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To see how the material of Chapter 9 can be used practically to test for the robustness of poverty comparisons, we focus for simplicity on classes of additive poverty indices denoted as II^s (z⁺), where s stands again for the "ethical order" of the class and where z⁺ will stand for the upper bound of the range of all of the poverty lines that can reasonably be envisaged. The additive poverty indices P(z) that are members of that class can be expressed as

where z is a poverty line and π(Q(p);z) is an indicator of the poverty status of someone with income Q(p).

We can also think of the function π(Q(p); z) as the contribution of an individual with income Q(p) to overall poverty P(z). Hence, we can also assume that π(Q(p); z) = 0 if Q(p) > z. This ensures that the poverty indices fulfill the well-known poverty focus principle, which simply states that changes in the incomes of the rich should not affect the poverty measure. The use of quantiles in equation (10.1) also ensures that the poverty indices P(z) obey the anonymity (see page 160) and population invariance principles (see page 160). For expositional simplicity, also assume that π(Q(p);z) is continuously differentiable in Q(p) between 0 and z up to an appropriate order, and denote the i^th-order derivative of π(Q(p); z) with respect to Q(p) as π⁽ⁱ⁾(Q(p);z).

The first class of poverty indices (denoted by II¹ (z⁺)) then regroups all of the poverty indices

that decrease when someone's income increases

and whose poverty line does not exceed z⁺.

Formally, indices within II¹ (z⁺) are such that:

where the Pareto principle (page 159) appears through the form of a non-positive first-order derivative π⁽¹⁾(Q(p);z).

The second class of poverty indices, II²(z⁺), contains those first-order indices that have a greater ethical preference for the poorer among the poor - recall the Pigou-Dalton principle of page 161. Increasing the income of a poorer individual is better for poverty reduction that increasing by the same amount the income of a richer person. The absolute value of the first-order derivative is therefore decreasing with Q(p), and the indices are thus convex in income. This class II² (z⁺) is then:

We will discuss further below the role of the continuity condition π(z, z) = 0. Clearly, II²(z⁺) ⊂ II¹(z⁺), but not the reverse.

Technically, obeying the "transfer-sensitivity" principle requires for the P(z) indices that their second-order derivative π⁽²⁾(Q(p);z) be decreasing in Q(p). Poverty indices belonging to the third-order class of poverty indices II³(z⁺ are then defined as:

As before, II³(z⁺)⊂ II²(z⁺).

Subsequent classes of poverty indices are defined in an analogous manner. Generally speaking, poverty indices P(z) will be members of class II^s(z⁺) if (-l)^s π^(s) (Q(p);z) ≤ 0 and if π⁽ⁱ⁾(z,z) = 0 for i = 0, l, 2..., s - 2. As the order s of the class of poverty indices increases, the indices become more and more sensitive to the distribution of income among the poorest. At the limit, and as mentioned above, only the income of the poorest individual matters in comparing poverty across two distributions. Increasing the order s makes us focus on smaller subsets of poverty indices, in the sense that II^s (z⁺) ⊂ II^s-l(z⁺).

All poverty indices seen in Chapter 5 fit into some of the classes defined above. The poverty headcount F(z) clearly belongs to II¹ (z⁺) (whenever z ≤ z+). As we will see, it also plays a crucial role in tests of first-order dominance. But it does not belong to the higher-order classes since it is not continuous at the poverty line. The average poverty gap belongs to II¹(z⁺) and to II²(z⁺), but not to the higher-order classes. The square of the poverty gaps index belongs to II¹ (z⁺), II²(z⁺) and II³(z⁺), but not to II⁴(z⁺). More generally, the FGT indices, for which π(Q(p); z) = g(p; z)^a, belong to II^s(z⁺) when a α ≥ s - 1 and z ≤ z⁺. The Watts index belongs to IIⁱ(z⁺) and to II²(z⁺), but not to II³(z⁺) since it does not obey the π¹(z,z) = 0 restriction. A transformation of the Watts index, by which π(Q(p); z) = g(p; z) [ln(z) - In (Q*(p))], would, however, belong to II³(z⁺). The Chakravarty and Clark et al. indices belong to II¹ (z⁺) and II² (z⁺), and so do as well the S-Gini indices of poverty.

We can now see how to determine whether poverty in A is greater than in B for all indices that are members of any one of these classes. For this, there exist two approaches: a primal and a dual one. We consider them in turn.

