Document(s) 13 of 20

11.1 Ethical welfare judgments

As for poverty, we may wish to determine if the ranking of two distributions of income in terms of social welfare is robust to the choice of social welfare indices. Of course, one way to check such robustness would be to verify the welfare ranking of the two distributions for a large number of the many social welfare indices that have been proposed in the literature. This, however, would certainly be a tedious task. Besides, new social welfare indices can always be designed.

A simpler and more powerful alternative is to apply tests of welfare dominance. Unlike for poverty, welfare dominance tests take into account the whole distributions of income, as opposed to just the censored distributions used for poverty comparisons.

As for poverty dominance, there are two testing approaches, a primal (income-censoring) and a dual (percentile-truncating) one. The primal approach has the advantage of being applicable to any desired (however large) order of dominance, and uses curves of the well-known FGT indices for an infinite range of "poverty lines" or income censoring points. The dual approach is practically convenient only for first and second order dominance, but it uses curves that are graphically instructive and that have been documented extensively in the literature. As for poverty dominance, if, for first and second order dominance, a welfare ranking is obtained using one of these two testing approaches, the same ranking will be obtained using the other approach. In other words, the two approaches are equivalent in terms of their ability to rank distributions robustly over classes of first- and second-order social welfare indices.

As for poverty dominance, for both of these approaches we will make use of classes of social welfare indices defined by the reactions of indices to changes in or reallocations of income. These social welfare indices do not need to be additive, but for expositional convenience assume that they are defined in the simple rank-dependent utilitarian format of W in (4.28):

The first-order class of social welfare indices regroups all of the symmetric (or anonymous) social welfare indices that are increasing in income. In terms of (11.1), this can be formulated as the class Ω¹ with

The second-order class of social welfare indices regroups all of the first-order indices that are increasing in mean-preserving equalizing transfers. Recall that such transfers redistribute one dollar of income from a richer to a poorer person. These indices thus obey the Pigou-Dalton principle of transfers. Using (11.1) again, this suggests the class of Ω² indices:

The third-order class of social welfare indices includes all of the second-order indices that further obey the transfer-sensitivity principle — requiring that equalizing transfers have a greater impact on social welfare when they occur lower down in the distribution of income. Expressed in terms of (11.1), this requirement forces ω(p) to be a constant and requires the concavity of individual utility functions to be decreasing in income. This suggests Ω³:

As hinted above on page 162, higher orders of classes can be defined analogously. Generally speaking, membership in a higher-order class of social welfare indices requires these indices to be more sensitive to the income of the very poor. Membership in Ωs implies membership in Ω^s-1, and for s-order additive welfare indices, we also need that (-1)⁽ⁱ⁾U⁽ⁱ⁾(Q(p)) ≤ 0 for i=1,..., s.

11.2 Tests of welfare dominance

As for poverty dominance, both primal and dual conditions can be used for testing first- and second-order welfare dominance. The two types of tests order social welfare on exactly the same class of indices.

First-order welfare dominance

The following conditions are equivalent:

1. Social we!lfare is larger in B than in A for any of the social welfare indices that obey the Pareto (p.159), the anonymity (p.160) and the population invariance principles (p.160);
   
   
2. W_B - W_A ≥ 0 for all W ε Ω¹;
   
   
3. P_A(z;α = 0) ≥ P_B(z; α = 0) for all z between 0 and infinity;
   
   
4. for all z between 0 and infinity;
   
   
5. Q_A(p) ≤ Q_B(p) for all p between 0 and 1¹.

First-order welfare dominance can thus be checked by verifying whether the headcount index is higher for A than for B for all poverty lines z. There is therefore a useful analogue between tests of poverty and welfare dominance. Ordering two distributions of incomes over the first-order class of social welfare indices can also be done by comparing the incomes of the two distributions over the entire range of percentiles. Graphically, it requires checking that "Pen's parade of dwarfs and giants" be everywhere higher in B than in A, whatever the percentiles being compared. The two distributions "parade" simultaneously alongside each other, and the distributive analyst observes if one parade dominates everywhere the other.

