Document(s) 14 of 20

12.1 The impact of targeting

For policy purposes, it is often as useful to assess the impact of reforms to a benefit or public expenditure program as it is to evaluate the effect of existing programs. For administrative or political reasons, it may indeed be impossible to eliminate or to amend dramatically the structure of existing programs. Hence, comparing a current tax or benefit program with a situation in which it is supposed not to exist may not be very useful for practical purposes. Marginal reforms to such programs are nevertheless often feasible, and we therefore focus on them in this chapter. As we will see, focusing on marginal reforms also has the advantage of making it possible to measure the welfare impact of such reforms independently of the behavioral adjustment that individuals may make in reaction to these reforms.

We consider five such marginal reforms in this chapter. The first reform one channels public expenditure benefits to members of specific and easily observable socio-economic groups. The main issue then is: for which socio-economic group is additional public money best spent to reduce aggregate poverty? The second type of reform consists in an increase in public expenditures that raises all incomes in some socio-economic groups by some proportional amount. Again, an important question is: For which socio-economic group would this increase in public expenditures reduce aggregate poverty the fastest? This second type of reform can also be thought as (for instance) a process that increases the quality of infrastructure and the quantity of economic activity in a particular group or region in a way that affects proportionally all incomes and that is thus distributionally neutral in the sense of not affecting inequality within the groups affected.

The third type of reform considers a change in the price of some commodities, either through some macroeconomic or external shocks, or through a change in commodity taxes or subsidies. How is the distribution of well-being, and poverty in particular, affected by such a price change? The fourth question we ask is: what type of reform to a system of commodity taxes and subsidies could we implement, with no change in overall government revenues, but with a fall in poverty? That is, which commodities should be prime targets for a reduction in their tax rate or for an increase in their rate of subsidy and which others should see their tax rate increase? The fifth and last type of reform affects proportionally all incomes of a certain type—such as some type of farm income, the labor income of some type of workers, etc... Which sort of income sources should the government attempt to bolster if the primary aim is to alleviate poverty?

For all such reforms, we measure their poverty impact by the change in the FGT poverty indices that they cause. Recall that the use of the FGT indices is closely connected to checks for stochastic dominance and for the ethical robustness of poverty changes. Hence, we can use the methods below to determine how the reforms affect poverty as measured not only by the FGT poverty indices, but also by all of the poverty indices that obey some ethical conditions. For instance, if we find that some form of targeting decreases a FGT index of some α value for a range [0, z⁺] of poverty lines, then we know that the reform will also decrease all poverty indices of ethical order α + 1, whatever the choice of a poverty line within [0, z⁺].

12.1.1 Group-targeting a constant amount

We consider first the effect of a transfer of a constant amount of income to everyone in a group k. For this, recall that the FGT index can be decomposed as:

The per capita cost to the government of granting an equal amount η(k) to each member of a group k is equal to:

Aggregate poverty after such transfers equals P(k; z; a):

To determine which group k should be of greatest priority for the targeting of government expenditures, we need to determine for which group k targeted government expenditures (in the form of ηk) reduce aggregate poverty the most per government dollar spent. In other words, we need to compare across k the aggregate poverty reduction benefits of targeting one government dollar to a group k.

When a ≠ 0, we can show that the marginal reduction of aggregate poverty per dollar of per capita government expenditures is given by¹:

and, for the normalized FGT, by:

E:18.8.45

To reduce P(z; α) the most, we must therefore target those groups for which P(k; z; α – 1) is the greatest. It is thus simply the FGT index with α – 1 that guides policy based on reducing FGT with α The greater the value of α, the greater the chance that we will favor those groups where extreme poverty is highest.

E: 18.8.34

When a = 0, the per dollar reduction of aggregate poverty is given by f(k; z), the group k's density of income at the poverty line²:

We must then target those groups with the greatest density of people just around the poverty line, regardless of how much poverty there is below that poverty line—another consequence of the insensitivity of the headcount index to the distribution of incomes below z.

Table 12.1 summarizes the marginal poverty impact of targeting a constant amount to everyone in the population, or only to those in a group k. The impact is shown both for the normalized and un-normalized FGT indices. The bottom part of the table shows the poverty impact relative to the overall per capita cost of the targeting program.

12.1.2 Inequality-neutral targeting

Consider now a transfer that increases by a proportion λ(k)– -1 the income Q(k;p) of each member of a group k. The increase in income is thus

¹ DAD: poverty|lump-sum Targeting.

² DAD: poverty|lump-sum Targeting.

