Document(s) 15 of 20

The lump-sum targeting schemes analyzed in Chapter 12 assumed that there exist characteristics on which governments can condition benefit transfers. For instance, we modelled the impact on poverty of giving $1 to everyone that belonged to some socio-demographic group k. These transfers were not decreasing with levels of income since we implicitly assumed that income levels were not directly observable. The tools derived in Chapter 12 enabled us, however, to identify on which observable socio-economic characteristics we should condition transfers to reduce poverty fastest.

We will suppose now that the distribution of population characteristics (including the levels of original income) can be observed without costs (for expositional simplicity), but that there exist costs to granting state support. We will see that the optimal targeting rules that follow are different from those of the traditional study of optimal income taxation, where labor supply and income generation are endogenous but where redistributive imperfections are generally ruled out. Instead, assume that the behavior of agents is fixed (e.g., constrained by labor market conditions) under alternative income support rules, except for the feature that such agents may freely choose whether to participate in the income support programs. Given the plausible presence of redistributive costs whose size may vary with individuals, the state then wishes to minimize the value of a poverty index, taking into account either the opportunity cost of government expenditures or the constraint of an aggregate redistributive budget for poverty alleviation. As we will see, the existence of redistributive costs leads to policy criteria that weigh efficiency as well as redistributive objectives. It also has important implications for the consideration of the principles of vertical and horizontal equity.

13.1 Poverty alleviation, redistributive costs and targeting

Redistributive costs can first arise from the efforts made by governments to monitor true levels of income. They can be interpreted in a sense as the certainty-equivalent costs of the presence of imperfect information. The more difficult is it to ascertain accurately someone's true income, the greater the expense of removing the associated information imperfections. Redistributive costs can also be incurred by benefit recipients and they may then have to be deducted from the gross impact of state support in order to yield net poverty relief. For expositional simplicity, we assume here that all costs take the form of a participation burden and that they are borne directly by the participating poor.

Assume that the poverty alleviation objectives of the state are to minimize the poverty index, P(z):

where y_i is the initial income of individual i, z is the poverty line, and NB_i is the net benefit to individual i of the availability of a non-negative gross benefit B_i*. As we shall define it below more precisely, NB_i is no greater than B_i* since it is reduced by the administrative and participation costs involved in transferring B_i*

The government allocates a total per capita budget to the minimization of poverty, such that

with B_i being the level of gross benefit actually expended to support individual i.

Let B_i* then represent the benefit offered to individual i, and denote by C_i the non-negative cost to i of accepting B_i*. If B_i* is less than C_i, then the benefit awarded B_i and the net benefit NB_i will be zero. When then and Define an indicator function I[x] that takes the value 1 when x is true and 0 when x is false. Then

costs C_i are only incurred when B_i* is taken up. Think for instance of c_i as an administrative cost necessary to grant support to i. B_i* is then the level of gross expenditures which the state would consider spending on i, B i is the level of gross expenditures actually spent on i, and NB_i is the level of benefit net of administrative costs that eventually reaches the individual.

The government thus wishes to choose the various B_i*, i = 1,..., n, to minimize (13.1) subject to (13.2). Note that NB_i and Bi are not differentiable with respect to B_i* at the point at which i just accepts state support, viz, when This causes no analytical difficulty since as we will see the optimum solutions for the B_i* never have to lie at these corner points.

Define λ as the Lagrange multiplier associated to the budget constraint, and π⁽¹⁾ as the non-negative derivative of π with respect to y. For now, assume that π is continuous, differentiable and convex — we will discuss later the important headcount case for which π does not fulfill these conditions. The government then wishes to ensure that the following condition is met at the optimum values of B_i and λ (given by B_i* and λ*) for each of the i in receipt of state support:

The optimum value of λ reflects the social opportunity cost of spending public resources. Note that a benefit offer B_i* below ci will not matter, for then B_i = NB_i = 0, that is, it has neither a cost nor a benefit.

Whether i should derive any net benefit at the optimum solution depends on its original income y_i and on the redistributive cost c_i that he faces. Figure 13.1 illustrates this dependency. The straight line λB_i* displays the opportunity cost in social welfare of granting i a benefit B_i*. The receipt of such a benefit will bring a net benefit NB_i that will decrease π_i (the contribution of i to the poverty index P) once the redistributive cost has been paid off. The shape of –π_i above 0 depends on the convexity of the function π(y_i + NB_i;z) and on the original income y_i. Individuals for whom it is possible to find a level of expenditure B_i* for which will be granted stated support. Whether eventually reaches λB_i*— and whether, therefore, the poverty alleviation benefit of granting state support to i is worth its opportunity cost — will thus also hinge on the size of ci, the size of the redistributive costs.