A similar result can be stated for second-order welfare dominance. To see this, first recall the definition of the Generalized Lorenz curve GL(p) (see (4.44) on page 65):

The Generalized Lorenz curve sums all incomes up to quantile Q(p), and is therefore the cumulative Pen's parade. We then obtain:

Second-order welfare dominance

The following conditions are equivalent:

1. Social welfare is larger in B than in A for any of the social welfare indices that obey the Pareto (p.159), the anonymity (p.160), the population invariance (p.160) and the Pigou-Dalton (p.161) principles;
   
   
2. W_B - W_A ≥ 0 for all W ε Ω²;
   
   
3. P_A(z;α = 1) ≥ P_B(z; α = 1) for all z between 0 and infinity;
   
   
4. for all z between 0 and infinity;
   
   
5. GL_A(p) ≤ GL_B(p) for all p between 0 and 1 ².

¹DAD: Curves|Quantile.An exactly similar result applies for higher-order welfare dominance. As for poverty dominance, the dual conditions are less convenient and are omitted here.

Higher-order welfare dominance

The following conditions are equivalent:

1. W_B - W_A ≥ 0 for all W ε Ω^s;
   
   
2. P_A(z;α = s - 1) ≥ P_B(z; α = s - 1) for all z between 0 and infinity;
   
   
3. for all z between 0 and infinity.

Checking for s-order welfare dominance thus simply requires comparing the FGT indices for α = s - 1 over all possible poverty lines.

11.3 Inequality judgments

As for poverty and welfare dominance, we can define classes of relative inequality indices over which to check the robustness of the inequality orderings of two distributions of income. As we will see, these classes of inequality indices have properties which are analogous to those of the classes of social welfare indices. They react to income changes or income reallocations in a manner that depends on the order of the classes to which the indices belong. Unlike social welfare functions, however, relative inequality indices also need to be homogeneous of degree 0 in all income. This means that an equiproportionate change in all incomes will not affect the value of these relative inequality indices.

Consider first the class ¹(1⁺) of inequality indices of the first-order. Recall that income shares (or normalized quantiles) are given by . ¹(1⁺)is a class of inequality indices that is not usually considered in the literature because it censors at l⁺ the effects of changes affecting income shares. Indeed, besides being homogeneous of degree 0 in income, the indices that are members of ¹(l⁺) are such that, for a given mean, inequality decreases when an individual's income increases, so long as that individual's income share does not exceed l⁺. Said differently, the inequality indices in ¹(l⁺) are decreasing in the income shares of those with If the income of an individual with income share greater than l⁺ changes, then an index that is a member of ¹(l⁺) cannot change. We can think of keeping mean income constant, following these changes, through a decrease in the income of those individuals with income shares exceeding l⁺, since that will not by definition affect the first-order inequality indices. In addition to being symmetric in income, these indices are therefore in some loose sense of the Pareto type.

²DAD: Curves| Generalized Lorenz.

The Pareto principle underlying ¹(l⁺) is thus an alternative ethical principle to the well-known Pigou-Dalton principle of transfers, which has been at the heart of inequality analysis for several decades. But the scope of this Pareto principle is censored: it only applies to income shares below l⁺. This makes the first-order class of inequality indices a poverty-like class. For this reason, we will not have that ² ⊂ ¹(l⁺)

The Pigou-Dalton principle will postulate that a mean-preserving transfer of income from a higher-income person to a lower-income person decreases inequality, whatever the income shares of those affected by this income reallocation. All of the inequality indices that belong to the class ² of second-order inequality indices obey this principle and decrease after a mean-preserving equalizing transfer. These inequality indices are also said to be Schur-convex. Almost all of the frequently used inequality indices (including the Atkinson, S-Gini and Generalized entropy indices, with the notable exception of the variance of logarithms) are members of ².

