Consistent Comparison of Pro-Poor Growth

Transcription

1 Consistent Comparison of Pro-Poor Growth Buhong Zheng Department of Economics University of Colorado Denver Denver, Colorado, USA April 2009 Abstract: This paper contributes to the literature of pro-poor growth measurement by introducing a growth-rate consistency axiom. The axiom states that if one growth pattern is judged to be more pro-poor than another growth pattern at a given growth rate, then the pro-poor ranking between the two growth patterns should remain the same at a higher growth rate. We show that summary pro-poor measures such as povertygrowth elasticities may violate this axiom. We then characterize a special dominance condition under which a given summary pro-poor measure will satisfy the growth-rate consistency axiom. Finally we establish a general growth-rate dominance condition under which all summary propoor measures will respect the growth-rate consistency axiom. JEL Classi cations: I32 Key Words: Pro-poor growth, growth-rate consistency, decomposable poverty measures, minimum distribution-sensitivity, generalized TIP curve. 1

2 Consistent Comparison of Pro-Poor Growth I. Introduction The Millennium Development Goals (MDGs) put forward by the United Nations General Assembly in 2000 declare poverty reduction as an important objective for economic development. This declaration has promoted economists and policy-makers to re-examine the link between economic growth and poverty-reduction and to identify e ective poverty-reduction policies. It is generally acknowledged that economic growth is a necessary but not a su cient condition for poverty reduction; a positive economic growth does not necessarily bene t the poor as the rich may absorb all the gains from growth. To evaluate how economic growth may bene t the poor, the new term pro-poor growth has been introduced and a growing literature has since been devoted to the characterization of this new concept and its measurement. 1 The concept of pro-poor growth, however, has been interpreted and measured di erently. To Kakwani and Pernia (2000), for instance, a growth pattern is propoor if the poor s income grows at a faster rate than the non-poor s does. In the literature, this de nition is referred to as the relative approach. The opposite view is represented by Ravallion and Chen (2003). According to them, a growth pattern will be pro-poor if and only if the poor s income grows regardless of how much the nonpoor may have gained. This de nition is referred to as the absolute approach. An intermediate de nition that is a combination between the two opposing de nitions has also been introduced (e.g., Osmani, 2005). The di erence among theses approaches lies in the choice of a benchmark for comparison. The benchmark for the relative approach is the so-called inequality-neutral growth, i.e., all incomes grow at the same rate. The benchmark for the absolute approach is the situation of zero growth for the poor regardless of what has happened to the non-poor. In contrast, the benchmark for the intermediate approach must re ect the absolute magnitude of poverty reduction, yet contain an element of bias in favor of the poor (Osmani, 2005). Clearly, how such a benchmark can be established is not a simple matter and is certainly subject to debate. These di erent approaches have given rise to various pro-poor measures. Mirroring the literature of poverty measurement, all these measures fall into two broad categories: summary indices and partial-ordering conditions. The measurement of pro-poor growth di ers, nevertheless, from the standard poverty measurement in a signi cant way the use of elasticity in measuring the pro-poorness of a growth pattern. In comparing di erent growth patterns, poverty elasticities with respect to mean income are computed and compared. A poverty-growth elasticity indicates the percentage decrease in poverty as the mean income increases by one percent. Clearly, the use of poverty-growth elasticity enables the comparison of growth patterns with 1 The new literature includes, for example, Kakwani and Pernia (2000), Ravallion and Chen (2003), Son (2004), Kakwani and Son (2006, 2008), Kraay (2006), Duclos (2009) and Essama-Nssah and Lambert (2009) which also provides a useful survey of the literature. 2