Table 12.1: The impact on poverty of targeting a constant marginal amount to everyone in the population or in a group k

(λ(k)–l)Q(k;p). The FGT index for group k after such a transfer is then³

For α ≠ 0, the marginal impact of a change in λ(k) is given by

and by

for the normalized FGT index

E:18.8.37

How (12.8) (and (12.9)) varies across values of k depends on two factors. First, there is the factor [P(k;z;α) – zP(k;z;α – 1). Groups in which there is a significant presence of extreme poverty will tend to see their P(k;z;α) poverty indices fall significantly with α, and will thus exhibit a large value of [P(k; z; α)–zP(k; z; α – 1)]. We may thus expect that these groups should be a priority for government targeting. However, those groups with considerable incidence of extreme poverty are also those for which a proportional increase in income has the least impact on the average income of the poor—since there is then little income on which growth may have an effect. Hence, whether those groups with a higher incidence of extreme poverty will exhibit a higher value of [P(k; z; α)–zP(k; z;α–1)] is ambiguous.

The second factor that enters into (12.8) is population share ø(k). Ceteris paribus, targeting government expenditures (in the form of an increase in λ(k)) to groups with a higher population share will naturally tend to decrease overall poverty faster. But this fails to take into account that a given increase in λ(k) will generally be more costly for the government to attain for groups with a large share of the population. Because of this, we may instead wish to compare across groups the ratio of the benefit in poverty reduction to the group per capita increase in income. Assume for simplicity that the cost of this group per capita income increase is entirely borne by the government. The per capita revenue impact of such a transfer on the government budget equals where:

³ DAD: Poverty|inequality-neutral Targeting

When α ≠ 0, the reduction of aggregate poverty per dollar spent per capita is then

and

for the normalized FGT index. To reduce P(z; α) the fastest, the government should therefore target those groups for which the term on the right is the greatest in absolute value. Compared to (12.8), (12.11) and (12.12) do not feature population shares since these shares are cancelled by the revenue impact of the government transfer. There now appears, however, the term μ(k) in the denominator. Indeed, if it must bear the entire cost of the income increase, the government will have to pay more to achieve a given increase in λ(k) for those groups with a high average income than for those groups with a lower average income level. Finally, and for the same reasons as those mentioned above, whether those groups with a higher incidence of extreme poverty will exhibit a higher value of [P(k; z; α)–zP(k; z;α– 1)] is ambiguous.

When α = 0, the per-dollar reduction of aggregate poverty following a proportional-to-income transfer is given by

Those groups with a high density of income at the poverty line, and whose average income is small, are then a prime target for poverty-efficient proportional-to-income transfer scheme.

Table 12.2 summarizes the marginal poverty impact of Inequality-neutral targeting either to everyone in the population, or to only those in a group k. The impact is shown both for the normalized and un-normalized FGT indices; as for Table 12.1, the bottom part of the table shows the poverty impact relative to the overall per capita cost of the targeting program.

12.2 The impact of changes in the poverty line

Variability of poverty line estimates across time, space, or poverty analyses and institutions can occur for several reasons. There may be methodological uncertainty and divergences as to how poverty lines should be estimated (recall Chapter 6). Estimation (sampling and non-sampling) errors also occur for purely statistical and survey reasons (see Chapters 16 and 17). Poverty lines may also be updated with time due to new data becoming available, or due

Table 12.2: The impact on poverty of inequality-neutral targeting within the entire population or within a group k

to the evolution of some form of socially representative or reference income. Whatever the reason, it may be useful, given this uncertainty, to assess how responsive poverty measurement will be to such variability in poverty line estimates.

To do this, consider first the case of the un-normalized FGT indices. We find that

For the headcount index, what matters is thus the income density at the poverty line. For higher-α indices, the sensitivity to the poverty line is given simply by P(z; α – 1}. The elasticity of FGT indices to the poverty line then follows as

Note that the elasticity of the headcount index has a useful graphical interpretation. Consider Figure 12.1 which shows the income density f(y) at different values of y. The area underneath the f(y) curve up to y = z gives the headcount P(α = 0; z) = F(z). The value of zf(k; z) is given by the size of the rectangle with width z and height f(z) in Figure 12.1. Hence, the elasticity of the headcount with respect to the poverty line is simply the ratio of the rectangular area zf(k; z) over the shaded area F(z). This elasticity is larger than 1 whenever the poverty line z is lower than the (first) mode of the distribution, and will in fact be above 1 in Figure 12.1 for any poverty line up to approximatively z'. For poverty lines larger than z', the poverty elasticity falls below 1. Thus, it is only for societies in which the headcount is initially high that we can expect the elasticity of the headcount with respect to the poverty line to be lower than 1. Otherwise, a change of 1% in the poverty line will cause a change of more than 1% in the headcount index. For normalized FGT indices, we obtain⁴

and

for the corresponding elasticities. Although expressed differently, the elasticities in (12.15) and (12.17) are the same.