Four cases are shown on Figure 13.1. Individual 1, with expenses c₁, will receive benefit B₁*, with a net benefit reward of . Individual 2, who faces the same redistributive cost but has a higher original income, will barely be deemed eligible, just as is the case with individual 3 with a lower y but a much higher c. Once benefit recipients, however, individuals 2 and 3 will receive what may be a sizeable net and gross benefit, thus showing an important discontinuity in the function of optimal state support. From the above optimality condition (13.4), we may indeed note that, when B_i* is received, the corresponding net benefit equalizes the post-benefit income (net of redistributive costs) of all benefit recipients. In other words, we have at the optimum that:

Individual 4, who enjoys a relatively large y and also faces high costs, does not benefit from the optimal program. Hence, all those individuals with:

original income greater than y₂ and costs greater than c₂,

or original income greater than y₂ and costs greater than c₂,

or original income greater than y₃ and costs greater than c₃,

or original income greater than y₃ and costs greater than c₃

ought not to receive income support. The greater the redistributive cost ci, the less the chance of receiving a positive B_i*, but the greater the optimal B_i* is if support should be granted. Furthermore, the greater his original income yi, the less likely an individual is to take up a positive B_i*and the smaller is B_i* if it is received.

13.2 Costly targeting

13.2.1 Minimizing the headcount

Consider now the case in which P(z) = F(z) is the headcount. π(y_i + NB_i;z) is then discontinuous at the point at which y + NB_i reaches z. This leads the state to distribute B_i in such a way as to raise to z as many of the individuals as possible. In order to do this, it will grant income support z – y_i + c_i first to that poor individual for which that amount is lowest, then to that poor individual with the second lowest z – y_i + c_i, and so on, until the budget has run out. This relatively straightforward optimal policy is similar to Proposition I of Besley and Coate (1992) in the absence of redistributive costs. We illustrate it on Figure 13.2 supposing that z = 1 and that the budget runs out at an individual with z – y + c = 0.5z, that is, when the government spends half of the poverty line on a recipient. It is clear from the Figure that individuals with original incomes closer to the poverty line are more likely to be optimal benefit recipients. Conditional on being an optimal recipient, however, the expense generated in being awarded a benefit decreases with income and increases with costs.

13.2.2 Minimizing the average poverty gap

Consider now the average poverty gap as P(z). It is continuous but not continuously differentiable everywhere in y_i = NB_i. Choosing to minimize the average poverty gap leads the state to choose benefit recipients such as to maximize the returns in poverty gap reduction per unit of government expenditure. In other words, the state wishes to minimize the aggregate level of redistributive costs incurred for a given total budget spent on the poor. Or, said again differently, the state attempts to fill as much as possible of the total poverty gap, avoiding as much as possible spending on wasteful redistributive costs.

Figure 13.1: Targeting and redistributive costs

Because there are fixed costs to granting income support, once an optimal benefit recipient has been identified, the state wishes to spend on him as much as is necessary to raise his net income to z. Thus, the government's optimal strategy is to compute an "efficiency" ratio (z – y_i) / (z – y_i + c_i,) of full poverty gap reduction over benefit expenditure for each individual i, and grant benefit first to that individual i with that greatest efficiency ratio, then to that individual j with the second highest ratio, etc., until the budget is depleted. Because some income support to some relatively poor individuals may yet involve relatively high redistributive costs, the state may find it preferable to grant income support to some richer individuals among the poor.

An individual i should then benefit from state support if the fall in his poverty gap does not fall below the opportunity cost of that fall. For all such benefit recipients, the state also wishes to raise their net income to the poverty line, z, with gross benefits and expenditures equal to B_i = z– y + c_i. Hence, at the optimum, an individual i will receive state support if

where λ* is the opportunity cost of government resources at the optimum. A decision-maker may feel, for instance, that the benefit of a $1 reduction in the poverty gap is at the margin worth $2 in taxes, with a consequent value of λ* =0.5. Thus, for individuals to be optimal benefit recipients, the social benefit of poverty gap reduction, net of the redistributive costs, must exceed the opportunity cost of gross state support. Otherwise, government expenditures would be better spent elsewhere than on poverty relief.

The identification of an optimal set of benefit recipients can thus be made on the basis of an opportunity cost, λ*, and on the interaction of z, y_i and C_i. From (13.6), we see that all i with

will be optimal recipients of state support B_i = z– y_i + c_i. A value of λ* equal to 1 would eliminate all i with c _i greater than zero. The lower the value of λ*, the lower the opportunity cost of government expenditures, and the easier it is for poor individuals to qualify for state support. Condition (13.7) above thus explicitly defines a set of income support recipients with a border fixed by a linear trade-off between c_iand y_i. To locate precisely that border, the opportunity cost of government expenditures (λ*) must be found or set. This can be done in at least three ways:

through setting λ* directly, taking into consideration the social welfare value of reducing the aggregate tax burden;

through setting a budget level that reflects the government's political or economic "capacity" to pay, and then deriving the implied value of λ*;

through identifying a point (y_i, c_i) that lies precisely on the border of the "eligibility set", and then calculating the implied λ*.

