AGGREGATE POVERTY MEASURES

Transcription

1 AGGREGATE POVERTY MEASURES Buhong Zheng University of Colorado at Denver Abstract. The way poverty is measured is important for an understanding of what has happened to poverty as well as for anti-poverty policy evaluation. Sen s (1976) pathfinding work has motivated many researchers to focus on the way poverty should be measured. A poverty measure, argued by Sen, should satisfy certain properties or axioms and the desirability of a poverty measure should be evaluated by these axioms. During the last two decades, many researchers have adopted the axiomatic approach pioneered by Sen to propose additional axioms and develop alternative poverty measures. The objective of this survey is to provide a clarification on the extensive literature of aggregate poverty measures. In this survey, we first examine the desirability of each axiom, the properties of each poverty measure, and the interrelationships among axioms. The desirability of an axiom cannot be evaluated in isolation, and some combination of axioms may make it impossible to devise a satisfactory poverty measure; some axioms can be implied by other axioms combined and so are not independent; some others are ad hoc and are disqualified as axioms for poverty measurement. Based on the interactions among axioms, we identify the core axioms which together have a strong implication on the functional form of a poverty measure. We then review poverty measures that have appeared in the literature, evaluating the interrelationships among different measures, and examining the properties of each measure. The axioms each measure satisfiesviolates are also summarized in a tabular form. Several good poverty measures, which have not been documented by previous surveys, are also included. Keywords. Poverty measurement; axiom; poverty measure; interrelationship; distribution-sensitive; deprivation A decent provision for the poor is the true test of civilization. The condition of the lower orders, the poor especially, was the true mark of national discrimination. (Samuel Johnson, 1770) 1 1. Introduction Poverty is in the news every year, especially when the U.S. Bureau of the Census releases the poverty statistics for the previous year. The problem of poverty in the U.S. became much more publicly recognized with Michael Harrington s (1962) vivid description of the plight of the poor and Lyndon Johnson s declaration of the war on poverty almost three decades ago. The concept of poverty seems straightforward and not in need of elaborate criteria, cunning measurement, or probing analysis (Sen, 1981) in order to recognize or understand it. However, when the focus on poverty is moved away from the extreme situation, i.e., /97/ JOURNAL OF ECONOMIC SURVEYS Vol. 11, No. 2 Blackwell Publishers Ltd. 1997, 108 Cowley Rd., Oxford OX4 1JF, UK and 350 Main St, Malden, BlackwellPublishersLtd.1997 MA 02148, USA.

2 124 BUHONG ZHENG starvation and famine, the answers to many seemingly simple questions may not be readily available and are far from obvious. What exactly do we mean by poverty in countries such as the United States and Western Europe? In what sense do we mean that poverty of a nation has increased over a certain time period? These questions have generated a great deal of controversy, and scholars have devoted many volumes to address the related issues. The first question is generally referred to as the perception of poverty identification of poverty. Research on the second question is called the measurement of poverty the aggregation of individual poverty. The objective of this survey is to provide a clarification of the extensive literature on aggregate poverty measures. 1.1 The importance of the way poverty is measured: the aggregation of poverty Over the past three decades in the U.S., many policies have been designed to help the people on the lower part of the income scale; Aid to Families with Dependent Children (AFDC), Medicaid, and housing subsidies are typical examples. At the same time, many anti-poverty policies have also been abolished for not properly functioning. Here one critical question emerges: how do we measure the efficacy of an anti-poverty policy, or, more fundamentally, how should we measure the poverty level of a community and its changes? The measure that has been used by most countries, including the United States, the United Kingdom, and the United Nations is the headcount ratio the fraction of people below the poverty line. This measure was criticized by Watts (1968) and Sen (1976) for not considering the income distribution of the poor. Sen was a pioneer in the study of poverty measurement. His path-breaking work (1976) has motivated many scholars to focus on the way poverty should be measured. The way poverty is measured is important for an understanding of poverty as well as for policy evaluation, because the efficacy of anti-poverty policies is evaluated by observing the changes in poverty statistics. The headcount ratio and another often-mentioned measure, the income gap ratio, which is the gap between the poverty line and the average income of the poor, may be misleading in indicating poverty status. Consider the comparison of poverty between two regions of the same population size. Imagine first that they have the same number of people below the common poverty line but the poor in the first region have almost no income while the poor in the second region are just marginally poor. It can be well argued that poverty in the second region is much less serious than in the first region. However, the headcount ratio will indicate that the two regions are equivalent in terms of poverty. Also imagine that the first region has only one person below the poverty line while the second has only one above the poverty line but the mean incomes of the poor are the same. In this case it can be strongly defended that the first region should have much less poverty than the second region (assuming there are at least three people in each region). However, the income gap ratio will judge them to have the same poverty level. Any combined measure of the headcount ratio and income gap ratio, though it may avoid the above absurd judgements, fails to reflect the difference between BlackwellPublishersLtd.1997