Those inequality indices that belong to the class ³ of third-order inequality indices also belong to ², and weakly decrease after a favorable composite transfer. This includes the Atkinson indices and some of the Generalized entropy indices, but not the S-Gini indices, classes ^s of higher order inequality indices can be similarly defined. For instance, to be members of the class of fourth-order inequality indices, inequality indices must be members of ³ and must be more sensitive to favorable composite transfers when they take place lower down in the distribution of income. Again, all of the Atkinson indices belong to ⁴. The higher the value of s, the more Rawlsian are the indices since the more sensitive they are to the income shares of the poorest.

Comparing the definitions of the classes Ω^s and ^s, note that when the means of the distributions are equal, the social welfare ranking is the same as the inequality ranking, in the sense that if I_A ≥ I_B for all I in Ω^s, then W_A ≤ W_B for all W in Ω^s, and vice versa. In such cases, checking for inequality dominance can be done by checking for welfare dominance. When the means are not equal, we can normalize all incomes by their mean (this does not affect relative inequality), and then use the welfare dominance results described in Section 1 1.2 for Ω^s to check for dominance over a class ^sof relative inequality indices. Hence, to check for inequality dominance, we can simply test for welfare dominance once incomes have been normalized by their mean. When B has more welfare than A at order s, we can say that I_B is lower than I_A for all of the inequality indices that belong to ^s.

11.4 Tests of inequality dominance

As indicated above, checking for inequality dominance can be done most easily by using the welfare dominance conditions of Section 11.2 and normalizing incomes by their mean. For the primal dominance curves, we will thus need the normalized stochastic dominance curve , defined as

is nicely linked to the normalized FGT indices :

where c is as before a constant that we can ignore. Thus, estimating the normalized dominance curve at lμ and order s is equivalent to computing the normalized FGT index for a poverty line equal to lμ and for α equal to s - 1. Similarly, for dual dominance conditions, we may use the poverty gaps normalized by mean income:

This leads to:

First-order restricted inequality dominance

The following conditions are equivalent³:

1. I_A - I_B ≥ 0 for all I in ¹(l⁺);
   
   
2. for all λ between 0 and l⁺;
   
   
3. for all p between 0 and 1.

Note that the condition is easily interpreted. It simply compares the proportion of those with income less than l times the mean in A and in B. If there are fewer such individuals in B than in A, for all l ≤ l⁺, inequality is greater in A for all of the indices in ¹(l⁺).

Second-order inequality dominance

The following conditions are equivalent⁴:

³ DAD: Dominance]Inequality Dominance.

⁴ DAD: Dominance|Inequality Dominance.

1. Relative inequality is larger in B than in A for any of the inequality indices that obey the anonymity (p. 160), the population invariance (p. 160), and the Pigou-Dalton (p. 161) principles;
   
   
2. I_A - I_B ≥ 0 for all I ε ²;
   
   
3. for all λ between 0 and infinity;
   
   
4. for all λ between 0 and infinity;
   
   
5. L_A(p) ≤ L_B(p) for all p between 0 and 1.

Testing for second-order inequality dominance can thus be done simply by comparing the usual normalized average poverty gap for A and for B for all possible proportions of the mean as poverty lines. An alternative equivalent test is that of comparing the Lorenz curves for A and B. This is the well-known and classical Lorenz test, which has long been considered the golden rule of relative inequality rankings. Dual conditions for higher-order (i.e., third-order) inequality dominance have also been proposed in the literature, but they are again less convenient to use than the primal conditions.

A general s-order inequality dominance condition is then simply stated as:

s-order inequality dominance

The following conditions are equivalent ⁵:

1. I_A - I_B ≥ 0 for all I ε ^s;
   
   
2. for all λ between 0 and infinity;
   
   
3. for all λ between 0 and infinity.

11.5 Inequality and progressivity

We can combine some of the results derived above to what we saw in Chapter 7 on the measurement of vertical equity in order to link progressivity and inequality dominance.