3 di erent rates of growth since all di erent patterns are normalized to have one-percent growth rate. One of the purposes of this paper is to point out a di culty with the use of poverty-growth elasticity as a pro-poor measure in assessing and comparing di erent growth patterns. The di culty is that a poverty-growth-elasticity pro-poor measure may be inconsistent in comparing growth patterns with respect to di erent growth rates. For instance, suppose a country faces two alternative growth patterns with the same growth rate (say ve percent). At this growth rate, suppose pattern A is judged to be more pro-poor than pattern B. Now if the growth rate is doubled (ten percent) while growth patterns remain inequality-neutral (i.e., the growth rate at each income level is doubled), will pattern A remain more pro-poor than pattern B? The answer should be in the a rmative. Yet, in certain circumstances, a poverty-growth-elasticity measure may reverse the direction of pro-poor comparison: at the higher growth rate, pattern B may become more pro-poor than pattern A. The second purpose of this paper is to introduce and characterize a growth-rate consistency condition. A main result of this paper is that for all possible growth patterns, no poverty-growthelasticity measure can be growth-rate consistent. The paper then investigates the circumstances in which a poverty-growth-elasticity measure can be used consistently. For the most commonly used decomposable poverty measures, the condition is the generalized TIP dominance that Essama-Nssah and Lambert (2009) recently de ned. Finally, this paper de nes a censored growth-rate curve and characterizes the growthrate dominance as the most general condition for growth-rate consistent comparison. II. Poverty-Growth Elasticity and Growth-Rate Consistency In this paper, we consider discrete income distributions with a xed population size n 2. The results can certainly be extended to continuous distributions and to distributions with variable sizes. Let x = (x 1 ; x 2 ; :::; x n ) 2 R n ++ be an income distribution where x i is ith individual s income. Without loss of generality, we assume that all incomes in a distribution are sorted in the increasing order, i.e., x 1 x 2 ::: x n. The mean income of x is denoted (x). The poverty line z 2 R ++ is pre-given and is the same for all distributions under consideration. A poverty measure P (x; z) indicates the poverty level associated with distribution x. All poverty measures considered in this paper are of the additively decomposable type. An additively decomposable poverty measure is P (x; z) = 1 n nx p(x i ; z) (2.1) where p(x; z) is the individual deprivation function with p(x; z) > 0 for x < z and p(x; z) = 0 for x z. We further require that p(x; z) be twice di erentiable with 3

4 respect to x for x < z with p x < 0 and xx p(x;z) > 0. 2 Examples 2 decomposable poverty measures are the Watts measure with p(x; z) = ln z ln x, the Clark et al. measure with p(x; z) = 1 [1 (x=z) ], the Foster et al. measure with p(x; z) = (1 x=z) ( 2), and the Kolm-type constant-distribution-sensitivity (CDS) measure with p(x; z) = e (z x) More-pro-poorness and the growth-rate consistency axiom Suppose economic growth takes place upon a given income distribution x. Consequently, the poverty level of x changes. If economic growth transforms x into w = (w 1 ; w 2 ; :::; w n ) with mean (w) >(x), 3 then for a given poverty measure P (x; z), the change in poverty as measured by the poverty-growth elasticity is (e.g., Essama-Nssah and Lambert, 2009) " P (x; w) = [P (w; z) P (x; z)]=p (x; z) [(w) (x)]=(x) = [P (w; z) P (x; z)]=p (x; z) (2.2) where = (w) (x) is the growth rate. In the literature, " (x) P (x; w) is then used, in combination with the elasticity calculated for the chosen benchmark, to de ne and measure the pro-poorness of economic growth. 4 It is useful for our purpose to note that the use of elasticity makes the actual rate at which the economy grows less relevant; di erent rates are normalized to the base of one percent before comparing pro-poorness. In this paper, we do not wish to address the question what is pro-poor? Instead, we are interested in measuring which is more pro-poor? That is, we investigate the circumstances under which a growth pattern is judged to be more pro-poor than another growth pattern. This exercise is very much in line with Osmani s suggestion that a more common concern will be a comparative one where a particular set of policies is likely to be more pro-poor than another and (s)uch a comparative exercise could command agreement, even if people disagree on the precise identi cation of the benchmark. To avoid using any pro-poor benchmark, we limit ourselves to the situation where all alternative growth patterns are growth-rate neutral, i.e., all postgrowth income distributions have the same mean income. Within this setting, it 2 These two conditions ensure that a poverty measure satis es the monotonicity axiom (poverty is reduced if the income of a poor individual is increased) and the transfer axiom (poverty is reduced by a progressive transfer of income among the poor). For surveys on poverty axioms, poverty measures and poverty orderings, see Zheng (1997, 2000b), Lambert (2001) and, more recently, Chakravarty (2009). 3 In this paper we consider positive economic growth ( > 0). Although all results derived are technically valid for the case of negative growth or contraction ( < 0), it seems an appropriate new term is needed to replace pro-poor growth in that case. 4 The benchmark growth used in the relative approach is the proportional and Lorenz-neutral growth, i.e., all incomes grow at the same rate. Denoting ~" P (x; w) the poverty-growth elasticity for the benchmark case, Kakwani and Pernia (2000) de ne their pro-poor measure as " P (x; w)=~" P (x; w). Using the same benchmark, Essama-Nssah and Lambert (2009) further consider, inter alia, " P (x; w) ~" P (x; w) as a pro-poor measure. 4