3 AGGREGATE POVERTY MEASURES 125 two income distributions among the poor. For instance, if two regions have the same number of the poor and the same mean income of the poor, any combined measure will show an equivalence in their poverty levels regardless of how income is distributed. In a case where the poverty line is far away from the hunger line, if income is equally distributed in one distribution but is shared by a few marginally poor in the other distribution, it can be well argued that the one with equal distribution has less poverty. A distribution-sensitive measure, which considers the income distribution of the poor, may present a better picture of poverty intensity and its changes. The headcount ratio and income gap ratio also fail to accurately evaluate the efficacy of anti-poverty policies. The headcount ratio only records those antipoverty policies which bring people out of poverty, i.e., only poverty-eliminating policies rather than any poverty-alleviating policies. As a consequence, the effectiveness of many anti-poverty policies cannot be accurately recorded because most policies such as AFDC are not designed to lift the poor out of poverty. The income gap ratio only record any policies that change the mean income of the poor. It does not distinguish any anti-poverty policies which aim at helping the bottom poor from others that help poor in general (e.g., the housing subsidies). The income gap ratio may also record an increase in the poverty level when some poor are lifted out of poverty or indicate a decrease in the poverty level when the poverty line is adjusted up. For a distribution-sensitive poverty measure, the efforts of any anti-poverty policies, eliminating, alleviating or redistributing, will be reflected in the overall poverty changes. Parallel to the above arguments, the headcount ratio, income gap ratio or their combinations, and the distribution-sensitive measures may imply different antipoverty strategies for an administration. To maximally alleviate poverty has been one of major goals for administrations since the war on poverty began, especially in the early years. A simple problem facing an administration is to minimize the poverty level, given the constraint of limited resources. Assuming no other constraints and that all poor are identical in other characteristics, it can easily be seen that the headcount ratio always has the tendency to channel the aid to the poor person next to the poverty line, i.e., the least needy person first and the most needy person last. The income gap ratio will suggest no specific direction for policies as long as no poor person crosses the poverty line. In contrast, any distributionsensitive poverty measure will channel assistance income to the poorest person first, the least poor last. The scheme works as follows: income is given to the poorest person until his income equals the next poorest person, then income is distributed to these two equal poor persons equally until each person s income equal the third poorest person s income. This pattern is repeated until all assistance income is distributed. Therefore, only distribution-sensitive poverty measures may lead to both horizontally and vertically equitable anti-poverty policies. 1.2 Other issues in poverty measurement The aggregation of poverty is not the only issue in poverty measurement. Other BlackwellPublishersLtd.1997

4 126 BUHONG ZHENG important issues include multi-dimensional poverty measurement and partial poverty orderings which arise from the multiplicity of poverty measures, poverty lines and equivalence scales. The concept of poverty is essentially multi-dimensional, income is only one of many factors in the identification of the poor. If we define poverty as the failure of basic capabilities to reach minimally acceptable levels (Sen, 1992, p. 109), and capacity is a function of income, health, housing, the provision of public goods, etc., then poverty should be measured from a multi-dimensional perspective. In fact, Sen viewed the capability method to be superior to the income method and the income method is at most a second best (Sen, 1981). While the research on the uni-dimensional poverty measurement has flourished in the last two decades, the multi-dimensional measurement of poverty remains an area to be explored. 2 Since the publication of Sen (1976), researchers have adopted the axiomatic approach to formulate new poverty measures. Unfortunately the axiomatic characterization of any particular poverty measure is somewhat ad hoc in that it is either built upon a prior functional form or based upon some arguable axioms. For each set of reasonable axioms there exist multiple poverty measures and poverty comparisons based upon different equally good poverty measures may lead to conflict conclusions about what has happened to poverty. Hence the poverty comparison based upon any single poverty measure is not robust. But if all poverty measures satisfying certain axioms lead to the same conclusion, then the poverty comparison will be much more robust. The first dimension of partial poverty ordering research is to determine the precise circumstances in which two distributions can be unambiguously ranked according to an entire class of poverty measures. The application of a poverty measure requires one to specify the poverty line which distinguishes the group of poor people from the rest of the population. As is well recognized, the choice of any specific poverty line is apt to be arbitrary and so is the poverty comparison based on a single poverty line. The second dimension of partial poverty ordering research is to describe situations in which a measure or a class of measures will indicate a consistent poverty ordering for all possible poverty lines. The measurement of poverty also needs to consider the difference in family composition. Typically this consideration is accomplished by the use of an equivalence scale which determines the relationship between the poverty lines for different types of family. Several methods, including minimum nutritional needs, expert judgement and questionnaires, have been used to determine a proper equivalence scale. However, the choice of a particular equivalence scale, much like the use of a single poverty line, is somewhat arbitrary and so is the comparison based upon a single equivalence scale. The third dimension of partial poverty ordering research is to find conditions under which two distributions can be unanimously ranked by a set of equivalence scales. Atkinson (1987), and Jenkins and Lambert (1993b) focused on the first dimension of partial poverty orderings and provided very useful ordering BlackwellPublishersLtd.1997