10.1 Primal approach

10.1.1 Dominance tests

We are interested in whether we may assert confidently that poverty in a distribution A, as measured by P_A(z), is larger than poverty in a distribution B, P_B(z), for all of the poverty indices P(z) belonging to one of the classes of poverty indices defined above. We are therefore interested in checking whether the following difference in poverty indices ΔP(z) = P_A(z) - P_B(z) is positive:

where on the second line a change of variable has been effected and where Δf(y) is the difference in the densities of income. To demonstrate the dominance conditions, we will make repetitive use of integration by parts of (10.5). This process will involve the use of stochastic dominance curves D^s(z), for orders of dominance s = 1, 2, 3,.... D¹(z) is simply the cdf, F(z), namely, the proportion of individuals underneath the poverty line z. The higher order curves are iteratively defined as

Thus, D²(z) is simply the area underneath the cdf curve for a range of incomes between 0 and z. This is illustrated in Figure 10.1. The curve shows the cdf F(y) at different values of y. The grey-shaded area underneath that curve (up to z) thus gives D²(z).

Defined as in (10.6), dominance curves may seem complicated to calculate. Fortunately, there is a very useful link between the dominance curves and the popular FGT indices, a link that greatly facilitates the computation of D^s(z).

Figure 10.1: Primal stochastic dominance curves

We can indeed show that

where c = 1/(s - 1)! is a constant that can be basically ignored. Therefore, to compute the dominance curve of order s, we need only compute the FGT index at α = s - 1, which is P(z; α = s - 1) (see (5.7)). Recall that P(z;α = 1) is the average poverty gap. Hence, the dominance curve of order 2 is simply the average poverty gap at different poverty lines. This can also be seen on Figure 10.1. The distance between z and y gives (when it is positive) the poverty gap at a given value of income y. For y = y¹, for instance, Figure 10.1 shows that distance z - y¹. dF(y¹) — as measured on the vertical axis — gives the density of individuals at that level of income. The rectangular area given by the product of (z - y¹) and dF(y^l) then shows the contribution of those with income y¹ to the population average poverty gap. Integrating all such positive distances between y and z across the population thus amounts to calculating the average poverty gap — again, this is the sum of individual rectangles of lengths (z - y) and heights dF(y), or simply the grey-shaded area of Figure 10.1.

Let us now integrate by parts equation (10.5). This gives:

where ΔD^s(y) is defined as D_AA(y)-D^s_B(y). If we wish to ensure that ΔP(z) is positive for all of the indices that belong to II¹(z⁺), we need to ensure that (10.8) is positive for all of the poverty indices that satisfy the conditions in (10.2), whatever the values of their first-order derivative π¹ (y;z), so long as that derivative is everywhere non-positive between 0 and z⁺. For this to hold, we simply need that (recall that D¹(y) = F(y))¹:

We refer to this as first-order poverty dominance of B over A. The result can be summarized as follows:

First-order poverty dominance (primal):

The dominance condition in (10.10) is relatively stringent: it requires the headcount index in A never to be lower than the headcount in B, for all possible poverty lines between 0 and z+. If, however, the condition is found to hold in practice, a very robust poverty ordering is obtained: we can then unambiguously say that poverty is higher in A than in B for all of the poverty indices in II¹(z⁺) (including the headcount index). Since (almost) all of the poverty indices that have been proposed obey this restriction, this is a very powerful conclusion indeed. Note again that this ordering is valid for any choice of poverty line up to z⁺.

¹DAD: Dominance|Poverty Dominance.

Moving to second-order poverty dominance, we integrate equation (10.8) once more by parts and find that:

Recall that the indices that are members of II²(z⁺) are such that π²(Q(p);z)≥ 0 when Q(p) ≤ z and with π(z,z) = 0. Hence, if we wish ΔP(z) to be positive for all of the indices that belong to II²(z⁺), we must have that:

This is second-order poverty dominance of B over A; it can be summarized as:

Second-order poverty dominance (primal)²:

Recall from 10.7 that D²(z) = P(z; α = 1). Second-order poverty dominance thus requires the average poverty gap in A to be always larger than the average poverty gap in B, for all of the poverty lines between 0 and z⁺. If the condition in (10.13) is found to hold in practice, then we can say that poverty is higher in A than in B for all of the poverty indices that are continuous at the poverty line and that are equality preferring (their second-order derivative is positive). That, of course, also includes the average poverty gap itself. Most of the indices found in the literature fall into that category, a major exception being the headcount and the Sen index. And that ordering is again valid for any choice of poverty line between 0 and z⁺.