5 seems quite reasonable to state that the growth pattern that reduces poverty more is more pro-poor (although the pro-poorness of each pattern can still be subject to debate). Before formally de ning the notion of more-pro-poorness, it is necessary to specify the exact meaning of growth pattern that we refer to in the previous paragraph. Roughly speaking, a growth pattern documents the distribution of individual income gains relative to the overall economic growth. That is, given the initial distribution x which grows into w, individual i s growth pattern is a continuous function of his income growth rate w i = w i x i x i and the overall growth rate. Let this function be f( w i ; ). A property that is pertinent to the notion of growth pattern is that if all incomes of w are multiplied by an > 1 then the growth pattern from x to w 0 = w should be the same as that from x to w. Since wi 0 = (1 + w i )x i and (w 0 ) = (1 + )(x), then f( w i ; ) = ~ f(1 + w i ; 1 + ) = ~ f[(1 + w i ); (1 + )] (2.3) for some continuous function ~ f and all > 1. It follows that 5 f( w i ; ) = ^f( 1 + w i 1 + ) (2.4) for some continuous function ^f. Letting ^f(t) = t, we have the following de nition of growth pattern: De nition 2.1. If x grows into w, the corresponding growth pattern is de ned as w = ( 1 + w ; 1 + w ; :::; 1 + w n 1 + ): With this de nition, we can de ne the notion of more-pro-poorness as follows: De nition 2.2. For a given distribution x, suppose there are two alternative growthrate-neutral growth patterns w and v which lead, respectively, to alternative income 5 If could take any positive value, then (2.4) is easily derived by choosing = In the case of > 1, (2.4) can be established as follows. Let g(; y) = f( ~ y ; y) for all y > 0 and > 0. Then ~f(x; y) = f[x; ~ y] becomes g( y x ; y) = g( y x ; y): De ne r(y; y) 1, then g( y x ; y)r(y; y) = g( y x ; y) which is a Sincov equation whose solution (Aczél 1966, p. 303) implies g(; y) = s(y)() and r(y; y) = s(y)=s(y) for some continuous functions s() and (). But r(y; y) 1, it follows that s(y) = s(y) = c a constant. Thus, f(x; ~ y) = c( y x ). Let ^f() = c(), x = 1 + and y = 1 + i completes the derivation. 5

6 distributions w = (w 1 ; w 2 ; :::; w n ) and v = (v 1 ; v 2 ; :::; v n ) with (w) =(v) >(x). Suppose economic growth does not completely eliminate poverty, i.e., w 1 < z and/or v 1 < z. 6 Then growth pattern w is said to be more pro-poor than growth pattern v according to " P if and only if " P (x; w) < " P (x; v): (2.5) To illustrate the notion of more-pro-poorness, consider the following numerical example. Suppose x = (1; 2; 3; 4; 5), w = (2; 3; 4; 5; 6) and v = (2:2; 2:8; 4; 4:8; 6:2). The poverty line is set at z = 5. The growth rate is = 0:3 and the two growth patterns (from x to w and from x to v) are, respectively, w = ( 3; 9 15 ; 1; ; 9 ) and v = ( 33; 21; 1; 9 ; 93 ). Let the poverty measure be the Watts measure (p(x; z) = ln z ln x). The poverty-growth elasticities for the two growth patterns are " P (x; w) = 1:481 and " P (x; v) = 1:468, respectively. Since " P (x; w) < " P (x; v), growth pattern w is thus more pro-poor than growth pattern v according to the de nition. Now suppose the economy grows at a faster rate while the growth patterns remain unchanged. Will this a ect the comparison of the more-pro-poorness between the growth patterns? In other words, if w is judged to be more pro-poor than v at 0, will w remain to be more pro-poor than v at a rate > 0? Since the growth rate is normalized in measuring pro-poorness, it seems reasonable that the more-pro-poorness should remain unchanged at a higher growth rate provided that poverty is not completely eliminated. To ensure a consistent and meaningful pro-poor comparison, we introduce the following consistency axiom on the notion of morepro-poorness. The growth-rate consistency axiom. For a given income distribution x and two alternative growth patterns w and v, if w is judged to be more pro-poor than v at growth rate 0, then w should remain more pro-poor than v at a higher rate > 0 provided that poverty is not completely eliminated. However, a poverty-growth-elasticity pro-poor measure may violate this consistency axiom. In the previous example, suppose now the growth rate is = 0:4 instead of = 0:3. The two post-growth income distributions become w 0 = 1:05w and v 0 = 1:05v. With the same growth patterns and the same Watts poverty measure, it is easy to verify that " P (x; w 0 ) = 1:347 > 1:367 = " P (x; v 0 ). That is, growth pattern v is now more pro-poor than w. The direction of more-pro-poor comparison is reversed when the economy grows a mere ve percent more! This reversal becomes even more telling when we consider the fact that the estimates of GDP growth rates are often revised upward or downward due to measurement error and in ation adjustment. The question that naturally arises is whether a poverty-growth-elasticity measure associated with a di erent poverty measure such as the Foster et al. measure can 6 If we allow the possibility that economic growth eliminates poverty completely in both distributions, then condition (2.5) needs to be weakened to " P (x; w) " P (x; v). In this case, all following derivations must be adjusted accordingly. 6