5 AGGREGATE POVERTY MEASURES 127 conditions. To some degree, Jenkins and Lambert (1993b) also allowed the poverty lines to be different among different distributions. Foster and Shorrocks (1988a, 1988b) and Foster and Jin (1994) addressed the second dimension of poverty orderings and provide important insights on poverty orderings. Atkinson (1992) and Jenkins and Lambert (1993a) derived useful dominance algorithms for the third dimension of poverty orderings. 1.3 The focus and organization of this survey This survey focuses on the literature of aggregate poverty measures. 3 The discussion is limited to income space or, in other words, we treat poverty as economic poverty as Foster (1984) termed. Also throughout the survey, a single fixed poverty line is assumed and no difference in family composition is considered. Although these make this survey more focused, it is necessary to keep in mind that other issues described in the previous section are also very important in poverty measurement. Twenty years have passed since the publication of Sen s (1976) pioneering work. By now it is widely accepted that poverty measurement needs to consider distribution-sensitivity in addition to counting the number of poor and calculating the average income gap. Following Sen s axiomatic approach, researchers have proposed additional axioms and developed alternative poverty measures which, including the one proposed by Sen, have been applied to address distributional issues in the literature. To date, several survey papers on poverty measurement have appeared in the literature. Among them, Foster s (1984) is very analytical and is the first comprehensive survey which has greatly influenced research on poverty measurement; Seidl s (1988) is the most comprehensive survey in that it also includes a discussion on poverty identification; Chakravarty (1990) surveyed poverty measurement within the broad category of social index number; other surveys such as Sen (1979, 1983 and 1992), Kundu (1981) and Borooah (1991) also provide many useful discussions on poverty measurement. Although some of these surveys are quite comprehensive, a new survey is now needed for the following two reasons. First, the desirability of each axiom, the properties of each poverty measure, and the relationships among axioms and poverty measures are not entirely clear. The clarification of interrelationships and the justification of axioms are very important for poverty measurement study since various axioms (sometimes even conflicting) have been used to construct and evaluate poverty measures. The desirability of an axiom cannot be evaluated in isolation, and some combinations of axioms may make it impossible to devise a satisfactory poverty measure. Second, there have been new developments, such as Foster and Shorrocks (1991), that have important implications for the device of poverty measures. There are also important elements in the literature which have been overlooked by previous surveys. The rest of this survey is organized as follows. Section 2 critically reviews all axioms in the poverty measurement literature and examines the logical BlackwellPublishersLtd.1997

6 128 BUHONG ZHENG interrelationships among different axioms. Based upon the interactions among axioms, this survey classifies all axioms into three groups: the core axioms, the implied axioms and nonrestrictive axioms, and the ad hoc axioms. The core axioms can be regarded as the basic properties a distribution-sensitive poverty measure should satisfy. Section 3 evaluates existent poverty measures, and examines the properties each measure satisfies and the interrelationships among measures. The properties each measure possessesviolates are summarized in a tabular form. This section also includes several good poverty measures that have not been documented in the previous surveys. Section 4 concludes the survey with a brief summary and a discussion of the application of distribution-sensitive measures. 2. Poverty axioms 2.1 Preliminary notation and definitions Consider discrete income distributions represented by vectors, x =(x 1,x 2,,x n ), drawn from the income space D = n =1 D n, where x i D which is some nondegenerate real interval and D n is the set of all n-tuples of elements from D. Without loss of generality, we further assume that the elements of x are pre-sorted in nondecreasing order, i.e., x 1 x 2 x n. For any given poverty line z D and distribution x D, the population can be separated into the poor and the nonpoor. According to Donaldson and Weymark (1986), two different definitions of the poor are possible, depending on how individuals at the poverty line are classified. They defined these two alternatives as the weak and the strong definitions of the poor. Weak definition of the poor: For all x D, the poverty domain is D p (z) {t D t<z}. Strong definition of the poor: For all x D, the poverty domain is D p (z) {t D tz}. Therefore, a person with income x is a poor person if x i D p (z) and a nonpoor person if x i DD p (z). Empirically, the use of one definition or the other may not produce any substantial difference for a fixed and somewhat arbitrary poverty line. 4 Theoretically, however, the choice between the two definitions may affect the properties a poverty measure satisfies. Donaldson and Weymark (1986) showed several impossibilities result when the strong definition of the poor is used; conflicts among some axioms arise when people at the poverty line are counted as the poor. In the literature these two definitions have been alternatively used and no agreement has been reached. In this survey, we suggest the use of the weak definition of the poor and defend that such a definition is consistent with the notion of a poverty line. If bringing all poor incomes to the poverty line is the goal of any poverty-eliminating policy, then the people at the poverty line should not BlackwellPublishersLtd.1997

7 AGGREGATE POVERTY MEASURES 129 be considered as the poor since no effort whatever will be needed to make them nonpoor (any small amount of additional income, say a penny, will do the trick). Hence, unlike Seidl (1988) and Chakravarty (1990), we use the weak definition of the poor throughout the survey and keep all discussions on the strong definition of the poor in footnotes. By so doing, the discussion and presentation are also greatly simplified. The population size corresponding to x is n(x) (or n), the number of poor q(x; z) (or q), the mean income of the poor is µ p (x; z) (or µ p ), and the income variance of the poor is σ 2 (x; z). Several other concepts are often used in the discussion of poverty measurement: Permutation: x D is obtained from y D by a permutation if x=yπ n(x) n(x) for some permutation matrix π n (x) n(x). A permutation matrix has elements of 0 and 1 only and each row and column sums up to one. Replication: x D is obtained from y D by a (k ) replication if n(x)=k n(y) and x =(y,y,,y) for some positive integer k. Simple increment (decrement): x D is obtained from y D by a simple increment (decrement) to a person j if x i =y i for all i j and x j >y j (x j <y j ). Progressive (regressive) transfer: x D is obtained from y D by a progressive (regressive) transfer if there exists i and j, i<j, such that x i y i =y j x j >0 (<0), x j >y i and x k =y k for all k i, j. Relative change: (x; z ) D D is obtained from (x; z) D D by a relative change if (x; z )=λ(x;z) for some positive λ. Absolute change: (x; z ) D D is obtained from (x; z) D D by an absolute change if (x; z )=(x;z)+(λ1 1 n(x) ; λ) for some positive λ, where 1 1 n(x) is a 1 n(x) vector of ones. A poverty measure, according to Watts (1968), is a function of individual incomes and the poverty line. Associated with this concept, the terms poverty value, poverty index, poverty level and the distribution-sensitive poverty measure are also used in the literature. A formal definition of these terms is as follows: 5 Definition: A poverty measure is a function P(x; z): D D2R + whose value poverty value indicates the degree of poverty intensity, or poverty level, associated with the distribution x and the poverty line z, where R + is the nonnegative real number set. Therefore, for a given poverty measure and poverty line, each income distribution is assigned a numerical number poverty index. A distribution-sensitive poverty measure is a poverty measure that satisfies the minimal transfer axiom (to be introduced below). The functional form of a poverty measure depends largely upon what we want to know about poverty. One has to first set up the purpose of measurement, then find a suitable measure within the framework. The axiomatic approach first used by Sen (1976) fits this framework. Following Sen s tradition, scholars specify the BlackwellPublishersLtd.1997