Let us illustrate the third way – which is both easy to follow and easy to interpret. At the borderline of eligibility, we note that:

Suppose, for instance, that we judge an individual with c_i/z = 0.25 and y_i/z = 0.5 to be deemed just barely eligible to state support. It follows from (13.8) that λ* = 2/3. This says that a $2 decrease in the average poverty gap is deemed, at the margin, socially worth a $3 increase in per capita taxes. With this information, the entire set of optimal benefit recipients can be identified. The derived value of λ* = 2/3 says, for instance, that all those with no original income at all would yet receive no state support if their redistributive costs exceeded 50% of z. All those deemed eligible will receive B_i = z - y_i + c_i, and will see their net income raised to z. This is illustrated on Figure 13.3 where z is again set to 1. Both the likelihood of being an optimal benefit recipient and the expense made when awarding a benefit are decreasing with incomes and redistributive costs.

13.2.3 Minimizing a distribution-sensitive poverty index

Consider finally the case in which the optimal state support policy must be geared towards minimizing the average of the squared poverty gaps, namely, p(z; a= 2). As for the above, individuals found to be optimal beneficiaries of state support will be those whose fall in poverty exceeds the opportunity cost of the gross expenditures needed to decrease their poverty, viz, those for whom we can find a Bi such that

For beneficiaries (recall (13.5)), we will have that y_i+ B_i*–c_i = e, where e is that constant to which the net income of all benefit recipients should be raised. Developing (13.9), we find that recipients will meet the condition that:

Because the return to decreasing the squared poverty gap decreases as net income approaches z, redistributive policy will benefit i only if λ* ≤ 2 (z –y_i), the initial marginal social, welfare return to raising i's income. If this condition were not satisfied, i would not receive income support even if c_iwere nil.

Equation (13.10) implicitly defines the set of the optimal recipients based on their values of y_iand c_i. Those with low y_i or low c_i will be granted support. The value e to which the level of all recipients' income will be raised depends implicitly on the opportunity cost λ*. The optimality condition requires that the marginal welfare gain of increasing B_i(when B_i= B_i*) is precisely equal to the opportunity cost λ* of such additional expenditure. If the welfare gain were higher than its opportunity cost, it would be preferable to increase support to the relevant i (instead of granting assistance to a new, additional recipient) since redistributive costs c_i would then already have been "sunk". Hence, it must be that

Using (13.10) and (13.11), the border of the eligibility set can now be defined by

To define the set of optimal recipients, we therefore need

either to set directly the opportunity cost of state expenditures, λ*;

to agree on a poverty alleviation budget

to identify one of the border points of the eligibility set;

or to rule on the value e at which the net income of all benefit recipients should be raised.

In everyone of these cases, a value judgement is expressed on the social value of using costly redistributive tools. This value judgement determines the set of the recipients as well as the level of their post-transfer income.

Take the same "borderline" individual as above, with c_i / z = 0.25 and y_i/z = 0.5. For such a border point, we find e/z = 0.809 and λ*/z = 0.382. Using (13.12), it follows that when y_i = 0, for instance, redistributive costs can go up to c_i / z = 1.71 as a proportion of the poverty line before income support is withdrawn. For all benefit recipients, net income will be raised to a proportion e / z = 0.809 of the poverty line, with state expenditure on i equal to B_i = 0.809_z – y_i + c_i. Incomes will not be raised to the poverty line since, above y_i + NB_i = 0.809_z, the marginal welfare gain of additional state expenditure is lower than its opportunity cost.

Figure 13.4 summarizes graphically these policy implications for the P(z;a = 2) index by showing the set of the optimal recipients as a function of their original income y and of the redistributive costs c that supporting them generates. The vertical axis shows the level B of expenditures which it is optimal to grant to individuals according to their value of y and c. For ease of reading, all variables are normalized by the poverty line z, which is equivalent to setting z = 1. The set of optimal recipients is clearly non convex, although as we will discuss below, the optimal level of state expenditure shows local linearities with respect to incomes and redistributive costs.

13.2.4 Optimal redistribution

There are several important lessons to be gained from the above discussion, and in particular from Figures 13.2, 13.3 and 13.4. First, state support for the eligible poor compensates them fully for their lower original income and/or higher redistributive costs. In other words, once they become recipients, they should receive support large enough to raise their net income to the level of that of all other optimal recipients.