⁴ DAD: Poverty|FGT Elasticity.

12.3 Price changes

The level of prices is an important determinant of the distribution of incomes, and can therefore matter significantly for poverty analysis. Governments can affect their levels directly or indirectly, through the use of sales and indirect taxes, competition policy, export taxes and import duties, subsidies on food, education, energy or transportation, etc..

To see how changes in prices (and therefore how price-changing reforms) can impact poverty, let y be a household-specific level of exogenous income, and express consumers' preferences as ν The indirect utility function is given by V(y, q; ν), where q is a vector of consumer and producer prices. We define a vector of reference prices as q^R—this is necessary to assess consumers' well-being at constant prices. Denote the real income in the post-reform situation by y^R, where y^R is measured on the basis of the reference prices q^R. y^Ris implicitly defined by v (y^R, q^R;ν) = v (y, q; ν,) and explicitly by the real income function y^R = R (y, q, q^R ν), where

By definition, y^R gives the level of income that provides under q^R the same utility as y yields under q.

We then wish to determine how real incomes are affected by a marginal change in prices. Let x_c (y, q; ν) be the net consumption of good c (which can be negative if the individual or household is a net producer of good c) of a consumer/producer with income y, preferences ν and facing prices q. Let q_c be the price of good c. We thus have y = Σq_cx_c(y,q;v). Differentiating (12.18), we find:

Using Roy's identity and setting reference prices to pre-reform prices, this leads to:

Equation (12.20) says that the observed pre-reform net consumption of good c is a sufficient statistic to know the impact on real income of a marginal change in the price of good c. This simple relationship is also valid for rationed goods. Equation (12.20) gives a "first-order approximation" to the true change

in real income that occurs following a change in the price of good c. The approximation is exact when the price change is marginal. It is less exact if the price change is non-marginal and if the compensated demand for good c varies significantly with q_c.

Assume that preferences ν and exogenous income y are jointly distributed according to the distribution function F(y, ν). The conditional distribution of ν given y is denoted by F (ν\y), and the marginal distribution of income y is given by F(y). Let preferences belong to the set Θ and assume income to be distributed over [0, a]. Expected consumption of good c at income y is given by x_c(y,q), such that

where Eν indicates that the expected consumption of good c is taken over all preferences in the set Θ. By (12.20), –x_c(y, q) is also proportional to the expected fall in real incomes of those with income y following an increase in q_c

Let x_c(q) then be the per capital consumption of good c, defined as . By (12.20), x_c(q) is also the average welfare cost of an increase in the price of good c. As a proportion of per capita consumption, consumption of good c at income y is expressed as

We can now see how the FGT indices are affected by a change in the price of good c. (For un-normalized FGT indices, we simply multiply the results by z^α.) Using (12.20), we find that:

E: 18.8.39

where f(z) is again the density of income at z. When graphed over a range of poverty lines z, this effect generates the so-called "consumption dominance" CD^c(z; a) curve of a good c⁵:

E: 18.8.40

Note that the impact on poverty depends on α and z. By (12.22), CD(z; α = 0) only takes into account the consumption pattern of those precisely at z. The impact of an increase in the price of good c on the headcount index will be large if there are many individuals bordering the poverty line (f(z) is then large) and/or if these individuals consume much of good c (x_c (z, q^R) is then large). The CD^c(z;α = 1) curve gives the absolute contribution to total consumption of good c of those individuals with income less than z. It is therefore an informative statistics on the distribution of consumption expenditures, similar in content to the generalized concentration curve GC_xc (p) for good c — which gives the absolute contribution to total x^c consumption of those below a certain rank p. For α = 2, 3,..., progressively greater weight is given to the shares of those with higher poverty gaps.

⁵ DAD: Curves|C-Dominance and DAD: Poverty|impact of price change.

12.4 Tax and subsidy reforms

The above section gave us the tools needed to assess the impact of marginal price changes on poverty. We may also use these tools to assess whether a revenue-neutral tax and subsidy reform could be implemented that would reduce aggregate poverty.