8 130 BUHONG ZHENG properties (or axioms) a desirable measure of the intensity of poverty should satisfy and then search for a measure possessing these axioms. Since 1976, scholars have developed more than a dozen new measures in this tradition. To evaluate different poverty measures, it is necessary to examine and evaluate various axioms and to select the basic axioms for a good poverty measure. 2.2 The axioms, interrelationships and the core axioms The axioms and justifications Sen was the first to formally propose axioms that a poverty measure should satisfy. The first set of axioms he suggested is still the core of poverty measurement today. The first axiom is the focus axiom which he implicitly used in his 1976 paper and explicitly expressed later (Sen, 1981). Focus Axiom: P(x; z)=p(y;z) whenever x D is obtained from y D by an increment to a nonpoor person. This axiom requires a poverty measure to be independent of the income distribution of the nonpoor. The desirability of this axiom, as argued before, rests upon the purpose of one s poverty measurement. If one regards poverty as an absolute deprivation of the poor, as suggested by Sen, then the focus axiom is perfectly appropriate. For other definitions of poverty, the focus axiom may be inappropriate. 6 Certainly, information on nonpoor incomes can be used to determine the poverty cutoff (such as relative poverty line), which is a completely different exercise than what we are pursuing here. 7 Note that this axiom does not assume the number of the nonpoor (therefore, the total population size) to be irrelevant to the poverty measurement. 8 The effect of changes in the nonpoor population size on the poverty value is captured by some other axioms which will be introduced later. Because of this axiom, researchers have frequently used the censored income distribution instead of the income distribution itself in the literature. A censored income distribution sets all incomes above the poverty line to the poverty line itself. 9 Before discussing Sen s other two axioms, we shall introduce several widely recognized axioms. Replication Invariance Axiom: P(x; z)= P( y; z) whenever x is obtained from y by a (k-) replication. Chakravarty (1983a) and Thon (1983b) first introduced this axiom into poverty measurement from the income inequality literature. 10 Because any two differentsized income distributions can be replicated to the same size, their inequality and poverty levels can be directly compared. For this intuitively appealing axiom, it is surprising to find that many early proposed poverty measures (including one of Sen s measures) violate it. BlackwellPublishersLtd.1997

9 AGGREGATE POVERTY MEASURES 131 Restricted Continuity Axiom: P(x; z) is left continuous as a function of x i on D p (z). 11 Continuity Axiom: P(x; z) is continuous as a function of x on D for any given z. 12 One consideration for requiring continuity is the inaccuracy of income data (Donaldson and Weymark (1986)). The restricted continuity axiom is quite reasonable. Given a very small change in a poor person s income, we should not expect a huge jump in the poverty level. The additional content of continuity over restricted continuity is the continuity of P(x; z) at the poverty line. If no one is at the poverty line, continuity and restricted continuity are the same for a focused poverty measure, because a focused poverty measure will always be continuous on the income of the nonpoor according to the focus axiom. Watts (1968) may have been the first to discuss this axiom. He argued that poverty is not really a discrete condition and one does not immediately acquire or shed the afflictions we associate with the notion of poverty by crossing any particular income line. Therefore, it would seem appropriate to maintain the graduation provided by a continuum (p. 325). We may treat his arguments as justifications for both restricted continuity and continuity. 13 Symmetry Axiom: P(x; z)=p(y;z) whenever x D is obtained from y D by a permutation. This axiom says that the names of income recipients do not matter for measuring the intensity of poverty. Symmetry does not impose any real restriction as any aggregate snapshot measure cannot avoid symmetry. 14 This simple axiom enables one to use an ordered income distribution. The second axiom Sen (1976) proposed was the monotonicity axiom which says a drop (increase) in a poor person s income should increase (decrease) the poverty level. The monotonicity axiom has two forms, i.e., the weak monotonicity axiom and the strong monotonicity axiom as Donaldson and Weymark (1986) originally distinguished. 15 Weak Monotonicity Axiom: P(x; z)>p(y;z) whenever x D is obtained from y D by a simple decrement to a poor person. Strong Monotonicity Axiom: P(x; z)<p(y;z) whenever x D is obtained from y D by a simple increment to a poor person. The contents of both monotonicity axioms are very appealing. Other things being the same, a decrease (increase) in a poor person s income should increase (decrease) the overall poverty level. However, these two axioms are not equivalent: strong monotonicity implies weak monotonicity, while the reverse is not always true. This non-equivalence arises in a situation when the increment of a small amount of income to a poor person lifts her out of poverty. In this case, weak monotonicity, together with continuity, imply strong monotonicity. Because continuity is very desirable for a poverty measure, both weak monotonicity and strong monotonicity can be well justified. 16 BlackwellPublishersLtd.1997