7 avoid the afore-described inconsistent situation. Indeed, as we will see later, for the two growth patterns given above, the inconsistency may be due to the speci c poverty measure used (the Watts measure in this case) and not all poverty-growth-elasticity measures will run into the same contradiction. In the following subsection, however, we document a general di culty with using a summary measure such as povertygrowth elasticity in measuring and ranking pro-poorness among growth patterns An impossibility result The general question is: For a given income distribution x, if a poverty-growthelasticity measure shows that v is more pro-poor than w at 0, can the measure also show that w is more pro-poor than v at any > 0 provided that poverty still exists in at least one distribution? Applying the poverty-growth elasticity de nition (2.2), the question amounts to asking that, for a poverty measure P (), if P (w) < P (v) then will P (w) < P (v) for all > 1 provided that P (v) > 0? If P (w) < P (v) always implies P (w) < P (v) for any such, then P (w) is necessarily an increasing function of P (w). In addition, the ranking between P (w) and P (v) is not a ected by the value that takes, thus and P (w) are independent in determining P (w). This reasoning can establish the following lemma: 7 Lemma 2.1. If a poverty measure P (; ) satis es the growth-rate consistency axiom then, for any distribution x 2 R n ++ and all > 1 such that P (x; z) > 0, there exists a continuous function h(; ) such that where h(; ) is also increasing in the second argument. P (x; z) = h[; P (x; z)] (2.6) To delineate the functional form of P (; ), we consider the situation where all individuals live in poverty and economic growth does not change any individual s poverty status, i.e., no individual crosses the poverty line. For a decomposable poverty measure, we then have: Proposition 2.1. For a given > 1 and for all x 2 (0; z=), a poverty measure (2.1) satis es the growth-rate consistency axiom if and only if for some constant t 2 R. p(x; z) = t p(x; z) (2.7) Proof. Since p(x; z) = t p(x; z) implies P (x; z) = t P (x; z) which satis es the growth-rate consistency axiom, it follows that the su ciency of the proposition is veri ed. 7 In formality, the growth-rate consistency axiom is similar to the unit-consistency axiom recently introduced in poverty measurement (Zheng, 2007) and a result similar to (2.6) has also been established there. The proof given in establishing the unit-consistency result can be modi ed to formally prove (2.6). For the sake of briefness, we omit the formal proof. 7

8 To prove the necessity of the proposition, we rst note that the growth-rate consistency of P (x; z) implies the growth-rate consistency of the deprivation function p(x; z). This is true since by setting x 1 = x 2 = ::: = x, P (x; z) collapses to p(x; z). For a given > 1, applying (2.6) to P (x; z) and to each p(x i ; z), we then have h[; 1 nx p(x i ; z)] = 1 nx h[; p(x i ; z)] (2.8) n n for all x i 2 (0; z=). Since for a xed z, p(x; z) is continuous and strictly decreasing in x, it follows that the domain for y i = p(x i ; z) is ( ~ ; ) where = p(z=; z) 0 and = lim x!0 p(x; z) which maybe nite or in nite. By further denoting h() ~ = h(; ), (2.8) becomes ~h( 1 nx y i ) = 1 nx ~h(y i ) n n for all y i 2 ( ~ ; ). The solution to this standard Cauchy equation is ~ h(y i ) = ay i for some constant a 2 R ++. In terms of function h and the deprivation function p, this solution means p(x i ; z) = h[; y i ] = a()p(x i ; z) (2.9) for some continuous function a(). For all possible values of and, (2.9) also entails a() = a()a(): The solution to this Cauchy equation is a() = t for some constant t. Substituting this back into (2.9) completes the proof of the proposition. Clearly, the result developed in the above proposition is also valid when some individuals live above the poverty line as long as no one changes poverty status as a consequence of economic growth. In this situation, the growth-rate consistency axiom actually characterizes a unique poverty measure. For any x < z, xing a x 0 such that x < x 0 < z and letting = x 0 =x in (2.7), we obtain or p(x 0 ; z) = ( x 0 x )t p(x; z) p(x; z) = (z)x t (2.10) for a continuous function (z) = [p(x 0 ; z)x0 t ]. To ensure that p(x; z) satis es p x < 0 and p xx > 0, we need to require t < 0. This characterization is formally reported below as a corollary. Corollary 2.1. If economic growth does not change any individual s poverty status, then a poverty measure (2.1) satis es the growth-rate consistency axiom if and only if the measure is a positive multiple of P (x; z) = 1 n 8 qx (z) x i (2.11)