For this, we need to take into account the government budget constraint, and more particularly the net revenues that the government raises from a policy of commodity taxes and subsidies. Let t be the vector of tax rates on the C goods. Setting producer prices to 1 and assuming them to be constant (for simplicity) and invariant to changes in t, we then have q = 1 + t and dq_c = dt_c, where t_c denotes the tax rate on good c. Let per capita net commodity tax revenues be denoted as R(q). They are equal to . Without loss of generality, assume that the government's tax reform increases the tax rate on the j^th commodity and uses the extra revenue raised to decrease the tax rate (or to increase the subsidy) on the Ith commodity. Revenue neutrality of the tax reform requires that

Now define γ as

The numerator in (12.25) gives the marginal tax revenue of a marginal increase in the price of good I, per unit of the average welfare cost that this price increase imposes on consumers. Equivalently, this is 1 minus the deadweight loss of taxing good l, or the inverse of the marginal economic efficiency cost of funds (MECF) from taxing l (see Wildasin (1984)). The denominator gives exactly the same measures for an increase in the price of good j. γis thus the economic (or "average") efficiency of taxing good I relative to taxing good j. We may thus interpret γ as the efficiency cost of taxing j relative to that of taxing l (the MECF for j over that for I). The higher the value of γ the less economically efficient is taxing good j.

By simple algebraic manipulation, we can then rewrite equation (12.24) as

which fixes dq_j as a revenue-neutral proportion of dq_l. This last relationship yields a nice synthetic expression for the impact on a FGT index P(z; a) of a revenue-neutral tax reform that increases the tax on a good l for the benefit of a fall in the tax on a good j⁶

We then wish to check whether such a tax reform would lead to a fall in poverty. For the fall to "ethically robust", we would want to check that it occurs for any one of the poverty indices of some ethical order and for a range of poverty lines. To test this, it is useful to define normalized CD curves and to denote them as (z; α). Normalized CD curves are just the above-defined CD_c curves for good c normalized by the average consumption of that good, x_c(q):

E: 18.8.30

curves are thus the ethically weighted (or social) cost of taxing c as a proportion of the average welfare cost. Comparing normalized (z; α) curves thus allows comparing the distributive benefits of decreasing tax rates (or increasing subsidies) across commodities, per dollar of average welfare benefit. If , then poverty falls faster per dollar of average welfare benefit if taxes on l are decreased (instead of taxes on j)⁷.

For overall social efficiency, we must also take into account the parameter of economic efficiency, γ This parameter translates tax revenue into average welfare changes. Suppose that we were to envisage a revenue neutral tax reform that decreases t_l but increases t_j. It follows from (12.27) that this tax reform is poverty reducing is and only if

⁶ DAD: Poverty|Impact of Tax Reform.

⁷ DAD: Curves|C-Dominance.

Recall from (12.25) that when γexceeds 1, the economic efficiency cost of taxing j exceeds that of taxing l. Considering economic efficiency alone then suggests increasing t_l and decreasing t_j.

The left-hand-side of (12.29) shows the distributive benefit of the reform. It compares the fall in poverty following a decrease in t_l versus that following of a fall in t_j, in each case per dollar of average welfare gain. Ignoring economic efficiency considerations, decreasing t_l and increasing t ^jis then poverty reducing if that difference is positive. Condition (12.29) therefore says that decreasing t^l but increasing t^j reduces poverty if the distributive benefit of such a reform is larger than its economic efficiency cost.

We may then check whether a tax reform is "poverty efficient" and ethically robust by verifying whether the following condition holds⁸:

E: 18.8.52

To interpret (12.30), it is useful to recall the general poverty dominance results of (10.14). Using (10.14), it follows that if condition (12.30) holds, then all of the poverty indices that are members of the class Π ^α+1 (z ⁺) (of ethical order α + 1) will decrease following a revenue-neutral fall in t_l and a rise in t_j. This can be summarized as:

s^th-order poverty dominant tax reform: A revenue-neutral marginal tax reform that decreases t_l and increases t_j will decrease all poverty indices that are members of Π^s(z ⁺) if and only if⁹

E: 18.8.41

Considering the relationship between poverty and welfare dominance (see page 184), a similar result holds for welfare dominance:

s^th-order welfare dominant tax reform: A revenue-neutral marginal tax reform that decreases t_l and increases t_j will increase all social welfare indices that are members of Π^sif and only if

12.5 Income-component and sectoral growth

It is just a matter of notational change to use the tools developed above to assess the poverty impact of growth in some income component, in some sector of economic activity, or for some socio-economic group. We will then be able to assess, for instance, by how much aggregate poverty would fall per percentage of growth rate in the industrialized sector (a sectoral change), or per dollar of growth in agricultural income (an income component that enters into aggregate income), or in some region.