First, in the absence of reranking, it is clear that a tax and/or a transfer that is TR- or IR-progressive, will decrease all of the inequality indices that are members of ². This is most easily seen by considering equations (8.11) and (8.13) and by noting that CN(p) = LN(p) when there is no reranking. For IR progressivity, this follows from the fact that a concentration curve for net income that lies above the Lorenz curve of gross income pushes the Lorenz curve of net income upward. This decreases inequality for all second-order indices of inequality.

⁵ DAD: DominanceJInequality Dominance.

Further, again in the absence of reranking, if a tax and/or transfer T₁ is more IR-progressive than a tax and/or transfer T₂ then T₁ necessarily reduces inequality by more than T₂ when inequality is measured by any of the inequality indices that belong to ². This can be seen by the sum of the IR- progressivity terms in (8.15) (see also equation (8.11)) and by noting again that C_N(p) = L_N(p) in the absence of reranking.

We may also be concerned about the impact of a tax and benefit system on the class of first-order inequality indices, viz, on those indices that are monotonic in some lower income shares, but not always monotonic in cumulative income shares. To check whether this impact reduces first-order inequality indices, we must check whether is always lower than μT/μX for all of the X that are below some censoring point l⁺μ. This supposes again, however, that the tax does not induce reranking. When it does, one way to account for the reranking effect is to compute "income growth curves", which are given by (N(p) - X(p))/X(p). (We will return to these curves in the context of the discussion of pro-poor growth in Section 11.8.) When these curves exceed the growth in average income — given by (μ_N - μ_X)/μ_X — for all p ≤ F_X(l⁺μ_X), then all of the first-order inequality indices in ¹(l⁺) will fall.

11.6 Social welfare and Lorenz curves

It often occurs that two income distributions A and B are compared using estimates of average income and inequality separately. Using second-order dual conditions, it is straightforward to combine these estimates to assess whether social welfare is greater in A than in B by noting from (4.44) that GL(p) = μL(p).

Say that we dispose of the entire Lorenz curves of each of the two distributions. Figure 11.1 shows four cases of comparisons of average income and inequality across these two distributions. In Case 1, A Lorenz-dominates B, and it also has a higher average income. Hence, there is generalized-Lorenz-dominance of A over B, and we are therefore assured that W_A - W_B ≥ 0 for all W ε Ω². In Case 2, A also dominates B according to the Lorenz criterion, but μ_A < μ_B; because of this, GL_A(P) crosses GL_B(p) and there can be no unambiguous second-order social welfare ranking. μ_A ≥ μ_B is indeed a necessary condition for welfare dominance of A over B for any order of dominance. Comparing the slopes of each of these two curves gives, however, the quantiles at various percentiles p. Since these quantiles are visibly larger in A than in B for a large lower range of p, A has less poverty than B for a large range of possible poverty lines and for many poverty indices. Case 3 depicts an ambiguous ranking of inequality across A and B. However, because μA is well above μ_B the Generalized Lorenz curve for A is above that for B. Finally, Case 4 shows a circumstance in which inequality and social welfare rankings clash. A has unambiguously less inequality than B according to the Lorenz criterion, but μ_A being significantly below μB has unambiguously less social welfare than B according to the generalized-Lorenz criterion and to second-order social welfare dominance.

11.7 The distributive impact of benefits

The impact of government benefits and transfers on the distribution of incomes can also be visualized using curves that are linked to the poverty, social welfare and inequality dominance curves.

Say, for instance, that the expected benefit at rank p of some government program — or some economic change — is given by (This could be estimated non-parametrically.) An impact indicator of the cumulative effect of that benefit up to rank p is given by:

with μ_B = GC_B(1). In analogy to the Generalized Lorenz curve, we may call GC_B(p) a Generalized concentration curve. GC_B(p) shows approximately the absolute contribution of the bottom proportion p of the population to the per capita benefits. The impact GC_B(p) is only approximate since it ignores the possible reranking of individuals by the program. The concentration curve of the benefit up to rank p can then be defined as:

Recall that the concentration curve C_B(p) at p gives the percentage of the total benefits that accrue to those with initial rank p or lower. Using C_B(p) and GC_B(p) can help assess the distributive effect of the program. For instance:

1. For understanding the approximate impact of the benefit on social welfare, we may wish to test whether is always positive, regardless of p. If so, then the benefit will tend to increase social welfare for all first-order welfare indices. If not, we can test if GC_B(p) is always positive regardless of p. If so, then the approximate impact of the benefit is to increase social welfare for all second-order welfare indices.
   
