yπ xπ ). We shall not, however, insist that Π be complete (which is the requirement that x, y X : xπy or yπ x ). Π, in other words, will be assumed to be a quasi-order. Throughout, we shall assume that the binary relation Π is anonymous, that is, for any given poverty line and all x, y X, if x is merely a permutation of y, then xπy - which is just another way of saying that poverty assessments are impervious to the personal identities of individuals. The anonymity axiom enables variable population (either cross-section or time-series) poverty comparisons to be performed in such a way as to suggest that, of a pair of distributions under comparison, one distribution can be seen to have been derived from the other through a population increment or decrement. 3 Axioms for poverty comparisons resented below is a set of axioms for comparisons of poverty across both fixed and variable populations. Since the axioms are generally well-known, and in any event their import is reasonably clear, we shall not spend much time in explaining their meaning and significance. It will be taken as read that in everything that follows, is a strictly positive scalar. Also, Π will be taken to belong to the set R of all anonymous quasiorders. Income-focus For all, and for all x, y X, if N ( x) = N( y) and x = y, then xπ y. This axiom is usually just called the focus axiom. It requires that any change in nonpoor incomes that leaves the numbers of individuals on either side of the poverty line unchanged, ought not to have any impact on the extent of poverty. Monotonicity For all, and for all such that x, y X, if N ( x) = N( y) and x = y i N( x) \ { j} for some j j Q(y ; ) & x j y, then xπ y. j The monotonicity axiom states that, other things equal, an increase in a poor person s income should reduce poverty. Transfer For all, and for all x, y X, if N ( x) = N( y), and xi = yi i N( x) \ { j, k} where j and k are such that j Q( y; ), k N \ Q( x; ), x y + δ, x = y δ, and 0 k j δ ( y y ) / 2, then xπ y. i i j = j k k The transfer axiom presented here is weaker than Donaldson and Weymark s (1986) weak downward transfer axiom. As we have stated it, the transfer axiom demands that, 3 ceteris paribus, a mutually rank-preserving transfer of income from a non-poor person to a poor person that keeps the former non-poor, should reduce poverty. Replication invariance For all, for all xπ y. x, y X, and any positive integer k, if y = (x,...,x) and n(y) = kn(x), then Again, replication invariance requires the extent of poverty to remain unchanged by any k-fold replication of a population. Weak poverty growth For all, and for all x, y X, if x is the q-vector (x o,, x o ) for some non-negative real number x o and positive integer q; R x = R y {}; then xπ y. y is the (q+1)-vector (x o,..., x o ); and This axiom was introduced in Subramanian (2002). It is a diluted version of the Kundu and Smith (1983) poverty growth axiom. It requires that in a situation where there is at least one non-poor person and all the poor have the same income,5 the addition of another poor person with this same income ought to increase poverty. Non-poverty growth For all, and for all x, y X, if R R x = y and y = ( x, x) (for any x ), then yπ x. This axiom is due to Kundu and Smith (1983). It stipulates that poverty should decline with the addition of a non-poor person to the population. Weak population-focus For all, and for all [ yπx]. x, y X, if R R x = y and y = ( x, x) (for any x ), then This axiom, which is diametrically opposed in spirit to non-poverty growth, corresponds to what Hassoun (2009) calls the no mere addition axiom : it says that poverty is not reduced by changes in the non-poor population which leave the income distribution amongst the poor unchanged. opulation-focus For all, and for all x, y X, if R R x = y and y = ( x, x) (for any x ), then xπ y. 5 The requirement of at least one person being non-poor is a weakening intended to give the headcount ratio some chance of surviving a strengthened version of the axiom: if everybody is poor, then the headcount ratio will be unvarying, at unity, irrespective of the dimensionality of the income vector. The additional stipulation that all the poor (including the addition to the population) have the same income, is to avoid any ambiguity that might arise from the additional poor person having an income greater than the initial average income of the poor: in this event, the headcount ratio would increase and the income-gap ratio would decline, and in principle, it may not be straightforward to suggest that poverty, on net, has increased. (See Subramanian 2002 for a discussion.) 4 opulation-focus is a strengthened version of weak population-focus, and corresponds to what axton (2003) calls the poverty non-invariance axiom. It requires an addition to the non-poor population to leave the extent of poverty unchanged (and not just to leave it un-reduced). Motivationally, it is very much in the spirit of the standard (income-) focus axiom, which says poverty is not affected by changes in non-poor incomes which leave the income distribution amongst the poor unchanged. 