We can repeat this process for any arbitrarily higher order of dominance, by successive integration by parts and by determining the conditions under which all of the poverty indices P(z) that are members of a class II^s (z⁺) will indicate more poverty in A than in B, and this for all of the poverty lines z between 0 and z⁺. This gives the following general formulation of s^th order poverty dominance:

²DAD: Dominance|Poverty Dominance.

s^th-order poverty dominance (primal)³:

This condition is illustrated in Figure 10.2 for general s-order dominance, where dominance holds until z⁺, but would not hold if z⁺ exceeded z_s. Checking poverty dominance is thus conceptually straightforward. For first-order dominance, we use what has been termed "the poverty incidence curve", which is the headcount index as a function of the range of poverty lines [0, z⁺]. For second-order dominance, we use the "poverty deficit curve", which is the area underneath the poverty incidence curve or more simply the average poverty gap, again as a function of the range of poverty lines [0, z⁺]. Third-order dominance makes use of the area underneath the poverty deficit curve, or the square-of-poverty-gaps index (also called the poverty severity curve) for poverty lines between 0 and z+. Dominance curves for greater orders of dominance simply aggregate greater powers of poverty gaps, graphed against the same range of poverty lines [0, z⁺].

10.1.2 Nesting of dominance tests

The condition (10.13) for second-order dominance is less stringent than (10.10) for first-order poverty dominance. To see why, consider (10.6) again. When first-order dominance over [0, z⁺] holds, then second-order dominance over [0, z⁺] must also hold. Hence, when we find that the poverty indices in II¹(z⁺) show more poverty in A, we also know that the poverty indices in II²(z⁺) will do the same. That is of course consistent with the fact that II²(z⁺) ⊂ II¹(z⁺}.

Suppose, however, that we have that ΔD²(y) > 0 for all y ∈ [0, z⁺], but not that ΔD¹(y) > 0 for all y ∈ [0, z⁺}. Hence, we have first-order, but not second-order, dominance. Poverty is larger in A for all of the indices in II²(z⁺) but not for all those in II¹(z⁺). This is possible since II¹(z⁺) is a larger set than II²(z⁺).

These relationships are in fact sequentially valid for higher orders as well. This is illustrated in Figures 10.3 and 10.4. Figure 10.3 shows that a class of indices H^s+l(z⁺) is a subset of the lower-class of indices II^s(z⁺). Whenever an ordering is made over II^s(z⁺), it is also necessarily valid over the subset II^s+1(z⁺). Figure 10.4 analogously illustrates the size of the sets of distributions (A, B) that can be ordered by the dominance condition in (10.14). The greater the value of s, the more likely can a couple (A, B) fall into those sets, and therefore the more likely can they be compared unambiguously by that dominance condition. Taken jointly, Figures 10.3 and 10.4 show the trade-off that exists between wishing to assert whether A really has more poverty than B (Figure 10.4), and wishing to assert this for as large a class of poverty indices and poverty lines as possible (Figure 10.3).

³DAD: Dominance|Poverty Dominance.

Figure 10.2: s-order poverty dominance

For a simple illustration of these relationships, consider a comparison of distributions A and B in Table 9.1 on page 156. The first-order dominance condition (10.10) only holds if z⁺ is lower than 9. Hence, we can conclude that A has more poverty than B for any choice of first-order indices so long as the poverty line is less than 9. Indeed, it is not hard to find some first-order indices that will show more poverty in B when z exceeds 9: the headcount between 9 and 11 clearly shows more poverty in B. We can, however, verify that the second-order condition is obeyed for any choice of z⁺. This then implies that all second-order indices (those that are members of II²(z⁺)) will show more poverty in A, regardless of the choice of poverty line. This is quite a robust statement, since it is valid for all distribution-sensitive poverty indices (the headcount is not distribution-sensitive, hence it does not always indicate more poverty in A) and again for any choice of poverty line. Again, as mentioned above, second-order poverty dominance is a criterion that is less stringent to check in practice than first-order dominance. The price of this, however, is that the set of indices over which poverty dominance is checked is smaller for second-order dominance than for first-order dominance.

10.2 Dual approach

There exists a dual approach to testing first-order and second-order poverty dominance, which is sometimes called a p, percentile, or quantile approach. Whereas the primal approach makes use of curves that censor the population's income at varying poverty lines, the dual approach makes use of curves that truncate the population at varying percentile values. The dual approach has interesting graphical properties, which makes it useful and informative in checking poverty dominance.