   
2. For understanding the approximate impact of the benefit on poverty, we proceed basically as in point 1 just above, with the only difference that we assess the curves and GC_B(p) only for all p ε [0,F(z⁺)]. If is always positive over that range of p, then the benefit will tend to decrease poverty for all first-order poverty indices Π¹ (z⁺), and if GC_B(p) is always positive for all p ε [0,F(z⁺)], then the approximate impact of the benefit is to decrease poverty for all poverty indices in Π²(z⁺).
   
   
3. For assessing the impact of the benefit on inequality and relative poverty, we may compare with X(p)/μX, and CB(p) with L_X(p). Comparing with X(p)/μ_X sheds light on the approximate impact of the benefit on first-order inequality indices, whereas comparing C_B(p) with L(p) shows the approximate impact of the benefit on second-order inequality indices. We compare these curves for all p ε [0, 1] if we are concerned about the whole population, for all p ε [0,F(z⁺)] if we are only concerned about the poor, or for all p ε [0, F(l⁺μ)] if we are concerned about first-order inequality indices.

11.8 Pro-poor growth

Assessing whether distributional changes are "pro-poor" has become increasingly widespread in academic and policy circles. We will see that it is relatively straightforward to use the tools developed above to make such an assessment. There are, however, two important issues that we must first discuss.

The first issue is whether our pro-poor standard should be absolute or relative. This is equivalent to asking whether we should be interested in the impact of growth on absolute poverty or on relative inequality. It is indeed important to distinguish between expectations that growth should change the incomes of the poor by the same absolute or by the same proportional amount — these expectations are conceptually not the same, and their empirical realization also varies significantly.

The second issue is whether pro-poor judgements should put relatively more emphasis on the impact of growth upon the poorer of the poor. This is equivalent to deciding whether our pro-poor judgements should obey higher-order ethical principles such as the Pigou-Dalton principle. We will consider two orders of pro-poor judgements: the first will obey the focus, the anonymity and the Pareto principles, and the second will also obey the Pigou-Dalton principle.

11.8.1 First-order pro-poor judgements

Let a distributive change entail a movement from a distribution X(p) to a distribution N(p), Let "income growth curves" be defined as the proportional change in income observed at various percentiles ⁶:

If the income-growth curve is positive everywhere over p ε [0, 1], then it is clear from the first-order welfare dominance results of page 183 that the change increases social welfare for all of the welfare indices that belong to Ω¹. It is also clear from the first-order poverty dominance results of page 174 that the change decreases poverty for all of the poverty indices that belong to Π¹(∞) (and thus for all those that obey the first-order — focus, Pareto and anonymity — ethical principles). This result is valid for any choice of poverty lines.

A test with a greater chance to succeed is to check whether the income-growth curve is positive everywhere over p ε [0,FX(z⁺)]. If so, then the distributive change decreases poverty for all poverty indices P(z) that belong to Π^l(z⁺). In such circumstances, the change can be called "absolutely pro-poor", in the sense that the poor benefit in absolute terms from the distributive change. We then have:

First-order absolute pro-poor judgements

The following statements are equivalent:

1. A movement from X to N is first-order absolutely pro-poor for all choices of poverty lines between 0 and z⁺;
   
   
2. Poverty is higher in X than in N for all of the poverty indices that obey the focus (p. 165), the population invariance (p. 160), the anonymity (p. 160) and the Pareto (p. 159) principles and for any choice of poverty line between 0 and z⁺;
   
   
3. P_X(z;α = 0) ≥ P_N(Z; α = 0) for all z between 0 and z⁺;
   
   
4. g(p) ≥ 0 for all p between 0 and F_X(z⁺).