9 where (z) > 0 is a continuous function, > 0, and q is the number of the poor in the distribution. But in reality, economic growth does lift people out of poverty. Once an individual s poverty status is allowed to change, the poverty measure derived above will necessarily violate the growth-rate consistency axiom. Consider, for example, the case of (z) = z and = 1, i.e., p(x; z) = z=x for x < z. If we also require p(x; z) = 0 for x z as we have assumed in this paper, then it is easy to verify that the numerical example given at the end of the last subsection still provides the needed contradiction. Even if we rede ne p(x; z) = 1 for x z, so that the poverty measure is continuous at the poverty line, the numerical example can be suitably modi ed to provide a counterexample. 8 This impossibility constitutes a main result of this paper. Theorem 2.1. There is no poverty-growth-elasticity measure that can satisfy the growth-rate consistency axiom for all possible growth patterns w and v. The impossibility can be seen directly from condition (2.7) when the individual income crosses the poverty line. For a given income x < z, this happens by choosing an so that x z. In this case, equation (2.7) does not hold since the left side is p(x; z) = 0 while the right side is t p(x; z) which is strictly positive. Even if we drop the requirement p(z; z) = 0 and replace it with p(z; z) = c 6= 0 (and hence p(x; z) = c for x z), it is easy to see that equation (2.7) still cannot hold. Before moving forward, it is important to emphasize that in reaching the above impossibility conclusion we require that growth-rate consistency hold for all possible growth patterns. It is certainly possible to give an example where a poverty-growthelasticity pro-poor measure will always give consistent ranking between two given growth patterns. This means that a poverty-growth-elasticity measure can satisfy the growth-rate consistency axiom if the growth patterns under investigation satisfy certain conditions. In the following section, we turn our attention to the task and identify a situation where the axiom of growth-rate consistency is satis ed A special dominance condition for growth-rate consistency The situation we characterize in this subsection depends on both the growth patterns to be compared and the type of poverty measures to be used. The type of poverty measures in our characterization is determined by the degree of distributionsensitivity. A poverty measure is said to be distribution-sensitive if it satis es the transfer axiom (a progressive transfer of income reduces poverty). For a decomposable poverty measure P (x; z) = 1 n P n p(x i; z), being distribution-sensitive means p x < 0 and p xx > 0 and the degree of distribution-sensitivity is de ned as (Zheng, 2000) s p (x; z) = p xx(x; z) p x (x; z) : (2.12) 8 For example, replace v in the previous example with v 0 = (2:2; 2:8; 4; 4:5; 6:5). 9