⁸ DAD: Dominance|Indirect Tax Dominance.

⁹ DAD: Dominance|Indirect Tax Dominance.

12.5.1 Absolute poverty impact

Assume that total income X is the sum of C income components, with quantile , where λ_c is a factor that multiplies income component and where (p) is the expected value of income component c at rank p in the distribution of total income. Again, (p) can be, for instance, agricultural or capital income, or the income of those living in some geographic area.

The derivative of the normalized FGT index with respect to λ_c is then given by¹⁰

E:18.8.42

where this CD_c(z;α) curve can now be interpreted as a " component dominance" curve for income component X^(c). It can be defined formally as¹¹:

Multiplied by a proportional change dλ_c, CD_c(z; a) gives the marginal change in the FGT indices that we can expect from growth in a component c. Note that the derivative of the un-normalized index P(z;α) is simply z^a CD_c(z; a).

We can intuitively expect, however, that a given percentage change will have a larger poverty impact when it applies to a larger sector or income component. To take this element into account and to normalize by the importance of the component, we may wish instead to compute the change in the FGT indices per dollar of per capita growth in the overall economy, when that growth comes exclusively from growth in a component c. This is given by the normalized CD curves for component c¹²:

or by for the un-normalized FGT index.

Note that the richer the society, the lower will the fall in poverty tend to be per dollar of per capita growth. This is so for two reasons. First, a richer society will tend to have a lower level of poverty and fewer poor, and hence there is less scope in such an environment for poverty to decrease significantly in absolute terms. This is captured in (12.34) by a lower value of f(z) and of [z–X(p)]₊. Second, in a richer society, a 1% increase in some component will generate a larger level of per capita growth in dollar terms. This is captured by a larger µX_(c). Both factors will thus tend to push (12.35) downwards. Thus, growth will arithmetically tend to have a smaller absolute poverty impact in richer societies.

¹⁰ DAD: Poverty|lncome-Component Proportional Growth.

¹¹ DAD: Curves|C-Dominance.

¹² DAD: Curves|C-Dominance.

12.5.2 Poverty elasticity

An alternative indicator of the poverty impact of growth is the elasticity of poverty with respect to overall growth, where again that overall growth comes strictly from growth in a component X_(c). From (12.35), this is given by

for both normalized and un-normalized FGT indices. Expressed as elasticities, the impact of income component and sectoral growth will tend to revert to comparable magnitudes between rich and poor countries. As shown by the right-hand-side of (12.36), that magnitude will mostly depend on the importance of component X_(c) among the poor (the term as a proportion of the importance of component X_(c) in total income (the term µX_(c)/µX).

Note, therefore, that the use of poverty elasticities as opposed to poverty changes will often give a different picture of where growth is (or has been) most effective in reducing poverty. Using absolute poverty changes (12.35) will usually suggest that growth reduces poverty most in poorer countries; using elasticities (12.36) may instead imply that growth reduces poverty most in richer countries.

12.6 Overall growth elasticity of poverty

How fast can inequality-neutral growth in the economy be expected to reduce poverty? On which group can inequality-neutral growth be expected to reduce aggregate poverty the fastest? And in which group would poverty fall the fastest due to such growth ?

Using (12.8) above, it can be shown that the elasticity of total FGT poverty with respect to total income–when growth in total income comes exclusively from inequality-neutral growth in group k – equals ε^y(k;z;α)¹³:

E:18.8.47

¹³ DAD: Poverty|FGT Elasticity.

for α≠0. When α = 0, (12.37) becomes:

Equations (12.37) and (12.38) can be used and interpreted in a number of interesting ways.

1. Replacing P(k;z;α) by P(z;α), P(k;z;α - 1) by P(z;α – 1), f(k;z) by f(z), and µ(k) by µ, in (12.37) and (12.38) gives as a special case the elasticity of total poverty with respect to inequality-neutral growth in the overall economy, ε^y(z;α).
   
   
2. Replacing P(z; α) by P(k; z; α), F(z) by F(k; z) and µby µ(k) in (12.37) and (12.38) yields the elasticity of poverty in group k with respect to inequality-neutral growth in the income of that same group.
   
   
3. As discussed above, the most beneficial source of growth (for overall poverty reduction) may not come from those groups with greatest poverty. Groups in which poverty is highest will tend to have a large [P(k; z; α)–zP(k; z; α–1)], but we also need to consider the ratio of µ(k) to µ: high poverty in a group can also be associated with a high level of average income.
   