Income growth curves can also be used to test whether a distributive change is "relatively pro-poor", in the sense that the change increases the incomes of the poor at a faster rate than that of the incomes of the rest of the population. For that purpose, we only need to compare the income growth curve g(p) at various percentiles to the growth in mean income. If the income growth curve at all p ε [0,F(z⁺)] is higher than the growth in mean income, then the change can be said to be first-order relatively pro-poor. An exactly equivalent test can be done by comparing the normalized quantiles for the initial and posterior incomes — recall that normalized quantiles are just incomes as a proportion of mean income. If the normalized quantiles of the poor are increased by the change, then the change is first-order relatively pro-poor. We thus have:

⁶ DAD: Curves|Pro-Poor.

First-order relative pro-poor judgements

The following statements are equivalent:

1. A movement from X to N is first-order relatively pro-poor for all choices of poverty lines between 0 and z⁺;
   
   
2. for all p between 0 and F_X(z⁺);
   
   
3. Q_X(p)/μ_X ≤ Q_N(p)/μ_N for all p between 0 and F_X(z⁺);
   
   
4. I_X - I_N ≥ 0 for all I in Π^l(z⁺/μ_X);
   
   
5. F_X(λμ_N) ≥ F_N (λμ_N) for all λ between 0 and z⁺/μ_X

11.8.2 Second-order pro-poor judgements

Testing for first-order pro-poor judgements can be demanding. It requires all quantiles of the poor to undergo a rate of growth that is either positive (for absolute judgements) or at least as large as the growth rate in average income (for relative judgements). We may want to relax this on the basis that a large rate of growth for the poorer among the poor may sometimes be deemed ethically sufficient to offset a low rate of growth for some percentiles of the not-so-poor. This therefore says that pro-poor judgements could give greater weight to the growth experience of the poorer among the poor. Implementing this is done by forcing pro-poor judgements to obey the Pigou-Dalton principie.

Second-order absolute pro-poor judgements

The following statements are equivalent:

1. A movement from X to N is second-order absolutely pro-poor for all choices of poverty lines between 0 and z⁺;
   
   
2. Poverty is higher in X than in N for all of the poverty indices that obey the focus (p. 165), the anonymity (p.160), the population invariance (p. 160), the Pareto (p. 159) and the Pigou-Dalton principles (p. 161) and for any choice of poverty line between 0 and z⁺;
   
   
3. P_X(z; α = 1) ≥ P_N(z;α = 1) for all z between 0 and z⁺;
   
   
4. G_N(p; z+) ≤ G_X(p; z⁺) for all p ε [0, 1].

Recall that the cumulative income up to rank p is given by the Generalized Lorenz curve. Denote its proportional change by ⁷

A sufficient condition for a second-order absolute pro-poor change is then that the growth in cumulative incomes be positive:

As for first-order pro-poor judgements, we may wish second-order judgements to require that the incomes of the poor at least keep up with those of the rest of the population. This yields:

Second-order relative pro-poor judgements

The following statements are equivalent:

1. A movement from X to N is second-order relatively pro-poor for all choices of poverty lines between 0 and z⁺;
   
   
2. for all λ between 0 and z⁺/μ_X

If the above conditions hold for z⁺ = ∞, then the change also reduces all of the inequality indices that are members of ². From the Theorem on second-order inequality dominance of page 186, this is therefore also equivalent to checking whether the Lorenz curve is pushed up by the distributive change.