10 For two poverty deprivation functions p(x; z) and q(x; z), the second measure is said to be more distribution-sensitive than the rst one if s q (x; z) > s p (x; z) for all x 2 [0; z). In that case, q(x; z) must also be a strictly convex function of p(x; z). The conditions we derive are related to the ordering condition by all poverty measures that are more distribution sensitive than P (; ). For ease of reference, we restate the ordering condition in the following proposition. Proposition 2.2 (Zheng, 2000). For two distributions w and v, the necessary and su cient condition for all poverty measures that are more distribution-sensitive than P (; ) to indicate that w has less poverty than v is that p(w i ; z) p(v i ; z); k = 1; 2; :::; n (2.13) with the inequality holding strictly for some k. Plotting the dominance condition (2.13) graphically, we obtain the generalized TIP-curve dominance that Essama-Nssah and Lambert (2009) recently introduced. With Proposition 2.2, we can now investigate the issue of growth-rate consistency in the pro-poorness comparison. Our second main result of the paper is stated in the following theorem: Theorem 2.2. For a decomposable poverty measure P (; ) and between two distributions w and v, if (i) P k p(w i; z) P k p(v i; z); k = 1; 2; :::; n, with the inequality holding strictly for some k; and (ii) p(w; z) is an increasing and convex function of p(w; z) for all w 2 [0; z=), then P (w; z) < P (v; z) implies P (w; z) < P (v; z) for all > 1 provided that poverty is not completely eliminated in w. Proof. If (i) holds, then by Proposition 2.2 all poverty measures that are more distribution-sensitive than P (; ) will indicate that w has more poverty than v. Clearly, p(w; z) is a continuous and nondecreasing function of p(w; z). This is because if p(w; z) increases for a xed z, then w must decrease and p(w; z) will increase if w < z= and remain unchanged if w z=. Next, p(w; z) can be viewed as a transformation of p(w; z): the latter is transformed into the former for w < z= and zero for w z=. Thus, if condition (ii) holds, then p(w; z) can be regarded as more distribution-sensitive than p(w; z). It follows that P (w; z) < P (v; z) must hold by Proposition 2.2. For all commonly used decomposable poverty measures, the condition that p(x; z) is a convex transformation of p(x; z) for all x 2 [0; z=) is satis ed. Denoting the degree of distribution sensitivity of p(x; z) as s p (x; z), it is easy to verify that for the Clark et al. measure (including the Watts measure as a special case), s p (x; z) = 1 = x s p (x; z); for the Foster et al. measure, s p (x; z) = ( 1)=(z= x) > ( 1)=(z x) = s p (x; z) for > 1; and for the CDS poverty measure, s p (x; z) = > = s p (x; z) for 10

11 > 1. For these poverty measures, the generalized TIP-curve dominance becomes a su cient condition for growth-rate consistency. Corollary 2.2. For a commonly used poverty measure P (; ) such as the Watts, the Clark et al., the Foster et al., and the CDS measures, and between two distributions w and v, if P k p(w i; z) P k p(v i; z); k = 1; 2; :::; n, with the inequality holding strictly for some k, then P (w; z) < P (v; z) implies P (w; z) < P (v; z) for > 1. In theory the condition s p (x; z) > s p (x; z) cannot be justi ed by other existing axioms of poverty measurement. This is because s p (x; z) = p xx (x; z)=p x (x; z) and even if we further assume that p xxx (x; z) < 0 as is usually the case, then together with p x (x; z) < 0 and p xx (x; z) > 0, we have p x (x; z) > p x (x; z) and p xx (x; z) < p xx (x; z) and thus it is not clear whether s p (x; z) > s p (x; z) can hold in general. But intuitively it is appealing to require that p(x; z) be more distribution-sensitive than p(x; z) since any transfer in x is augmented by in x A general dominance condition for growth-rate consistency The conditions developed in the previous section are for a selected poverty measure and the conditions may not be applicable to a di erent poverty measure. In this sense, the conditions are weak conditions for growth-rate consistency. A strong condition can be established by choosing the special poverty measure with p = 1 x=z, i.e., the poverty gap ratio, which does not belong to the class of poverty measures that we considered. Since all distribution-sensitive poverty measures are all more distributionsensitive than the poverty gap ratio, the dominance condition for the measure should imply growth-rate consistency for all distribution-sensitive poverty measures. As pointed out by Essama-Nssah and Lambert (2009), the condition for p = 1 x=z is the normalized TIP-curve dominance which is also equivalent to censored generalized Lorenz dominance (Zheng 2000b). In this section, we characterize this dominance in terms of a comparison between di erent growth patterns. We will also compare it with other proposed pro-poor dominance conditions (Ravallion and Chen, 2003; Son, 2004; Duclos, 2009). Again, suppose the initial distribution x is evolved into either distribution w or distribution v under two alternative growth-rate-neutral growth patterns. Then the growth pattern that leads to distribution w is more pro-poor than the growth pattern that leads to v if min(w i ; z) min(v i ; z); k = 1; 2; :::; n: (2.14) Dividing both sides of the inequality by P k min(x i; z) for each respective k, we have P k min(w P i; z) k P k min(x i; z) min(v i; z) P k min(x ; k = 1; 2; :::; n: (2.15) i; z) 11