10.2.1 First-order poverty dominance

To illustrate this second approach, we focus on indices that aggregate poverty gaps using weights that are functions of p:

Note that using aggregates of poverty gaps as in (10.15) is more restrictive than using functions π(Q(p); z) defined separately over Q(p) and z, as is done in (10.1). When the poverty lines are the same across distributions (as was implicitly assumed above for the primal approach, and as is almost always assumed to be the case in practice), the dominance rankings are, however, the same for the two approaches, as we will see below.

Membership in the (dual) first-order class II¹(z⁺) of poverty indices only requires that the weights ω(p) be non-negative functions of p:

If we want to check whether ΔΓ(z) = Γ_A(z) - F_B(z) is positive for all of the indices that belong to Il^l(z⁺), we need only assess whether gA(p;z⁺) ≥ gB(p;z⁺) for all p ∈ [0,1]. This yields the following dual first-order poverty dominance:

First-order poverty dominance (dual)⁴:

Condition (10.17) requires poverty gaps to be nowhere lower in A than in B, whatever the percentiles p considered. It thus amounts to ordering the poverty gap curves. It is not difficult to show that this is also equivalent to checking the primal first-order poverty dominance condition in (10.10). In other words, if we can order poverty over II¹ (z⁺), then we can also do so overII¹(z⁺), and vice versa. In fact, first-order poverty dominance (primal or dual) implies ordering all poverty indices (additive or otherwise) that are (weakly) decreasing in income. To check for such a wide degree of ethical robustness, we can use either the primal or the dual first-order poverty dominance condition.

First-order poverty dominance

The following conditions are equivalent:

1. Poverty is higher in A than in B for any of the poverty indices that obey the focus (see p.165), the anonymity (p.160), the population invariance (p.160) and the Pareto (p.159) principles and for any choice of poverty line between 0 and z⁺;
   
   
2. P_A(z;α = 0) ≥ P_B(z;α = 0) for all z between 0 and z⁺;
   
   
3. g_A(p;z⁺) ≥ g_B(p;z⁺) for all p between 0 and 1.

10.2.2 Second-order poverty dominance

Membership in the dual second-order class II2(z) of poverty indices requires that the weights ω(p) be positive and decreasing functions of the ranks

⁴DAD: Curves|Poverty Gap.

p:

To show what dominance condition applies to (10.18), recall that G(p;z) is the Cumulative Poverty Gap (CPG) curve, and integrate by parts (10.15):

For ΔΓ(z) to be positive for all of the indices that belong to II²(z⁺) (and therefore also for all poverty lines z ≤ z⁺), we need to order the CPG curves. The result is summarized as:

Second-order poverty dominance (dual)⁵:

Again, we can show that the condition in (10.20) is equivalent to the primal second-order poverty dominance condition in (10.13). In other words, if and only if Γ_A(z) - Γ_B(Z) ≥ 0 for all p(z) ∈ II²(z⁺), then P_A(z) - P_B(z) > 0 for all P(z) e II²(2⁺). Thus, to check robustness of poverty ordering over all distribution-sensitive poverty indices, we can use either the primal or the dual second-order poverty dominance condition. This is summarized as follows:

Second-order poverty dominance

The following conditions are equivalent:

1. Poverty is higher in A than in B for any of the poverty indices that obey the focus (see p.165), the anonymity (p.160), the population invariance (p. 160), the Pareto (p. 159) and the Pigou-Dalton (p. 161) principles and for any choice of poverty line between 0 and z⁺;
   
   
2. P_A(z; α = 1) ≥ P_B(z; α = 1) for all z between 0 and z⁺;
   
   
3. G_A(p; z⁺) ≥ G_B(p; z⁺) for all p between 0 and 1.

10.2.3 Higher-order poverty dominance

Dual conditions for higher-order poverty dominance are not as convenient and simple as those just stated for first- and second-order dominance. It is therefore usual to check higher-order dominance using the primal conditions of (10.14). Stated in terms of ethical principles, third-order dominance reads for instance as:

⁵DAD: Curves|CPG.