10 132 BUHONG ZHENG The third axiom Sen (1976) proposed was the transfer axiom which requires the poverty measure to be sensitive to the redistribution of the income within the poor. 17 Donaldson and Weymark (1986) distinguished four different transfer axioms by incorporating the possible effects and directions of transfers. Minimal Transfer Axiom: P(x; z)<p( y; z) (P(x; z)>p( y; z)) whenever x D is obtained from y D by a progressive (regressive) transfer between two poor persons with no one crossing the poverty line as a consequence of the transfer. Weak Transfer Axiom: P(x; z)<p(y;z) (P(x; z)>p(y;z)) whenever x D is obtained from y D by a progressive (regressive) transfer with at least the recipient (donor) being poor with no one crossing the poverty line as a consequence of the transfer. Regressive Transfer Axiom: P(x; z)>p(y;z) whenever x D is obtained from y D by a regressive transfer with at least the donor being poor. Progressive Transfer Axiom: P(x; z)<p(y;z) whenever x D is obtained from y D by a progressive transfer with at least the recipient being poor. 18,19 The core of these four transfer axioms is that an equalizing transfer (from a richer person to a poor person) should decrease the poverty value, while a disequalizing transfer (from a poor person to a richer person) should increase the poverty value. By definition, minimal transfer is the weakest form among these four axioms while progressive transfer is the strongest form, i.e., progressive transfer regressive transfer weak transfer minimal transfer. The difference between the weak forms (minimal transfer and weak transfer) and the strong forms (regressive transfer and progressive transfer) lies in whether the transfer makes anyone cross the poverty line. The difference between minimal transfer and weak transfer is that minimal transfer restricts the transfers within the poor group (and, of course, no one becomes nonpoor from the transfer) while weak transfer extends to include the transfers between a poor person and a nonpoor person, i.e., it treats monotonicity as a transfer axiom between the poor and nonpoor. In the case of transferring income from a poor person to a nonpoor person, the transferred income is wasted according to the focus axiom, resulting in a pure loss to the poor. Hence weak transfer is equivalent to minimal transfer and weak monotonicity. Progressive transfer differs from regressive transfer in that a progressive transfer may make the recipient nonpoor while regressive transfer cannot imply progressive transfer without the additional assumption of continuity. The contents of these axioms are very appealing. However, the justification is a bit less direct (Foster (1984)). Sen, according to Foster, offered two general lines of argument for the weak form of the transfer axiom. One based upon the comparisons of utility gains and losses in a world where the marginal utility of income is positive but diminishing. The other is made in terms of a notion of relative deprivation: when a regressive transfer takes place from a more deprived BlackwellPublishersLtd.1997

11 AGGREGATE POVERTY MEASURES 133 poor person to a less deprived poor person, in a straightforward sense the overall relative deprivation is increased. (Sen (1981), p. 31). The justification for regressive transfer and progressive transfer has proven to be the most troublesome (Thon, 1983a). Sen originally proposed a version of regressive transfer in his 1976 paper. Later he found that the poverty measure he proposed violated the axiom, and hence he maintained only minimal transfer. 20 Sen (1981, 1982) viewed regressive transfer as a perfectly suitable requirement for an income inequality measure, but less compelling as an axiom for a poverty measure. Any poverty measure satisfying the strong version of the transfer axioms considers the poverty-alleviating role of crossing the poverty line less crucial. However, Sen s arguments can hardly defend the following inconsistency: any transfer from a poor person to a richer poor person (remaining in poverty after the transfer) or to a nonpoor person increases the poverty level, while the transfer may lower the poverty level if the recipient is next to the poverty line and the transfer lifts him out of poverty. As a matter of fact, when continuity is maintained, any poverty measure satisfying weak transfer will also satisfy regressive transfer and progressive transfer (Donaldson and Weymark (1986)). 21 Because continuity and weak transfer have been justified to be very reasonable, both regressive transfer and progressive transfer can therefore be well justified. 22 Subsequently, we use regressive transfer as a basic property for a distribution-sensitive poverty measure. 23 Besides the focus axiom, the monotonicity axiom, and the transfer axiom, Sen (1976) also proposed several other more specific axioms. Because those axioms served only for the formulation of the measure proposed by Sen and have not been widely recognized, we will not discuss them here. For a good discussion on those axioms, see Foster (1984). Kakwani noticed the lack of sensitivity of the Sen measure to the income level of transfer. He argued that a poverty measure should be more sensitive to what happens among the bottom poor. 24 Kakwani (1980a) proposed three sensitivity axioms, two on income transfer and one on income incrementdecrement. Monotonicity Sensitivity Axiom: P(x; z) P(x;z)>P(x;z) P(x;z) whenever x and xd are obtained from y D by the same amount of decrement to poor incomes y i and y j, respectively, where y i <y j. 25 This axiom says that a poverty measure should be more sensitive to a drop in a poor person s income, the poorer the person is. It is interesting to note that this axiom is identical to minimal transfer (Kakwani (1980a)). Therefore, minimal transfer has another interesting interpretation, and the justification for minimal transfer can serve for monotonicity sensitivity as well. Just like the independence between the monotonicity axioms and minimal transfer, monotonicity sensitivity does not necessarily imply weak monotonicity. By definition, monotonicity sensitivity concerns the sign of the difference between P(x; z) P(x;z) and P(x ; z) P(x;z) not the sign of P(x; z) P(x;z) or P(x; z) P(x;z). Therefore, it is not safe to assume that a poverty measure satisfying monotonicity sensitivity BlackwellPublishersLtd.1997

