Document(s) 7 of 20

5.1 Poverty indices

Two approaches have been used to devise cardinal indices of poverty. The first uses the concept of equally distributed equivalent (EDE) incomes, and applies it to distributions whose incomes have been censored at the poverty line. It then compares those EDE incomes to the poverty line. The second approach transforms incomes and the poverty line into poverty gaps, and aggregates these gaps using social-welfare like functions. We look at these two approaches in turn.

5.1.1 The EDE approach

For the EDE approach to building poverty indices, we start with the distribution of income Q(p). Since, for poverty comparisons, we want to focus on those incomes that fall below the poverty line (the "focus axiom"), the incomes Q(p) are censored at the poverty line z to give Q*(p; z). The censored incomes are then aggregated using one of the many social welfare functions that have been proposed in the literature, such as the Atkinson or S-Gini ones. A poverty index is obtained by taking the difference between the poverty line and the EDE income. For instance, for the social welfare functions proposed in section 4.3, this procedure leads to the following class of poverty indices:

where ξ*(z; p, ε) is the EDE income of the distribution of censored income Q*(p; z) and where we need ρ ≥ 1 and ε ≥ 0 for the Pigou-Dalton transfer principle not to be violated. P(z; ρ, ε) can then be interpreted as the "socially representative" or EDE poverty gap.

Examples of such poverty indices include a transformation of the Clark, Hemming and Ulph's (CHU) second class of poverty indices, given by P(z; ε) = P(z; ρ = 1, ε)¹:

The CHU indices are then obviously closely related to the Atkinson social welfareatk functions and inequality indices. When ε = 1, the CHU poverty index is also the EDE poverty gap corresponding to the Watts poverty index, an index which is defined as²:

For 0 ≤ ε < 1, the CHU indices also correspond to the EDE poverty gap of the class of poverty indices proposed by Chakravarty, PC(z; ε):

Moreover, if we choose ε = 0 for the class of indices defined in (5.1), we obtain the class of S-Gini indices of poverty, P(z; ρ)³:

P(z; ρ = 2) is then a "Gini-like" index of poverty.

5.1.2 The poverty gap approach

The second approach to constructing poverty indices uses the distribution of poverty gaps, g(p; z) = z - Q*(p; z). Once this distribution is known, no other use of the poverty line is needed for the aggregation of poverty. Because of this, the poverty gap approach to constructing poverty indices is slightly more restrictive and also puts more structure on the shape of the allowable poverty indices than the previous EDE approach. After the distribution of poverty gaps has been computed, we may use aggregating functions analogous to those used in Section 4.3 for the analysis of social welfare. Like social welfare functions, where we normally want an increase in someone's income to increase social welfare, we would normally wish the poverty indices to be increasing in poverty gaps. Unlike social welfare functions, however, where an equalizing Pigou-Dalton transfer would often increase the value of a social welfare function, we would typically wish a poverty index to decrease when such an equalizing transfer of income takes place.

¹ DAD: Poverty|CHU Index.

² DAD: Poverty|Watts Index.

³ DAD: Poverty|S-Gini Index.

A popular class of poverty gap indices that can obey these axioms is known as the Foster-Greer-Thorbecke (FGT) class. It differentiates its members using an ethical parameter α ≥ 0 and is generally defined as⁴

E: 18.7.4

for the normalized FGT poverty indices and as

for the un-normalized version (which can sometimes be more useful than the more usual normalized form). Note that poverty gap indices other than the FGT ones can also be easily proposed, simply by using other aggregating functions of poverty gaps that obey some of the desirable axioms (such as that of being increasing and convex in poverty gaps) discussed in the literature.

5.1.3 Interpreting FGT indices

When α = 0, the FGT index gives the simplest and most commonly used poverty index. It is called the poverty headcount ratio, and is simply the proportion of a population that is in poverty (those with a positive poverty gap), F(z) ⁵. The shorter expression "poverty headcount" is sometimes meant to indicate the absolute (as opposed to the relative) number of the poor in the population. Since our population size is normalized ton 1 in the this book, we will use the two expressions "headcount" and "headcount ratio" interchangeably.