12 Denoting ^" i = min(x i ; z)= P k min(x i; z) the cumulative censored share of individual i s income in distribution x and ^ w i = [min(w i ; z) min(x i ; z)]= min(x i ; z) the censored growth rate of w i, then the condition can be expressed as or ^" i (1 + ^ w i ) ^" i^ w i ^" i (1 + ^ v i ); k = 1; 2; :::; n; (2.16a) ^" i^ v i ; k = 1; 2; :::; n: (2.16b) These derivations establish directly our third main result of the paper. Theorem 2.3. For a decomposable poverty measure P (; ) and between two distributions w and v, if condition (2.16a) or (2.16b) is satis ed, then P (w; z) < P (v; z) implies P (w; z) < P (v; z) for all > 1 provided that poverty is not completely eliminated in w. The censored income share ^" i in (2.16) is straightforward to interpret but the censored growth rate ^ w i requires some elaboration on how the growth rate is censored. The relationship between ^ w i and w i depends upon whether or not the poverty line is reached faster in w than in x (or the number of the poor in w is smaller or larger than that in x). Denote the number of the poor in x as q and the number of the poor in w as l, then if l < q, 8 < w ^ w i i l z x i = i x : i l i q ; (2.17a) 0 i q if l > q, 8 < ^ w i = : i q q i l 0 i l w i w i z z : (2.17b) Hence the censoring rule: If income remains below the poverty line before and after the economic growth, then the growth rate is not censored; if income remains above the poverty line before and after economic growth, the growth rate is set to zero; if income crosses the poverty line then the growth rate w i = (z x i )=x i is censored to (z x i )=x i for positive growth and to (w i z)=z for negative growth or contraction. The dominance condition (2.16b) is di erent from all other proposed pro-poor dominance conditions. The condition developed by Ravallion and Chen (2003), applying to our setting of more-pro-poor comparison, would be w i v i ; k = 1; 2; :::; q; (2.18) where q is the number of the poor in distribution x. Since we require all poverty measures to satisfy the condition p x < 0 and p xx > 0, the dominance condition we 12

13 have developed is a second-order condition. Ravallion and Chen s condition is of rst-order and holds for all poverty measures satisfying p x < 0. Our corresponding rst-order condition would be ^ w i ^ v i ; k = 1; 2; :::; n: (2.19) The di erence between (2.18) and (2.19) is clear: our growth rates are censored at the poverty line in each distribution involved while Ravallion and Chen s are truncated only at the poverty line in the initial distribution x. The condition developed by Son (2004), in our terms, would be " i w i " i v i ; k = 1; 2; :::; n: (2.20) That is, growth rates are not censored. Our condition (2.16b) would become (2.20) if the (maximum) poverty line is greater than the largest income in all three distributions considered (i.e., x, w and v). Finally, our condition is di erent from the dominance conditions that Duclos (2009) recently developed. In our more-pro-poor setting in which the benchmark distribution of pro-poor is not relevant, his secondorder condition would lead to " i w i " i v i ; k = 1; 2; :::; q: (2.21) Compare (2.21) with (2.16b), the di erence between his condition and ours is also very clear. III. Concluding Remarks Summary pro-poor measures such as poverty-growth elasticities provide a convenient way to tell whether or not a growth pattern is more pro-poor than another growth pattern. A poverty-growth elasticity normalizes poverty reduction by the rate of economic growth and thus makes pro-poor comparison growth-rate independent. Clearly this independence is reasonable only if some consistency condition over di erent growth rates in pro-poor comparison is satis ed. In this paper, we have introduced an appealing consistency requirement on propoor comparison. But we showed that a poverty-growth-elasticity pro-poor measure will necessarily violate this requirement if all possible growth patterns are considered. We then characterized the circumstances in which the growth-rate consistency axiom is satis ed. These circumstances are represented by two sets of dominance conditions one set is a special condition for a given poverty-growth-elasticity pro-poor measure and the other set is for all measures. An implication of our characterization is that dominance conditions should always be checked in more-pro-poorness comparison 13