   
4. The growth elasticity of the headcount, , has a nice graphical interpretation. To see this, consider Figure 12.1 where the density f(y) of income at different y is shown. Recall that the area underneath the f(y) curve up to y = z gives the headcount F(z). The term z. f(z) in (12.38) is the area in Figure 12.1 of the rectangle with width z and height f(z). Hence, the elasticity (in absolute value) of the headcount with respect to inequality-neutral growth is given in Figure 12.1 by the ratio of the rectangular area z. f(z) over the shaded area F(z).
   
   

5. The growth elasticity of the average poverty gap, ε^y(z; α = 1), also has a nice interpretation. Denote the average income of the poor by µP(Z)= can then be expressed as:

   

    

It is clear, then, that this elasticity is larger than one whenever the poverty line z is lower than the (first) mode of the distribution. In fact, it will be above one in Figure 12.1 for any poverty line up to approximatively z'. For poverty lines larger than z', the growth elasticity will in absolute value fall below 1.

This can have important policy consequences. For societies in which the poverty line is deemed to be lower than the mode (which is usually not far from the median), then the headcount in these societies will fall at a proportional rate that is faster than the growth rate in average incomes. But for societies in which the headcount is initially high (larger than 0.5, say), we can expect the growth elasticity of the headcount to be lower than 1. This implies that inequality-neutral growth can be expected to have a proportionately smaller impact on the number of the poor in poorer societies than in richer ones.

Figure 12.1: Growth elasticity of the poverty headcount

Hence, the growth elasticity of the average poverty gap is simply (minus) the ratio of the poor's average income to the poor's average distance to the poverty line. Because this only takes into account the average income of the poor, however numerous or few they may be, the elasticity ε^y(z; α–1) can easily be misleading. A society A with a small headcount and with a given ε^p(z) and a society B with a much larger headcount but the same ε^p(z) will exhibit the same growth elasticity, although intuitively we might feel that growth would decrease poverty more in B than in A.

12.7 The Gini elasticity of poverty

12.7.1 Inequality and poverty

It may also be of interest to predict how changes in inequality will affect poverty. The immediate difficulty here is that, unlike the case of growth in mean income, it is not immediately obvious which pattern of changing inequality we should consider. Indeed, as discussed above, a natural reference case for analyzing the impact of growth is the case of inequality-neutral growth–all incomes then vary proportionately by the same growth rate in mean income. For inequality changes, which inequality index should we use to measure inequality? And, supposing that we were to agree on the choice of such a summary inequality index, which of the many different ways in which that index can change by a given amount should we choose? Each of these different ways can have a dramatically different impact on poverty.

To make this difficulty slightly more concrete, suppose that we wish to understand the impact of an increase in the Gini index on the poverty headcount (this is often done in aggregate "inequality-poverty-growth regressions"). Also suppose that this increase in the Gini comes from a mean-neutral increase by some constant in the gap between two quantiles Q(p₁) and Q(p₂), with P2–P1 = η > 0. From (4.12), note that the impact of this on the Gini is the

same, whatever the value of p1. There are, however, several possible reactions of the headcount following this increase in the Gini:

1. If p₁ is well above F(z), the headcount will not change;
   
   
2. If p₁ is just above F(z), the headcount will increase;
   
   
3. If p₁ is below F(z) and p2 is above F(z), the headcount will not change;
   
   
4. If p₂ is just below F(z), the headcount will fall;
   
   
5. If p₂ is well below F(z), the headcount will not change.

Clearly, even in this very special setting, the relationship between poverty and inequality is far from being unambiguous.

So trying to predict the effect on poverty of a process of changing inequality, through the use of a single inequality index, is really to ask too much of summary indices of inequality. There cannot exist any stable structural relationship between inequality indices and poverty, even assuming mean income to be constant. This in fact casts serious doubt on the structural soundness of the many studies that regress past changes in poverty indices upon past changes in inequality indices, and which then try to explain or predict the impact of changing inequality on poverty.