Third-order poverty dominance

The following conditions are equivalent:

1. Poverty is higher in A than in B for any of the poverty indices that obey the focus (see p.165), the anonymity (p.160), the population invariance (p.160), the Pareto (p.159), the Pigou-Dalton (p.161) and the transfer-sensitivity principles (p.161) and for any choice of poverty line between 0 and z⁺;
   
   
2. P_A(Z; α = 2) ≥ P_B(Z; α = 2) for all z between 0 and z⁺.

10.3 Assessing the limits to dominance

Whether we use the primal or the dual approach, testing for poverty dominance involves specifying an upper bound z⁺ for the ordering of the dominance curves. This bound can presumably be obtained from empirical or ethical work on what reasonable range of poverty lines should be used to compare poverty. It can of course also be specified arbitrarily by the researcher. An alternative strategy is to use the available sample information and estimate directly from that information the upper bound up to which a distributive comparison can be inferred to be robust. We can then interpret this statistics as a "critical" bound. In the light of the results above, this critical bound will limit the range of poverty lines over which we will be able to order poverty across A and B.

Assume for instance that a primal poverty dominance curve D^sA(y) for A is initially higher than that for B for low values of y, but that this ranking is reversed for higher values of y. Let be the first crossing point of the curves, such that Distribution B then has less poverty than distribution A for all of the poverty indices in II^s(z⁺), so long as As the notation implies, this calculation can be done for any desired order s of poverty dominance.

It may be, however, that we feel (for some order s) that is too low. Said differently, being able to order poverty only over a relatively narrow range may seem unsatisfactory. We may change this by moving to a higher order of dominance. Indeed, we can show that is increasing in s, with , whenever for some and for all . We may thus increase the range of poverty lines over which a poverty ranking is robust by moving up to a higher class of indices.

This is illustrated in Figure 10.5, where z⁺ < z⁺⁺. For the sake of illustration, suppose that the first-order dominance curves of A and B cross somewhere between 0 and z⁺⁺. It is then impossible to order poverty over all of the indices that belong to II^l(z⁺⁺). Assume, however, that decreasing the upper bound from z⁺⁺ to z⁺ does rank the distributions over II¹(z⁺), and that increasing the order of dominance from 1 to 2 while maintaining the up­per bound at z⁺⁺ also ranks the distributions in terms of poverty. In either case, poverty is now ordered, but over different sets. The alternative is then to choose between an ordering on indices that are ethically more restrictive (such as II²(z⁺⁺)), and an ordering on indices with a more restrictive range of poverty lines (such as II¹ (z⁺)).

10.4 References

Methods for testing poverty dominance are relatively recent, and postdate much of the literature on inequality and social welfare dominance. One of the early influential papers is Atkinson (1987) — that paper also introduced the idea of "restricted" dominance. The theoretical poverty dominance conditions have been further and rigorously explored in Foster and Shorrocks (1988b) and Foster and Shorrocks (1988c). Bounds to poverty dominance are discussed in Davidson and Duclos (2000). Zheng (2000a) provides a different approach based on "minimum distribution-sensitivity" poverty indices.

The Pigou-Dalton principle has been framed alternatively as a strong and as a weak axiom for the study of poverty indices (see Donaldson and Weymark (1986) and Zheng (1999a)). In the weak version, the axiom says that the poverty index must increase following a transfer from one individual to another wealthier individual, providing that both are initially below the poverty line and that the transfer does not lift the wealthier person above this threshold. The strong axiom postulates that the index must increase even if this transfer pushes the higher-income recipient above the poverty line. The strong formulation of the axiom is usually preferred.

Del Rio and Ruiz Castillo (2001), Jenkins and Lambert (1998a), Jenkins and Lambert (1998a), Jenkins and Lambert (1998b) and Spencer and Fisher (1992) discuss the use of CPG (or "TIP") curves (initially proposed by Jenkins and Lambert 1997) for second-order poverty dominance, surveys and integrative reviews of the literature can be found in Zheng (1999a), Zheng (2000b) and Zheng (2001a). US applications include Bishop, Formby, and Zeager (1996) (for the marginal impact of food stamps on US poverty) and Zheng, Gushing, and Chow (1995) (for another US application).

Figure 10.3: Poverty indices and ethical judgements — The sets of poverty indices that belong to the classes IIⁱ(z⁺). i = 1, 2, 3

Figure 10.4: Poverty dominance and income distributions — The sets of distributions that are ordered by the dominance conditions ΔDⁱ(y) ≥ 0, y ≤ z⁺, and i = 1, 2, 3

Figure 10.5: Classes of poverty indices and upper bounds for poverty lines

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