E:18.1.1

The next simplest and most commonly used index, μg(z), is given by the average poverty gap, P (z; α = 1), and is the average shortfall of income from the poverty line:

To see how to interpret the form of the FGT indices for general values of α consider Figure 5.1. It shows the (absolute) contributions to total poverty (z; α) of individuals at different ranks p. These contributions are given by (g(p; z)/z)α. For α = 0, the contribution is a constant 1 for the poor and 0 for the rich (those whose rank exceeds F(z) on the Figure, or equivalently those whose income Q(p) exceeds z). The headcount is then the area covered by the dotted rectangle on Figure 5.1. For α = 1, the contribution of someone at p equals his normalized poverty gap, g(p; z)/z. Poverty is then the area underneath the g(p; z)/z curve drawn on Figure 5.1. The same reasoning is valid for higher values of α For instance, the absolute contribution to (z; α = 3) of individuals at rank p is given by (g(p; z)/z)3 on Figure 5.1, and (z; α = 3) equals the area underneath the (g(p; z)/z)3 curve.

⁴ DAD: Poverty|FGT Index.

⁵ DAD: Poverty|FGT Index.

Notwithstanding the above, interpreting the numerical value of FGT indices for α different from 0 and 1 can be problematic. We can easily understand what is meant by a proportion of the population in poverty or by an average poverty gap, but what, for instance, can a squared-poverty-gap index actually signify? And how to explain it to a government Minister?... A further difficulty with such indices emerges from a closer look at Figure 5.1, which indicates that the absolute contribution of poverty gaps to poverty decreases with α — the contribution curves (g(p)/z)α move down as α rises. This also implies that the normalized FGT indices necessarily fall as α increases. This is paradoxical since it is usually argued that the higher the value of α, the greater the focus on those who suffer most "severely" from poverty. It would thus be more natural if an increase in α also increased (z; a).

5.1.4 Relative contributions to FGT indices

One partial solution to these interpretive problems is to switch one's focus from the absolute to the relative contribution to an FGT index of individuals with different poverty gaps. Such a relative contribution is depicted on Figure 5.2 for α = 0, 1 and 2. It shows the ratio of the absolute contributions g(p)α to total poverty P(z; α) — these ratios are the same for normalized and unnormalized FGT indices. Since this graph shows relative contributions to total poverty, the area underneath each of the three curves must in all cases equal 1.

For α = 0, each poor contributes relatively the same constant 1/F(z) to the poverty headcount. The poor's relative contribution to the average poverty gap increases with their own poverty gap, as shown by the curve g(p)/P(z; α = 1). That relative contribution equals 1 for those individuals whose own poverty gap is precisely equal to the average poverty gap. The rank of such individuals is given by F(μg(z)), as is also shown on Figure 5.2. Thus, those located at p = F(μg(z)) have a poverty gap that is representative of the average poverty gap in the population. Increasing α from 1 to 2 decreases the relative contribution of the not-so-poor, but inversely increases the contribution of those with the highest poverty gaps as shown by the curve g(p; z)/P(z; α = 2). This then becomes consistent with the general view that, in the aggregation of individual poverty, higher values of α put more emphasis on those who suffer most severely from poverty — those with lower values of p and higher values of g(p;z).

Figure 5.1: Contribution of poverty gaps to FGT indices

5.1.5 EDE poverty gaps for FGT indices

Figure 5.2 does not, however, solve the main interpretation problems associated with the FGT indices. As mentioned above, explaining to non-technicians or policymakers the practical meaning of FGT indices for general values of α is difficult since these indices are averages of powers of poverty gaps. They are also neither unit-free nor money-metric (except for α = 0 and 1). An another already-mentioned difficulty is that the usual FGT indices will generally fall with an increase in the value of their poverty aversion parameter, α.