12 134 BUHONG ZHENG must also possess at least weak monotonicity. 26 An example is the Takayama poverty measure, which we will discuss later. Weak Transfer Sensitivity Axiom: P(x; z)>p(x;z) whenever x and xd are obtained from y D by transferring income δ (>0) from poor incomes y i to y j and from poor incomes y k to y l respectively with y j y i =y l y k >δ, y k >y i with no one crossing the poverty line after the transfers. 27 The axiom given here is due to Kakwani (1980a). 28 The basic idea of this axiom is that the poverty assessment should give more emphasis to transfers taking place down in the distribution, other things being equal. Although the transfer sensitivity axiom has been used in measuring income inequality and poverty, a complete definition for it was not given until Shorrocks and Foster (1987). They considered the weak form of transfer sensitivity as placing too many constraints on transfers and relatively few transfers satisfy the requirements (Shorrocks and Foster, 1987). They gave a general definition for transfer sensitivity for the measurement of income inequality, and subsequently introduced it into poverty measurement (Foster and Shorrocks, 1988a). Transfer Sensitivity Axiom: P(x; z)<p(y;z) whenever x D is obtained from y D by a favorable composite transfer (FACT): a progressive transfer of income δ (>0) from y j to y i and a regressive transfer of income ρ (>0) from y k to y l, i.e., x=y+δ(e i e j )+ρ(e l r k ) with σ 2 (x; z)=σ 2 (y;z), y i <y j y k y l <z, and x i x j x k <k l <z. The difference between weak transfer sensitivity and transfer sensitivity is that the former requires P(x; z)<p( y; z) to be true only for all equal-amount and equal-distance transfers among the poor; the latter requires P(x; z)<p( y; z) to be satisfied for any variance-preserving and mean-preserving composite transfer. The amounts of two transfers for transfer sensitivity do not have to be the same and the distances between two pairs of persons involved do not have to be equal. To put it more precisely, the amount of transfer from y j to y i, ρ, the amount of transfer from y k to y l, δ, and y i, y j, ψ λ, y k must observe the following relation: ρ 2 + ρ( y j y i )=δ 2 +δ(y l ψ k ) for y i <y j y k y l <z. (2.1) Although transfer sensitivity is stronger than and theoretically superior to weak transfer sensitivity, the additional content is minimal and a poverty measure satisfying the latter will most likely fulfil the stronger requirement. When evaluating a poverty measure, one may choose to use weak transfer sensitivity instead of transfer sensitivity because the former is more intuitive and is easier to be used to screen poverty measures. One must also exercise caution in interpreting the transfer sensitivity axioms. A transfer sensitive poverty measure does not necessarily satisfy transfer axioms, and transfer sensitivity is not a stronger version of the weak transfer axiom as some authors claimed. This may be a surprise to many casual observers. However, BlackwellPublishersLtd.1997

13 AGGREGATE POVERTY MEASURES 135 as Shorrocks and Foster (1987) noted, it makes little sense to assume a measure is transfer sensitive if transfer axioms are violated. Practically, such a concern does not pose much problem. Note that the above sensitivity axioms are only applicable to comparisons of transfers within the poor where no one crosses the poverty line as a result of the transfers. Following Donaldson and Weymark (1986) one may introduce other axioms involving transfers which change the number of poor. For any focused, continuous poverty measure, weak transfer sensitivity is equivalent to the following stronger axiom: (Stronger) Weak Transfer Sensitivity Axiom: P(x; z)>p(x;z) whenever x and xd are obtained from y D by transferring income δ (0 < δ z y l ) from y i to y j and from y k to y l respectively with y j y i =y l y k >δ, y k >y i and y l <z. The axiom stated above allows a poor person to cross the poverty line as a result of the regressive transfer. However one has to be cautious not to generate weak transfer sensitivity to situations involving transfers between poor and nonpoor because such an axiom may have absurd policy implications: to reduce the poverty level government should always equalize the poor incomes first rather than redistribute income from the rich to the poor! Compared with transfer axioms, (weak) transfer sensitivity demands that a poverty measure be even more sensitive to the income changes among the bottom poor. It seems that no higher level of transfer sensitivity has been proposed in the literature since Kakwani (1980a) and Shorrocks and Foster (1987), though one certainly could do so in a similar fashion. 29 Since a higher level sensitivity axiom requires a measure to be even more sensitive to what happens to the bottom poor, it is an open question as to the extent this sensitivity should be. The suggested justifications for transfer sensitivity (or any higher level of sensitivity) are very meagre and are less convincing than those for the transfer axioms. Furthermore, transfer sensitivity may have somewhat arguable redistribution policy implications: the most effective transfers are those among very poor incomes, and government should equalize the two poorest incomes first and then equalize them with the third poorest income and so on. 30 However, since this axiom is independent of other existing axioms we keep it as a core axiom. Two popular axioms suggested in recent literature are the subgroup consistency axiom and the decomposability axiom. Foster and Shorrocks (1991) discussed the former, while decomposability first appeared in Hamada and Takayama (1977) and was subsequently discussed by Kakwani (1980b) and Foster, Greer, and Thorbecke (1984) in different senses. 31 The necessity for a measure of decomposability arose from practical considerations (for instance, Anand (1977) and van Ginneken (1980)). The decomposability axiom most often used today is due to Foster, Greer, and Thorbecke (1984). Subgroup Consistency Axiom: P(x; z)<p(y;z) whenever x =(x,x)d is obtained from y =(y,y)d with n(x )=n(x), n( y)=n(y) and P(x; z)<p(y;z), P(x; z)=p(y;z). BlackwellPublishersLtd.1997