Second, the case of c = 0 is clearly a special case in which there are fewer support discontinuities. In the more general framework in which redistributive costs c > 0 are allowed, however, some largely intuitive results do not hold any more. It is not true, for instance, that the state is indifferent as to the identity of the poor with the same y_i. as seen above, values of c affect who should be targeted for poverty relief. Figure 13.4 also shows that all individuals with zero costs are optimal recipients of state support regardless of their own resources. With higher costs, however, optimal eligibility quickly becomes restricted to the very poor. As redistributive costs rise, the social gain of supporting those with relatively high incomes rapidly falls below the opportunity cost of state resources. Hence, as long as there prevails at least some redistributive cost, not all individuals should be raised to the same final net income, but an optimal selection needs to be made on the basis of original income and levels of redistributive costs.

This last result does not require variability in the redistributive costs across individuals. The more positive the correlation between levels of original income and redistributive costs, the greater the chance that poor individuals would be deemed optimal recipients of state support. But so long as redistributive costs are strictly positive, there will be some poor who will not be optimal benefit recipients. This can be seen on Figure 13.4 for those individuals with y/z at or slightly below 0.8, who become suddenly ineligible with small increases in their c/z. This discontinuity of the optimal level of state support as a function of original income also naturally occurs when using poverty indices that are discontinuous in income (such as the poverty headcount). Redistribu­tive costs introduce these discontinuities for continuous poverty indices as well.

Third, the model above suggests some features of optimal redistribution policy that are somewhat disturbing, at least when considered in the context of the usual discussions of efficiency and equity. On account of the variability of redistributive costs across individuals, some relatively richer individuals might be deemed optimal recipients of income support whereas some poorer individuals might be denied such support. Supporting the poorer and not the richer may generate a greater level of vertical equity and of redistribution, but this is clearly not necessarily optimal if individuals differ in ways (other than their original income) that are relevant to the redistributive effectiveness of the state.

Finally, note in Figure 13.4 that all optimal recipients will receive enough support to raise their net income to 0.809z. There are, however, many individuals with original income less than 0.809z who will not qualify for state support and whose final income will consequently have to remain below 0.809z. Once optimal state support has been allocated, therefore, some of the originally poorer individuals will enjoy a level of net income above that of formerly richer individuals.

This reranking of individuals in the dimension of net incomes and welfare is especially likely when richer individuals present high levels of redistributive costs. It will also occur among those richer and poorer individuals that face identical redistributive expenses. Even more significantly, there are some originally richer individuals with a relatively low c_i that will be denied support and end up worse off than some initially poorer individuals with higher c_i. Classical horizontal inequity also occurs: individuals with the same original incomes are not all treated alike by the state. If deemed to be socially important, the consideration of horizontal inequity as a social evil would thus necessarily put a constraint on such policies.

13.3 References

The literature on optimal income taxation is large and varied. A review can be found in Slemrod (1990), Stern (1984) and Tuomala (1990) – see also Kanbur, Keen, and Tuomala (1994a). The literature typically allows for labor supply and income generation to be endogenous, but generally supposes the absence of redistributive imperfections – see Stern (1982) for an exception to this.

Budgetary rules under the more specific objective of poverty reduction are discussed in Bourguignon and Fields (1990), Bourguignon and Fields (1997), Kanbur (1985), and Chakravarty and Mukherjee (1998). Additional works on optimal income taxation and optimal benefit provision include Besley (1990) (for a comparison of means testing and universal provision of public assis­tance), Besley and Coate (1992) and Besley and Coate (1995) (on the desirability of workfare constraints), Creedy (1996) (for a comparison of means testing and linear taxation for poverty reduction), Fortin, Truchon, and Beauséjour (1990) (on comparing workfare and negative income tax systems), Glewwe (1992) (for designing benefit allocation rules when income is not observed), Haddad and Kanbur (1992) (for the potential role of intra-household allocation issues), Immonen, Kanbur, Keen, and Tuomala (1998) (for a comparison of means testing and categorical benefit provision), Kanbur, Keen, and Tuomala (1994b) (for differences in the optimal rules implied by welfarist and non- welfarist social objectives), Keen (1992) (for the link between needs and optimal allocations of benefits), Thorbecke and Berrian (1992) (for general-equilibrium optimal budgetary rules), Viard (2001) (for a theory of optimal categorical transfer payments), and Wane (2001) (for optimal taxation when poverty generates negative externalities on society).

Figure 13.2: Optimal set of benefit recipients and levels of state expenditure B, with α = 0 and z = 1

Figure 13.3: Optimal set of benefit recipients and levels of state expenditure B, with α = 1 and z = 1

Figure 13.4: Optimal set of benefit recipients and levels of state expenditure B, with α = 1 and e = 0.809.

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