A simple solution to these two problems is to transform the FGT indices into EDE poverty gaps. An EDE poverty gap is that poverty gap which — if it were assigned equally to all individuals — would yield the same aggregate poverty index as that which is currently observed. An EDE poverty gap can then usefully be interpreted as a socially-representative poverty gap. This transformation provides a money-metric measure of poverty which can be usefully compared across different poverty indices and/or across different values of α. As we will see later, it also allows the analyst to determine the impact of poverty-gap inequality upon the level of poverty. For the un-normalized FGT indices, the EDE poverty gap is given simply by (for α > 0)⁶

For the normalized FGT indices, it is just ξ-9 (z; a) = ξg (z; a)/z. An EDE poverty gap cannot be defined for α = 0.

Figure 5.3 shows such socially-representative poverty gaps ξg (z; α) for different values of α. In each case, we obtain a socially-weighted money-metric indicator of the distribution of deprivation in the population. This summary aggregate indicator can also be compared to the individual distribution of poverty, given by the g(p; z) curve. Those whose g(p; z) exceeds ξg (z; α) experience more poverty than the socially representative average. Those exactly at ξg (z; α) are located exactly at the socially representative poverty gap. Those representative individuals are thus found at the ranks given by F (ξg (z; α)), which are also shown on Figure 5.3 for different values of α.

An important point to note is that an increase in α moves the socially-representative poverty gap closer to that experienced by the poorest individuals. This is since ξg(z; α + 1) ≥ ξg(z; α) for any α > 0. (This is unlike the usual definition of the FGT indices, for which we have P (z; α + 1) ≤ P(z;α)

⁶ DAD: Poverty|FGT Index.

for any α > 0.) Hence, we can readily interpret increases in α as leading to increases in the socially-representative poverty gap, and thus in the relative weight given to the poorer of the poor. The larger the value of α, the more important are the most severe cases of deprivation in computing a socially-representative aggregate level of poverty.

Note finally that, besides being already in an EDE poverty gap form, the S-Gini index of poverty also has the property of being a poverty gap index. Indeed, by (5.5), we have that

5.2 Group-decomposable poverty indices

Much of the early literature on the construction of poverty indices focussed on whether indices were decomposable across population subgroups. This has led to the identification of a subgroup of poverty indices known as the "class of decomposable poverty indices". These indices have the property of being expressible as a weighted sum (more generally, as a separable function) of the same poverty indices assessed across population subgroups. They most commonly include the FGT and the Chakravarty classes of indices as well as the Watts index.

Let the population be divided into K mutually exclusive population subgroups, where φ(k) is the share of the population found in subgroup k. For the FGT indices, we then have that:

where P(k; z; α) is the FGT poverty index of subgroup k⁷. The Watts and Chakravarty indices are expressible as a sum of the poverty indices of each subgroup in exactly the same way as for the FGT indices in (5.11).

E: 18.6

To illustrate the practical implications of the group-decomposition property, consider the following two-group (K = 2) example. Let the first group contain 40% of the total population, and let poverty in group 1 be 0.8 and that of group 2 be 0.4. Poverty in the total population is then a simple weighted mean of group poverty, and is immediately computable as 0.4 · 0.8 + 0.6 · 0.4 = 0.56. Estimates of total poverty in a population can then be constructed in a decentralized manner, first by estimating poverty within communities or regions, and then by averaging over these decentralized estimates, without there being a need for all of the micro data to be regrouped in one single register.

⁷DAD: Decomposition|FGT: Decomposition by Groups.

Subgroup decomposability also implies that an income improvement in one of the subgroups will necessarily improve aggregate poverty if the incomes in the other groups have not changed. It will also mean that the optimal design of social safety nets and benefit targeting within any given group can be assessed independently of the income distribution in the other groups: only the distributive characteristics of the relevant group matter for the exercise. If targeting succeeds in decreasing poverty at a local level, then it must also succeed at the aggregate level.