A sufficient condition for second-order relative pro-poorness can also be implemented by comparing the growth in the cumulative incomes of the poor to the growth in average income. If, for all p lower than F(z⁺), the percentage growth in the cumulative incomes of a bottom proportion p of the population is larger than the percentage growth in mean income, then the change can be said to be second-order relatively pro-poor:

Income growth curves and cumulative income growth curves may also be used to assess the impact of a distributive change on relative poverty. The procedure is similar to that of checking whether the change is pro-poor — we compare income growth for the poor to the growth of some central tendency of the income distribution. One difference with the measurement of pro-poor growth is that the central tendency of interest may be some quantile (such as median income) if the relative poverty line is set as a proportion of that quantile

⁷ DAD: Curves|Pro-Poor.

11.9 References

Methods for establishing inequality dominance surprisingly predate those for establishing welfare dominance in welfare economics. The seminal works are those by Atkinson (1970), Dasgupta, Sen, and Starret (1973) and Kolm (1969) for inequality dominance, and Shorrocks (1983) and Foster and Shorrocks (1988c) for welfare dominance. Foster and Shorrocks (1988a) explore the links between relative poverty and relative inequality dominance (see also Davidson and Duclos 2000 and Formby, Smith, and Zheng 1999). Welfare economists have made extensive use of the literature on the ranking of distributions under risk aversion — see among many others Fishburn and Vickson (1978), Pratt (1964), Whitmore (1970) and Yitzhaki (1982b).

Descriptions and theoretical foundations of dual stochastic dominance tools can be found inter alia in Pen's parade of "dwarfs and giants" (Pen 1971, Chapter 3), in Yaari (1987), in Moyes (1999) (for links with Lorenz curves), and in Davies and Hoy (1995) and Muliere and Scarsini (1989) (for when Lorenz curves intersect).

Empirical tests for inequality and welfare dominance are numerous; they include inter alia Bishop, Formby, and Smith (1991d) (Lorenz dominance in the US), Bishop, Chow, and Formby (1991b) (first-order and truncated dominance), Bishop, Formby, and Thistle (1991e) (Pen or "rank" dominance), Bishop, Formby, and Smith (1991c) (Lorenz dominance across 9 countries), Bishop, Formby, and Thistle (1992) (convergence of US regional distributions), Bishop, Formby, and Smith (1993) (welfare and inequality dominance using LIS data), Chen, Datt, and Ravallion (1994) (comparisons of 44 less developed countries), Gouveia and Tavares (1995) (Portuguese distributions), Makdissi and Groleau (2002) (Canadian distributions), Ravallion (1992) (Indonesia), Sahn and Stifel (2002) (applied to nutritional data), and Wang, Shi, and Zheng (2002) (comparing inequality and social welfare in China).

Numerous methods and indices have been proposed recently for assessing whether distributive changes are pro-poor. See, for instance, McCulloch and Baulch (1999) for the difference between a post-change poverty headcount with that headcount which would have occurred if all had gained equally; Kakwani, Khandker, and Son (2003) for a "poverty equivalent growth rate" which computes (using estimates of poverty elasticities) the growth rate that would have been needed to achieve some poverty change without a change in the distribution of relative incomes, and then compares that growth rate to the growth rate in mean income; Kakwani and Pernia (2000) for a pro-poor index given by the ratio of the actual change in poverty over the change that would have been observed under distributional neutrality, and then compares its value to 1; Dollar and Kraay (2002) for a comparison of the growth rate in average income to the growth rate in the incomes of the lowest quintile; Ravallion and Chen (2003) for a comparison of the growth rate in average income to a "population weighted" average growth rate of the initially poor percentiles of the population — this can also be done using the area underneath the income growth curve g(p) defined in (11.11); Klasen (2003) for a comparison of the growth rate in average income to "population" and "poverty weighted" average growth rates; Essama-Nssah (2004) for the use of an ethically-flexible weighted average of individual growth rates that does not make use of poverty lines; Datt and Ravallion (2002) for an example of the popular use of growth elasticities of poverty measures; and Son (2004) for a "poverty growth curve" that displays the growth rate in the mean income of a bottom proportion p of the population — the cumulative income growth curve G(p) of (11.12) — and compares it to the growth rate in mean income.

Figure 11.1: Inequality and social welfare dominance

Document(s) 13 of 20