14 136 BUHONG ZHENG Decomposability Axiom: For x =(x,x)d with n(x)=n(x)+n(x), P(x; z) = n (x ) n(n) P(x ; n z) + (x ) n(x) P(x ; z), (2.2) or equivalently, P(x; z) = 1 n p(x i, z), p(x i, z) = 0 for x i z. (2.3) n(x) i = 1 The subgroup consistency axiom is desirable for a poverty measure. Foster and Shorrocks (1991) compared this axiom to the monotonicity axiom. While the latter is concerned with the change in an individual s poverty status, subgroup consistency is about the change in a subgroup s poverty level. They argued that a subgroup consistent poverty measure is very useful for policy purposes because consistency is needed to coordinate the effects of a decentralized strategy typically involving a collection of activities targeted at specific subgroups or regions of the country. One typical feature of a decomposable measure is that it can decompose overall poverty into that of subgroups according to certain characteristics. A decomposable poverty measure is also subgroup consistent and meets the need of a decentralized strategy towards poverty alleviation. The use of a decomposable measure allows policy-makers to identify subgroups particularly susceptible to poverty and to design effective, consistent national and regional anti-poverty strategies. Furthermore, it can be used to construct profiles of poverty and to evaluate each subgroup s contribution to overall poverty. The marginal contribution of a subgroup s poverty to overall poverty is recorded by its population share the coefficient in (2.2). Foster and Shorrocks (1991) found that a closer link exists between the class of subgroup consistent poverty measures and the class of decomposable poverty measures. They showed that for any subgroup consistent poverty measure that satisfies continuity (may be restricted continuity), replication invariance and is nondecreasing in income, there exists some decomposable poverty measure, P (x; z), and a continuous and increasing function F such that P(x; z)=f[p(x;z)] for all x D. (2.4) A direct, and perhaps the most important, implication of the result in (2.4) is that it shows a direct relationship between subgroup consistent poverty measures and decomposable poverty measures. All decomposable poverty measures are subgroup consistent and all subgroup consistent poverty measures, under some reasonable conditions, are increasing transformations of some decomposable poverty measures. Some may regard decomposability as putting too detailed a restriction on the functional form of a poverty measure rather than on its properties. Foster and Shorrocks (1991) above finding between the subgroup consistent poverty measure and the decomposable poverty measure justifies the use of a decomposable measure given that a poverty measure is unique up to an increasing transformation. BlackwellPublishersLtd.1997

15 AGGREGATE POVERTY MEASURES 137 Recently, subgroup consistency (decomposability) has gained wide recognition in the literature and decomposable poverty measures have become very popular. Nevertheless, Sen (1992, p. 106, n 12) questioned the appropriateness of decomposability. He believed that one group s poverty may be affected by what happens to other groups. Since the additional requirements of subgroup consistency over decomposability are not substantial and not controversial, one can rephrase Sen s criticism as a doubt on subgroup consistency the overall poverty level, ceteris paribus, does not have to go up (may even go down) as a consequence of an increase in the poverty level of a subgroup. However, such an argument is quite inconsistent with the general perception of poverty and of changes in poverty. We believe that subgroup consistency is consistent with the notion of poverty changes and hence can be well justified as a basic axiom. Kundu and Smith (1983) proposed two population monotonicity axioms. They were concerned with the behaviour of a poverty measure when it is used to compare income distributions of different population sizes. Before Kundu and Smith (1983), all discussions assumed a fixed population. The two population monotonicity axioms are Poverty Growth Axiom: P(x; z)<p(y;z) whenever x D is obtained from y D by deleting a poor person from the population. Nonpoverty Growth Axiom: P(x; z)<p(y;z) whenever x D is obtained from y D by adding a nonpoor person to the population. Kundu and Smith (1983) showed that these two population monotonicity axioms are not compatible with regressive transfer: there is no poverty measure that can possess poverty growth, nonpoverty growth and regressive transfer. 32 A poor person can be made nonpoor via two routes: a simple increment and a transfer from poorer persons. The intuition behind the conflict between the population monotonicity axioms and the transfer axiom is that they predict the opposite directions about the change in the poverty level as a result of the linecrossing; poverty growth and nonpoverty growth suggest that the poverty level should go down while the transfer axiom implies that the poverty level should go up (or not go down). Kundu and Smith blamed this impossibility on regressive transfer, which fails to distinguish those income transfers which actually alter the size of the poor population. They defended both poverty growth and nonpoverty growth, there appears to be less room for modification, while considering regressive transfer as the most conspicuous candidate for modification toward some weaker form in order to construct a possible poverty measure. However, such a modification is not warranted because several weaker requirements combined may necessarily imply a stronger one, e.g, the weaker form of regressive transfer, say weak transfer, and continuity jointly will imply regressive transfer as shown by Donaldson and Weymark (1986). One might argue that by adding poor persons who are richer than the average poor person, the degree of poverty may thereby be decreased. Kundu and Smith believed that such an argument ignores the relevance of a poverty line altogether. The poverty BlackwellPublishersLtd.1997