Subgroup decomposability is therefore useful, although it is certainly not imperative for poverty analysis. In particular, it is not because an index facilitates poverty profiling and targeting analysis that this index is necessarily ethically fine. Ease of computation and ethical soundness are also two different an potentially conflicting criteria. Among other things, imposing the decomposability and additivity property can mean sacrificing some important ethical features in the aggregation of poverty. In that context, Ravallion (1994) notes that when measuring poverty "one possible objection to additivity is that it attaches no weight to one aspect of a poverty profile: the inequality between subgroups in the extent of poverty". This can be an important flaw if for instance between-group relative deprivation is considered ethically significant.

5.3 Poverty and inequality

Expressing poverty indices in the form of EDE poverty gaps enables the decomposition of poverty as a sum of average poverty and inequality in poverty. Let ξg (z) be the EDE poverty gap and g(z) be the cost of inequality in poverty gaps. We then have:

or, alternatively,

For instance, for the popular FGT indices, we have that the cost of inequality in poverty gaps is given by:

When α = 1, we have that the socially representative poverty gap ξ^g(z) is just the average poverty gap μ^g(z); inequality in poverty gaps is thus not taken into account in assessing poverty. The poverty cost of inequality is then nil. Since μ^g(z) is insensitive to α, and since ξ^g(z; α) is increasing in α, it follows that ^g(z; α) is also increasing in α; the larger the value of α, the larger the impact of inequality on the level of aggregate poverty. This can be checked on Figure 5.3. We can thus interpret α as a parameter of inequality aversion in measuring poverty. For 0 < α < 1, we have that ξ^g(z; α) < μ^g(z), and inequality in poverty is then deemed to reduce poverty: ^g(z, α) < 0. Ceteris paribus, we then have that the greater the level of inequality, the lower the socially representative level of poverty. For α > 1, we have that ^g(z; α) > 0 and inequality has therefore a positive poverty cost.

A similar decomposition can be done using (5.1) and the EDE level of censored income. The EDE poverty gap corresponding to that approach is defined as

where *(z; ρ, ε) = μ*(z) · I*(z; ρ, ε) is the cost of inequality in censored income and where I*(z; ρ.ε) is the index of inequality in censored income.

5.4 Poverty curves

It is often informative to portray the whole distribution of poverty gaps on a simple graph, in a way which shows both the incidence and the inequality of income deprivation. Particularly useful is the poverty gap curve, which plots g(p; z) as a function of p — see again Figure 5.3. The curve naturally decreases with the rank p in the population, and reaches zero at the value of p equal to the headcount. The integral under the curve gives the average poverty gap, and its steepness indicates the degree of inequality in the distribution of poverty gaps.

Another percentile-based curve that is graphically informative and that is useful for the measurement and comparison of poverty is called the Cumulative Poverty Gap (CPG) curve (also sometimes referred to as the inverse Generalized Lorenz curve, the "TIP" curve, or the poverty profile curve). The CPG curve cumulates the poverty gaps of the bottom p proportion of the population. It is defined as:⁸

E:18.7.8

A CPG curve is drawn on Figure 5.4. The slope of G(p; z) at a given value of p shows the poverty gap g(p; z). Since g(p; z) is non-negative, G(p; z) is non-decreasing. G(p = 1; z) equals the average poverty gap μ^g(z). The percentile at which G(p; z) becomes horizontal (where g(p; z) becomes zero) yields the poverty headcount. Furthermore, since the higher his rank p in the population, the richer is an individual, and therefore the lower is his poverty gap, G(p; z) is therefore concave in p. Because of this, the CPG curve exhibits for poverty analysis the same descriptive interest as the Lorenz and Generalized Lorenz curves for the analysis of inequality and social welfare. The distance of G(p; z) from the line of perfect equality of poverty gaps (namely, the line 0B in Figure 5.4) shows the inequality of poverty gaps among the total population. The distance of G(p; z) from the line of perfect equality of poverty gaps among the poor (namely, the line 0A in Figure 5.4) displays the inequality of poverty gaps among the poor. Finally, the concavity of G(p; z) is inversely related to the density of poverty gaps at p.