12.7.2 Increasing bi-polarization and poverty

What can be done, however, is to illustrate how some peculiar and simplistic pattern of changing inequality can affect poverty. Such an illustration can be made using the single-parameter (λ) process of bi-polarization shown by equation (4.15). How does poverty change when inequality changes due to this bi-polarization ? For this, we use the most popular indices of poverty and inequality, the FGT and the Gini indices (the result is exactly the same if we use the broader class of S-Gini indices). Assume that the change in inequality comes from a λ that moves marginally away from 1. The impact on the normalized FGT index is given by

Thus, the elasticity of the (normalized and un-normalized) FGT poverty indices with respect to the Gini index is obtained as ε^G(z; α)¹⁴,

¹⁴ DAD: Poverty|FGT Elasticity.

for α > 0. When the headcount is used, we have

and thus¹⁵

Note that even with this highly simplified process of changing inequality, the impact on poverty is ambiguous. It depends in part on the sign of (µ–z). When mean income is below the poverty line, an increase in the Gini index can—and, for the headcount index, will—imply a fall in poverty.

12.8 The impact of policy and growth on inequality

12.8.1 Growth, fiscal policy, and price shocks

We may now turn to the impact of policy and growth on inequality. The approach we use enables us to consider the impact on inequality of several ways in which income changes may occur. One is growth that takes place within a particular socio-economic group. Another is growth that affects the value of some income sources—such as agricultural income or informal urban labor income. Another is the impact of price changes, which affect real income and its distribution. One more is the impact of changes in some tax or benefit policies, such as changing the subsidy rates on some production or consumption activity, or increasing the amount of monetary transfers made to some socio-economic groups.

For each such income-changing phenomenon, we may be interested in the absolute amount by which inequality will change, or in the absolute amount by which inequality will change for each percentage change in mean real income, or in the elasticity of inequality with respect to mean income.

Assume that we have as above that total income X is the sum of C components, X_c), to which we apply again a factor λ_c to yield . We then have that . If we are interested in total consumption, then we may think of the X_(c) as different types of consumption expenditures. If we are thinking of tax and benefit policy, then some of the X_(c) may be transfers or taxes. If we are alternatively concerned with the impact of sectoral growth on income inequality, then we may think of the X_(c) as different sources of income, or of the income of different socio-economic groups. By how much, then, is inequality affected by variations in λ_c?¹⁶

¹⁵ DAD: Poverty|FGT Elasticity,

We will consider two ways of measuring inequality, the Lorenz curve and the S-Gini inequality indices—of which the traditional Gini is again a special case. The derivative of the Lorenz curve of X with respect to λc is given by:

E:18.8.49

Equation (12.44) therefore gives the change in the Lorenz curve per unit ofλ_c, that is, per 100% proportional change in the value of X_(c). Say that we predict that income component X_c will increase by approximately 10% over the next year¹⁷. We can then predict that the Lorenz curve L_x(p) will move by approximately 10% of (12.44) over that same period. How big an impact this will be on inequality will depend of course on the size of the proportional change, on the importance of the component (µX_(c)), and on the concentration of the component relative to that of total incomes (the difference Cx_(c) (p) -Lx(p)).

A similar result is obtained for the Gini indices¹⁸:

E:18.8.50

Thus, if for instance the removal of a subsidy or the advent of an external shock is foreseen to increase by 10% the price of a good X(c), the Gini index can be predicted to move by approximately– [10%.µx_(c)/µx (Cx_(c)(p)-Lx(p))]. (The negative sign comes from the fact that an increase in the price of a consumption good leads to a fall in the real value of the expenditures made on that good.) The impact per dollar of change in per capita income is then given by

We may also wish to assess the impact on inequality of a change in λ_c per 100% of mean income change. This is given by

¹⁶ DAD: Inequality|Income-Component Proportional Growth.

¹⁷ DAD: Curves|Lorenz and DAD: Curves|Concentration.

¹⁸ DAD: Inequality|Gini/S-Gini index and DAD: Redistribution|Coefficient of Concentration.

for the Lorenz curve and by

for the Gini indices. These expressions are simple to compute and have a nice interpretation. Multiplying the above two expressions by the proportional impact that some change in X_(c) is predicted to have on total per capita income gives the predicted absolute change in inequality. For instance, if we predict that growth in rural areas will lift mean income in a country by 5%, then the Lorenz curve of total income L_x(p) will shift by approximately 0.05 (Cx_(c)(p)–Lx(p)), where X_(c) is rural income. If rural income is more concentrated among the poor than total income, this will push the Lorenz curves up; otherwise, growth in rural income will increase inequality.