4 Three impossibility results We first present and prove three propositions on the existence of quasi-orders satisfying specified sets of variable- and fixed-population properties, and then comment on these results. roposition 1. There exists no Π R which satisfies replication invariance (RI), weak poverty growth (WG), and weak population-focus (WF). roof. Let = 2. Consider the income distributions a = (1,3), b = (1,3,3) and c = (1,1,3,3). By WG, By RI, bπ c. (1.1) cπ a. (1.2) From (1.1) and (1.2), transitivity of Π dictates: bπ a. (1.3) However, by WF, [ bπa]. (1.4) (1.4) contradicts (1.3). roposition 2. There exists no Π R which satisfies monotonicity (M), replication invariance (RI), and weak population-focus (WF). roof. Let = 2. Consider the income distributions a = (1,1), b = (1,3) and c = (1). By M, By RI, bπ a. (2.1) aπ c. (2.2) 5 Transitivity, in conjunction with (2.1) and (2.2), yields bπ c. (2.3) However, WF requires that [ bπc], (2.4) and we have a contradiction. roposition 3. There exists no Π R which satisfies transfer (T), replication invariance (RI), and population-focus (F). roof. Let = 2. Consider the income distributions a = (1), b = (1,3), c = (1,3,3), d = (1,1,5), and e = (1,1). By F: and aπ b ; (3.1) bπ c. (3.2) Given (3.1) and (3.2), by transitivity of Π over the triple { a, b, c} : By T: aπ c. (3.3) cπ d. (3.4) Given (3.3) and (3.4), by transitivity of Π over the triple { a, c, d} : By F: aπ d. (3.5) dπ e. (3.6) Given (3.5) and (3.6), by transitivity of Π over the triple { a, d, e} : aπ e. (3.7) RI, however, dictates that: aπ e, (3.8) in contradiction of (3.7). 6 Remark 1. roposition 1 is (effectively) a slightly strengthened version of one in Subramanian (2002), which (again effectively) shows that it is impossible to combine weak poverty growth, replication invariance and population-focus note that weak population-focus is weaker than population-focus. (Also, Subramanian employs a realvalued measure of poverty, whereas here we employ only a quasi-order on which more in Remark 3.) Remark 2. It is interesting to note that Kundu and Smith (1983) came very close to proving our roposition 1. The Kundu-Smith impossibility theorem asserts that there is no real-valued representation of poverty which simultaneously satisfies three properties, which they call, respectively, upward transfer, poverty growth, and non-poverty growth. In the course of commenting on their result, Kundu and Smith observe, in footnote 7 of their paper (1983: 431), that their impossibility outcome would be retained if nonpoverty growth were replaced by replication invariance. This, as they point out, is because poverty growth and replication invariance together imply non-poverty growth. [To see this, consider again the example presented in the proof of roposition 1: by poverty growth (which is a strong version of what we have called weak poverty growth and requires, simply, that, other things equal, an addition to the poor population should increase poverty), bπ c, and by replication invariance, cπ a, whence, by transitivity, bπ a - which is exactly what non-poverty growth demands.] From here, it is but a single short step to the impossibility result of roposition 1: our weak population focus axiom is in direct opposition to non-poverty growth, and thus our roposition 1. Remark 3. In regard to ropositions 3 and 4, it is relevant to note that Sen s (1976) seminal work, reflected in the quest for income-responsive and distribution-sensitive poverty measures, was motivated precisely by the failure of the headcount ratio to satisfy fixed-population axioms like monotonicity and transfer. However, in a variable population context, the headcount ratio is the archetypal replication invariancesatisfying measure, and this note has shown that when we employ a population-focus axiom, it is impossible to hold replication invariance and monotonicity (roposition 2) or transfer (roposition 3). Hassoun (2009) suggests that similar results hold for specific real-valued poverty indices which incorporate the headcount ratio: she shows that a decline in a poor person s income, or a regressive transfer of incomes between two poor persons, is nevertheless compatible with a decline in poverty if these changes are accompanied by a sufficiently large decline in the headcount ratio owing to an increase in the non-poor population. Remark 4. In respect of all three of our results, it should be reiterated that these impossibilities are, in one important way, different from the Kundu-Smith impossibility mentioned above (Kundu and Smith 1983). The Kundu-Smith theorem points to a representational problem, not an ordering problem.6 As the authors state explicitly (Kundu and Smith 1983: 429): the weight of the argument rests directly on the structural properties of the real number system. We ourselves do not insist on anything as demanding as real-valued representation: the fact that there does not exist even a quasi-order satisfying sets of specified properties points to a much more foundational conflict in the axiom systems we employ. No doubt our results are mathematically very 6 Although Subramanian employs a real-valued measure of poverty, his result does not reflect an essentially representational problem. 7 simple a
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