⁸DAD: Curves|CPG.

5.5 S-Gini poverty indices

When weighted by K(p; ρ), the area underneath the CPG curve generates the class of S-Gini poverty indices⁹:

Recall that K(p; ρ) = ρ(ρ–1) (1 – p)^ρ–2 · P(z; ρ = 1) thus equals the average poverty gap, μ^g(z), P(z; ρ = 2) is the poverty index that is analogous to the standard Gini index of inequality, and the well-known Sen index of poverty is given by:

An interesting feature of the P(z; ρ) indices is their link with absolute and relative deprivation. Let absolute deprivation, AD(z), be given by the average shortfall from the poverty line, that is, by μ^g(z). Recalling (4.25) and (4.26), we can define relative deprivation in censored income at percentile p as:

Average relative deprivation across the whole population is then:

It is then possible to show that:

The larger the value of ρ, the larger is relative deprivation, RD(z; ρ), and the larger are P(z;ρ) and the contribution of relative deprivation and inequality to poverty. This provides an alternative link between inequality and poverty.

⁹DAD: Poverty|S-Gini Index.

5.6 Normalizing poverty indices

Most of the poverty indices discussed above have initially been introduced in the literature in a normalized form, that is, by dividing censored income and poverty gaps by the poverty line. The FGT indices, for instance, are generally expressed as¹⁰:

(see (5.6)). Normalizing poverty indices will make no substantial difference and little expositional difference for poverty analysis when the distributions of income being compared have identical poverty lines. This will typically be the case, for instance, when incomes are expressed in real (or constant) values, and when the focus is on absolute poverty with constant real poverty lines. Normalizing poverty indices by the poverty line will

make the EDE poverty gap lie between 0 and 1,

make poverty indices insensitive to and independent of the monetary units (e.g., dollars or cents) used in assessing income, and

make the indices invariant to an equi-proportionate change in all incomes and in the poverty line.

Normalizing poverty indices is particularly useful if the poverty lines serve as price indices, and thus used to enable comparisons of nominal income across time and space (recall that price indices are used to convert nominal incomes into base-year real incomes).

Normalized poverty indices are usually referred to as "relative poverty indices"; changing all incomes and the poverty line by the same proportion will not affect the value of relative poverty indices. FGT and other poverty gap indices that are not normalized are often called "absolute" poverty indices; it can be checked that equal absolute additions to all incomes and to the poverty line will not affect their value. Increasing all incomes and the poverty line by the same proportion will, however, increase the value of such absolute poverty indices.

When poverty lines are different across distributions, and when their ratio across time or space cannot be interpreted simply as a ratio of price indices, the normalization of poverty indices by these poverty lines can, however, be problematic, and is surely open to debate. This is the case, for instance, when we are interested in comparing the absolute shortfalls of "real" income from a "real" poverty line, when these real poverty lines vary across populations or population subgroups. Examples can arise, inter alia, in comparing the poverty of families of different sizes and composition, or in comparing poverty across distributions with different social or cultural bases for the definition of a poverty line.

¹⁰DAD: Poverty|FGT Index.

To see this more clearly, consider the following example in which all incomes and poverty lines are expressed in real terms (namely, they have been adjusted for differences in the cost of living, and they are therefore comparable). In country A, the poverty line is $1,000, and a poor person i has an income of $500. Because, say, of cultural and/or sociological differences (these differences may exist across time or space), the poverty line in country B is larger and is equal to $2,000, and a poor person j in it has an income equal to $1,100. Who of i and j is poorer? If we adopt the relative view to building poverty indices, i will be considered the poorer since as a proportion of the respective poverty lines he is farther away from it than j. If, instead, absolute poverty indices are used, j will be deemed the poorer since his absolute poverty gap ($900) is by far larger than that of i ($500). Which of these two views should prevail is then open to debate.