Finally, we may prefer to know the elasticity of inequality with respect to µX, when growth comes entirely from X_(c). It is given by

for the Lorenz curve and by

for the Gini indices. Thus, a proportional increase in taxes that reduces total mean net income by 1 percent will change the Gini index by 1 – ICx_(C)(ρ)/Ix(ρ) percent, where ICx_(c) is the concentration index of taxes X_(c). This will decrease inequality if taxes are more concentrated than net income: ICx_(c)(ρ)/Ix(ρ) > 1. The elasticity of the Lorenz curve and of the Gini indices with respect to µx_(c) when growth comes entirely from a proportional change in X(c) is finally given by

and

12.8.2 Tax and subsidy reforms

As in the case of poverty (recall Section 12.4), it is useful to assess the impact of a price reform (through consumption and production taxation) on inequality. Assume that we are interested in the effects of a revenue-neutral marginal tax reform that increases the tax on a good j for a benefit of a fall in the tax on a good l. Recall that γis the MECF for j over that for l — the larger the value of γ, the lower the fall in t_l that we can generate for a given revenue-neutral increase in t_j. Denoting real income by y^R, the impact of a marginal revenue-neutral increase in the price of good j is then

on the Lorenz curve and an impact

on the Gini indices¹⁹. When γ = 1, viz, when the marginal economic effciency of taxing l and j is the same, expressions (12.54) and (12.56) reduce to a proportion of the difference between the concentration curves and the concentration indices for the two goods. For instance, the change in the S-Gini inequality indices is then given by:

E:18.8.48

It is then better for inequality reduction to tax more the good that is less concentrated among the poor, for the benefit of a reduction in the tax rate on the other good, which is less concentrated among the rich.

We may also wish to express the above changes in inequality per 100% change in the value of per capita real income. This is then given by

¹⁹ DAD: Curves|Lorenz and DAD: Curves|Concentration.

for the Lorenz curve and

for the Gini indices.

12.9 References

The literature on the empirical effectiveness of targeting schemes has grown substantially in the last years. See for instance Bisogno and Chong (2001) (on the effectiveness of proxy means tests), Hungerford (1996) (on the effectiveness of the targeting of social expenditures in the US), Gueron (1990) (on the effectiveness of targeted employment programs in the US), Park, Wang, and Wu (2002) (on the effectiveness of targeting in China), Ravallion, van de Walle, and Gautam (1995) (on the effect of the targeting of social programs on persistent and transient poverty in Hungary), Ravallion (2002) (on the variability of targeting across economic cycles in Argentina), Schady (2002) (on the potential for geographic targeting in Peru), and Moffitt (1989) and Slesnick (1996) (on whether in-kind transfers are efficient for poverty reduction).

The recent literature has also queried whether "finer" geographical targeting schemes lead to more equitable and more effective poverty reduction. Evidence on this issue–which is linked to the broader context of the benefits and costs of decentralization–is discussed inter alia in Alderman (2002) (for Albania), Bigman and Srinivasan (2002) (for India), and Ravallion (1999).

The targeting literature has often resorted to an analysis of programs' "targeting errors" and how they vary with program reforms. These errors are variably called "leakage" and "undercoverage" errors, "E" and "F" mistakes, and "Type I" and "Type II" errors. Discussion and use of them can be found in van de Walle and Nead (1995), Cornia and Stewart (1995) and Grosh (1995). See also van de Walle (1998b) (on the virtues and costs of "narrow and broad" targeting), and Wodon (1997b) (for use of "ROC curves" to study the performance of targeting indicators).

Work on the impact of marginal price changes on well-being and welfare includes: Ahmad and Stern (1984), Ahmad and Stern (1991), Creedy (1999b), Creedy (2001), Newbery (1995) and Stern (1984), for the impact of indirect marginal indirect tax reforms on some parametric social welfare functions, making use among other things of the "distributional characteristics" of goods; Mayeres and Proost (2001), for the impact of marginal indirect tax reforms in the presence of an externality (peak car transport); Besley and Kanbur (1988), for the impact of marginal changes in food subsidies on FGT poverty indices; Creedy and van de ven (1997), Creedy (1998a) and Creedy (1998b), for the impact of price changes and inflation on well-being and social welfare; Liberati (2001), Mayshar and Yitzhaki (1995), Mayshar and Yitzhaki (1996), Yitzhaki and Thirsk (1990), Yitzhaki and Slemrod (1991) and Yitzhaki and Lewis (1996), for the impact of marginal indirect tax reforms on classes of social welfare indices using "marginal" dominance analysis; Lundin (2001); for marginal dominance analysis for a marginal tax reform affecting the importance of an externality (the presence of carbon dioxide); Makdissi and Wodon (2002), for the use of CD curves in the analysis of marginal poverty dominance; and Yitzhaki (1997), for the impact on inequality of marginal price changes.

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