14 among di erent growth patterns. If one is contented with using a selected poverty measure, then the special condition corresponding to that measure should be checked. If one wants the result to be robust to all poverty measures, then the general condition should be veri ed. Once the respective condition is satis ed at a given growth rate, then we know that the pro-poor ranking will hold at any higher growth rate and hence the consistency is guaranteed. If a dominance condition fails, then the morepro-poorness of a growth pattern is growth-rate dependent for the respective povertygrowth elasticity measure(s) considered. It is possible to locate the exact ranges of growth rates over which a growth pattern is more (or less) pro-poor than another growth pattern. In pro-poor studies, the location of these ranges should be of interest to practitioners as well as to policy-makers. In our theoretical investigation, we have avoided invoking any pro-poor de nition by assuming a growth-rate-neutral condition. That is, all growth patterns to be compared assume the same rate of growth. In empirical pro-poor comparison among di erent countries or the same country in di erent phases of economic development, the growth rates will likely be di erent. In that case, it seems that one may have to de ne pro-poor as a relative, absolute or some intermediate concept before more-propoorness can be compared. Again, the aim of this paper is not to settle the issue of what is pro-poor?, rather it makes a theoretical contribution about the consistency in the pro-poor comparison. But if the relative view is adopted, the more-pro-poor comparison can be performed by scaling all growth patterns to the lowest growth rate before applying the relevant dominance condition. This is the right procedure since the conclusion will then hold at a higher rate of growth. For other views, however, appropriate mechanisms must be established to make the di erent growth patterns have the same base before the more-pro-poor criteria can be applied. Finally, a word on the axiom proposed. The growth-rate consistency axiom requires the more-pro-poor ranking to hold at all growth rates > 0 if it holds at 0. One may be tempted to extend the axiom to 0 < < 0, and thus to require the ranking to hold for all > 0. We do not intend to generalize the axiom in that direction since the cases of > 0 and < 0 are very di erent and not symmetric. The asymmetry arises from the fact that a poor income and a nonpoor income are treated di erently in poverty measurement an increase in a poor income reduces poverty while an increase in a nonpoor income has no e ect on poverty. Consider three income distributions that originate from x by following the same growth pattern but at di erent growth rates: w (at 0 ), w 0 (at 1 > 0 ) and w 00 (at 2 < 0 ). The growth from x to w 0 (w 00 ) can be decomposed into the growth from x to w and the growth from w to w 0 (w 00 ). Since 1 > 0, the growth from w to w 0 may lift some poor out of poverty but no individual from nonpoor is pulled into poverty. It follows that the information set in the growth from x to w 0 is contained in the information set in the growth from x to w. In other words, one can logically make inference about the change from x to w 0 based upon the change from x to w. In contrast, the (negative) growth from w to w 00 drags some nonpoor into poverty and 14

15 thus the growth from x to w 00 contains more information than the growth from x to w; the exact income and distribution information of the newly poor is not available in w since they were nonpoor there. Consequently, one cannot make inference about the change from x to w 00 based upon the change from x to w. Hence, the axiom is de ned only for > 0. 15

16 References Aczél, J. (1966): Lectures on Functional Equations and Their Applications, Academic Press, New York. Duclos, J. (2009): What is Pro-Poor? Social Choice and Welfare 32, Chakravarty, S. (2009): Inequality, Polarization and Poverty, Springer. Essama-Nssah, B. (2005): A Uni ed Framework for Pro-poor Growth Analysis, Economics Letters 89, Essama-Nssah, B. and P. Lambert (2009): Measuring Pro-Poorness: A Unifying Approach with New Results, The Review of Income and Wealth, forthcoming. Kakwani, N. and E. Pernia (2000): What is Pro-poor Growth? Asian Development Review 18, Kakwani, N. and H. Son (2006): Pro-Poor Growth: The Asian Experience, UNU- WIDER Research Paper No. 2006/56. Kakwani, N. and H. Son (2008): Poverty Equivalent Growth Rate, The Review of Income and Wealth 54, Kraay, A. (2006): When is Growth Pro-poor? Evidence from a Panel of Countries, Journal of Development Economics 80, Lambert, P. (2001): The Distribution and Redistribution of Income, 3rd edition, The Manchester University Press, Manchester and New York. Osmani, S. (2005): De ning Pro-Poor Growth, One Pager Number 9, International Poverty Center, Brazil. Ravallion, M. and S. Chen (2003): Measuring Pro-Poor Growth, Economics Letters 78, Son, H. (2004): A Note on Pro-poor Growth, Economics Letters 82, Zheng, B. (1997): Aggregate Poverty Measures, Journal of Economic Surveys 11, Zheng, B. (2000a): Minimum Distribution-Sensitivity, Poverty Aversion, and Poverty Orderings, Journal of Economic Theory 95, Zheng, B. (2000b): Poverty Orderings, Journal of Economic Surveys 14, Zheng, B. (2007): Unit-Consistent Poverty Indices, Economic Theory 31,