5.7 Decomposing poverty

5.7.1 Growth-redistribution decompositions

It is often useful to determine whether it is mean-income growth or changes in the relative income shares accruing to different parts of the population that are responsible for the evolution of poverty across time. Investigating this can also help assess whether these two factors, mean-income changes and inequality changes, work in the same or in opposite directions when it comes to the behavior of aggregate poverty. Similarly, we may wish to assess whether differences in poverty across countries or regions are due to differences in inequality or to differences in mean levels of income.

There are several ways to do this. To illustrate them, assume that we wish to compare distributions A and B to determine if it is the difference in their mean income (" growth") or the difference in their income inequality (" redistribution") that accounts for their difference in poverty. The common feature of all existing growth- redistribution decomposition procedures is

1. first, to scale the two distributions A and B such that they have the same mean, and interpret the difference in poverty across these two scaled distributions as the impact on poverty of their difference in inequality;
   
   
2. and second, to interpret the difference in poverty between one of the distributions (say, A) and that same distribution scaled to the mean income of the other distribution (B) as the impact on poverty of their difference in mean income.

Starting from this, the precise growth- redistribution decomposition procedures that are chosen differ by the solution they apply to a basic problem known generally in the national-accounts literature as the "index problem". Specifically here, should we scale A to the mean of B, or B to the mean of A, to assess the impact of differences in inequality? And, in estimating the impact of differences in mean incomes, should we compare A with A-scaled-to-the-mean-of-B, or B with B-scaled-to-the-mean-of-A?

The first paper that implemented a growth- redistribution decomposition of poverty differences (Datt and Ravallion 1992) used the initial distribution as a reference "anchor point". To see how, it is easiest to use the normalized FGT indices (z; a) defined in (5.6), although the growth- redistribution decomposition methodologies can be used with any relative poverty indices, additive or not. The change in poverty between A and B is expressed as a sum of a "growth" (difference in mean income) effect and of a "redistributive" (difference in relative income shares) effect, plus an error term that originates from the above-mentioned index problem. This gives¹¹:

The first expression in the first term on the left of (5.23), A, is poverty in A after A's incomes have been scaled by μB/μA to yield a distribution with mean μB and inequality unchanged. is thus the difference between two distributions with the same relative income shares but with (possibly) different mean incomes. When μB > μA, this growth term is negative — this simply says that growth reduces poverty. The first expression in the second term, B , is poverty in B after B's incomes have been scaled by μA/μB to yield a distribution with mean μA. is thus the difference between two distributions with identical mean incomes but with (possibly) different inequality. When the Lorenz curve for B is everywhere above the Lorenz curve for A, this redistribution term is necessarily negative when α ≥ 1, but it can also be positive when α < 1.

The error term in (5.23) can be expressed as:

¹¹DAD: Decomposition|FGT: Growth & Redistribution.

This error term can be shown to be either the difference between the growth effect measured using B as a reference distribution and that using A as the reference distribution,

or the difference between the redistribution effect measured using B as the reference distribution and the redistribution effect using A as the reference distribution,

An alternative decomposition uses the posterior distribution B as the reference distribution for assessing the growth and redistribution effects. This yields:

Clearly, a middle way between these two alternative decomposition procedures is to measure the growth effect as the average of the two growth effects, in (5.23) and (5.27), and likewise to measure the redistribution effect as the average of the two redistribution effects. Proceeding this way has the advantage of eliminating the error term in the poverty decomposition, since the error terms of each of the two alternative decompositions sum to zero. This middle way is in fact what would be given by the use of the Shapley value to perform a growth- redistribution decomposition — see the Appendix 4.7 for more details on the Shapley value. This leads to the following growth- redistribution decomposition¹²:

5.7.2 Demographic and sectoral decomposition of differences in FGT indices

Equation (5.11) shows how poverty can be expressed as a sum of the poverty contributions of the various subgroups that make a population. Each subgroup contributes in proportion to its share in the population and to the level of poverty found in that subgroup. Hence, we may wish to express changes in poverty across time or space as a function of differences in these factors. More precisely, we want to see whether differences in poverty across distributions can be attributed to differences in demographic or sectoral composition across these distributions, or to differences in poverty across these demographic or sectoral groups. We may express this as follows¹³:

Note that the decomposition in (5.29) suffers from the same index number problem as the earlier one in (5.23). For example, one could prefer to use φB(k) instead of φA(k) to compute the within-group poverty effects. It may also seem more convenient to weight the within-group poverty effects by the average population shares, and to weight the demographic and sectoral effects by the average poverty index. This yields ¹⁴:

¹²DAD: Decomposition|FGT: Growth & Redistribution.

¹³DAD: Decomposition|FGT: Sectoral.

where(k)= 0.5 (φA (k) + φB (k)) and (k; z; α) = 0.5 (A(k;z;α) +B(k;z;α)). Note from (5.30) that this decomposition procedure removes the error term. Depending on the context, the decomposition in (5.30) could serve to show, for instance, how variations in the size and in the poverty of various sectors of the economy account for variations of total poverty across economies, how differences in the size and in the poverty of various demographic groups explain differences in total poverty across societies, how migration and differential poverty across regions account for changes in poverty across time, etc..

5.7.3 The impact of demographic changes

An alternative use of the decomposition in (5.11) computes the impact of a change in the proportion of the population that is found in a group k, this change being accompanied by an exactly offsetting change in the proportion of the other groups. This may be useful, for instance, if one wishes to predict the impact of migration or demographic changes on national poverty, keeping out within-group poverty. Let the population share of a group t, φ(t), increase by a proportion λ to φ(t)(1 + λ), with a proportional fall in the other groups' population share from φ(k) to φ(k) (1 – φ(t)λ/(1 – φ(t))). Note that the new population shares will add up to 1 since

The net impact of this on poverty is then¹⁵

¹⁴DAD: Decomposition|FGT: Sectoral.

¹⁵DAD: Poverty|Impact of Demographic Change.

We may instead wish to predict the impact of an absolute increase in the population share of a group t. Let this change be from φ(t) to φ(t) + λ, with a corresponding fall in the other groups' population share that is proportional to their initial share (a fall from φ(k) to φ(k) (1 – λ/(1 – φ(t)))). The resulting change in poverty is analogously given as

Note that the only difference between (5.31) and (5.32) comes from the size in the increase in φ(t), which is φ(t)λ in (5.31) and λ in (5.32).

5.7.4 Decomposing poverty by income components

Let C income components add up to total income X(p), with X(p) = and being the expected value of income component c at rank p in the distribution of total income. can be, for instance, agricultural or capital income, or the income of those living in some geographic area, or some type of expenditure that enters total expenditure X ¹⁶.

We may wish to know by what amount total poverty is reduced by the presence of an income component. Clearly, we would expect those components with a large mean μX_(c) to be more effective in helping to alleviate total poverty. But we must also take into account the distribution of . Suppose for instance that urban capital income is larger than rural capital income, but that poverty is low in urban areas because urban labor income is large there. Then, it is unclear whether relatively high capital income in urban areas is more effective at alleviating poverty than the relatively low capital income in rural areas, where poverty is more concentrated.

The contribution of an income component c to poverty alleviation can be given by the fall in poverty after is added to initial income. But this fall depends on what this initial income is. Does it include some of the other income components? This path-dependency difficulty can again be circumvented by the use of the Shapley value. We start by assuming maximum poverty, that is, poverty when total income is nil for everyone. We then estimate the contribution of component c to poverty alleviation as the expected value of its marginal contribution when it is added to anyone of the various subsets of income components that one can choose from the set of all the components.

¹⁶DAD: Decomposition|FGT: Decomposition by Sources.