16 138 BUHONG ZHENG growth axiom Kundu and Smith proposed goes to another extreme. It ignores the distributional changes resulting from the adding of a poor person completely. Imagine a society where everyone is poor with almost no income. One day people of the same population size from another country migrate to this isolated land. These new people are not rich but have incomes close to the poverty line, which is not zero. It is peculiar to argue that the new society becomes poorer because of the migration of these people. Even consider the case where there is at least one person to be nonpoor as demanded by Kundu and Smith, it is still difficult to maintain poverty growth. The consideration of others of the aforementioned axioms may throw some light on these two population monotonicity axioms. The replication invariance axiom, strong monotonicity (or weak monotonicity and continuity) and the focus axiom combined will necessarily imply nonpoverty growth, while poverty growth will hold if the entrant has an income no higher than the poorest person. 33,34 The reason that these two axioms cannot be equally justified is the asymmetric treatment of the poor and the nonpoor: when adding a nonpoor person we know that he has at least an income above the poverty line and the exact amount of income does not matter; while for the adding of a poor person, the exact income does matter. We pointed out that poverty growth may be a quite appropriate axiom for measuring the scope of poverty but not for measuring the intensity of poverty and poverty in general. As a matter of fact, the only decomposable poverty measure that satisfies both poverty growth and nonpoverty growth is the linear transformation of the headcount ratio. 35 Therefore, this result and the one given by Kundu and Smith suggest that there is no strong reason to maintain poverty growth. Sen (1981, p. 193) believed that the two population monotonicity axioms are really very demanding, we have shown here that nonpoverty growth is not demanding at all while poverty growth cannot and should not be treated as an axiom for a measure of poverty intensity. Note that nonpoverty growth (or strong monotonicity and replication invariance) implies that a poverty measure is a decreasing function of the population size of the nonpoor. Therefore a poverty measure is independent of the income distribution of the nonpoor (by the focus axiom) but is dependent on the population size of the nonpoor. Two other invariance axioms concern the behaviour of a poverty measure when all poor incomes are uniformly changed. They are Scale Invariance Axiom: P(x; z )=P(x;z) whenever (x; z ) D D is obtained from (x; z) D D by a relative change. Translation Invariance Axiom: P(x; z )=P(x;z) whenever (x; z ) D D is obtained from (x; z) D D by an absolute change. Blackorby and Donaldson (1980) incorporated these two concepts into the study of poverty measurement. 36 They referred to the poverty measures satisfying scale invariance as relative measures and the measures satisfying translation invariance as absolute measures. Because most existing poverty measures are relative, it is useful to discuss the contents of scale invariance. The scale BlackwellPublishersLtd.1997

17 AGGREGATE POVERTY MEASURES 139 invariance axiom has two implications: one for nominal income and the other for real income. The axiom in the first sense says that the poverty value is unaffected by either the unit or currency against which income is measured. It does not matter whether one uses cents versus dollars, or U.S. dollars versus British pounds; the poverty value should be unit-free. The axiom in this sense may sound reasonable and would be well justified. However, one may argue that such a requirement is only a practical convenience and not a theoretical necessity. We may always compare physical good (real income) rather than nominal income. The justification of the axiom in terms of real income is even more questionable. The axiom in this sense requires the doubling of all real incomes and the real poverty line to leave the poverty value unchanged. While no solid justification for this axiom in this sense has been made, it is certainly nonsensical in a case where some people have no incomes or negative incomes. If all incomes and the poverty line are doubled, then those with negative incomes become poorer, and those with incomes of zero experience no change. Income inequality and poverty should increase rather than remain unchanged in this case. While these two senses of scale invariance can be distinguished economically, it is impossible to separate them mathematically. One way to avoid the nonsensical situation is to require a positive income, which is not an unreasonable requirement as long as we treat poverty value as a function of income only. 37 There has been no economically solid justification for translation invariance. Hence, according to the criteria for an axiom in the general measurement theory (e.g. Krantz et al., 1971), neither scale invariance nor translation invariance can be regarded as a basic axiom for a poverty measure; each requirement only identifies a special class of poverty measures. Be that as it may, there is a widespread interest in these two classes of measures for their special functional forms and it is useful to investigate the properties each class of measures possess and the interactions between them. In investigating a class of subgroup consistent poverty measures, Foster and Shorrocks (1991) proved that the only restricted continuous, replication invariant poverty measures that are both relative and absolute are continuous, increasing transformations of the headcount ratio. One implication of their result, though not stressed by them, is that there is no distribution-sensitive poverty measure that can be both relative and absolute within the class of subgroup consistent, replication invariant, and continuous poverty measures. 38 In a recent paper, Zheng ( 1994) showed that Foster and Shorrocks implication can be greatly strengthened: the only focused, restricted continuous poverty measures that are both relative and absolute are those related to head counts, i.e., P(x; z)= f(q(x;z), n(x)). Therefore, there is no distribution-sensitive poverty measure, not just within the subgroup consistent subgroup, that can be both relative and absolute. Increasing Poverty Line Axiom: P(x; z)<p(x;z) whenever z<z. This axiom is very reasonable: between two identical societies, the one with the higher poverty line must also have higher poverty level. The first authors to list it, implicitly and explicitly, as a requirement for a poverty measure were Clark, BlackwellPublishersLtd.1997