When a component is missing from that set for an individual, we assume that its value is 0.

5.8 References

Rowntree (1901) predated by far the modern quantitative approach to poverty measurement. General and recent references include Chen and Ravallion (2001) (for wide empirical evidence on poverty), Constance and Michael (1995) (for the US debate on poverty measurement), Deaton (2001) (for the empirical difficulties associated with "counting the poor"), Glewwe (2001) (for a very extensive coverage of the nature, evolution, and causes of poverty), Jantti and Danzinger (2000) (for poverty in more advanced countries), Lipton and Ravallion (1995) (for poverty and policy), Ravallion (1994) and Ravallion (1996) (for a non-technical overview and discussion of poverty measurement issues), Smeeding, Rainwater, and O'Higgins (1990) (for early results using Luxembourg Income Study data) and Zheng (1997) (for a review of poverty indices).

The papers by Watts (1968), Sen (1976) and Foster, Greer, and Thorbecke (1984) influenced greatly much of the subsequently large literature on poverty indices. Relatively early contributions on poverty measurement are found in Anand (1977), Blackorby and Donaldson (1980), Chakravarty (1983a), Chakravarty (1983b), Clark, Hamming, and Ulph (1981), Donaldson and Weymark (1986), Foster (1984), Hagenaars (1987), Kakwani (1980), Kundu and Smith (1983), Takayama (1979), and Thon (1979). More recent works include Chakravarty (1997), Myles and Picot (2000), Osberg and Xu (2000) and Shorrocks (1995) on a revisited and improved form of the Sen (1976) poverty index; Duclos and Gregoire (2002) on the link between linear poverty indices and relative deprivation; Morduch (1998) and Zheng (1993) on the Watts index; Pattanaik and Sengupta (1995) on the original Sen index; and Shorrocks (1998) on "deprivation profiles".

Applied poverty studies using these developments have been almost innumerable. A small subset of the studies that have been published includes Coulombe and McKay (1998) (Mauritania), Coulombe and McKay (1998) (Ghana), Davidson and Duclos (2000) (using LIS data), Gustafsson and Nivorozhkina (1996) (Northern countries), Grootart and Kanbur (1995) (Côte d'Ivoire), Gustafsson and Shi (2002) (China), Hagenaars and De Vos (1988) (the Netherlands), Hill and Michael (2001) (US), Iceland, Short, Garner, and Johnson (2001) (US), Milanovic (1992) (Poland), Osberg and Xu (1999) (Canada), Osberg (2000) (Canada and the US), Pendakur (2001) (Canada), Rady (2000) (Egypt), Ravallion and Bidani (1994) (Indonesia), Ravallion and Chen (1997) (67 less developed countries), Rodgers and Rodgers (2000) (Australia), and Szulc (1995) (Poland).

The empirical links between growth, poverty and inequality have also often been analyzed in recent years. Studies on whether growth is beneficial to the poor, both absolutely and relatively speaking, include Bigsten and Shimeles (2003) (for Ethiopian evidence), Datt and Ravallion (2002) (for a survey of the Indian evidence), Dollar and Kraay (2002) (for an influential study of the experience of 42 countries over 4 decades), Essama Nssah (1997) (for Madagascar evidence), Ravallion and Chen (1997) (where growth is found to decrease inequality as often as it increases it), Ravallion (2001) (where a warning against the use of cross-country regressions is made), and Ravallion and Datt (2002) (for differential evidence across Indian states). De Janvry and Sadoulet (2000), Deininger and Squire (1998) and Ravallion (1998a) also apply causal tests to determine whether inequality favors or impedes growth. See also Ravallion and Chen (2003) and Tsui (1996) for the use of the average poverty gap and the Watts index as indices of whether growth is beneficial to the poor.

Figure 5.2: The relative contribution of the poor to FGT indices

Figure 5.3: Socially-representative poverty gaps for the FGT indices

Figure 5.4: The cumulative poverty gap (CPG) curve

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