SOCIALLY IMPROVING TAX REFORMS*

1. introduction

Policies that affect consumer and producer prices have an impact on welfare, and there are many such policies. Some governments maintain high import tariffs or fail to implement regulations to foster competition. This may protect national producers by maintaining high domestic producer prices, but it also raises consumption prices, which hurts consumers. Most governments use sales and indirect taxes to raise tax revenues, thus affecting consumer and producer prices. This is particularly important for developing countries, which rely heavily on commodity taxation to generate tax revenues. Price subsidies on food, education, energy and transportation are also common in developing and developed countries alike.

Policy analysts must routinely evaluate the impact of such pricing policies on poverty and social welfare. Important information problems stand, however, in the way of anyone wishing to do this. The main objective of the article is to show how these problems can be (somewhat) circumscribed. We are particularly interested in demonstrating the empirical applicability of a “social improvement” approach. This approach makes it possible to identify marginal price changes that will be deemed socially desirable by wide spectra of social welfare and poverty analysts. The term “social improvement” thus encompasses both the classical welfare-dominance and the poverty-dominance approaches.

For expositional simplicity, we will focus in this article on the analysis of the social impact of indirect tax reforms. We will think of two goods, j and l, and will ask whether it is socially desirable (in a sense to be defined precisely below) to increase a tax rate on good j to decrease a tax rate on good l. Note that tax rates may be positive or negative, which also says that decreasing the tax on a good may mean increasing its rate of subsidy. In doing this, we keep producer prices constant for expositional simplicity, but this framework can be extended to the general equilibrium analysis of any government intervention that changes relative prices.

The first measurement difficulty is then to estimate the impact of price changes on consumer welfare. It is well known that this can be a complicated exercise and that its results can be sensitive to a number of important theoretical and econometric assumptions. The task is particularly problematic when the goal is to find globally optimal tax systems, since the tax analyst then needs reliable demand responses over extended price intervals for each household. Keeping in mind that actual changes in the tax system are “slow and piecemeal,”² and that it would be unwise to ignore the role of the actual tax system as a departure (or anchor) point for the identification of desirable tax reforms, we focus here on the effect of marginal tax reforms. One immediate advantage is that evaluating the distributive impact of marginal tax reforms does not require estimates of individual demand and utility functions, but can instead be assessed directly from the observed data alone,³ as we review in Section 2.

The second difficulty resides in the choice of a social evaluation function with which the impact of the tax reform is to be measured.⁴ This choice poses a fundamental problem, since any particular selection of functional form and parameters for a social evaluation function necessarily embodies arbitrary value judgements. The strategy we follow in this article is instead to define classes of social evaluation indices that incorporate increasingly stronger judgements on the importance of distributive issues in designing tax policy. For this purpose, we consider social evaluation indices that take into account (though not necessarily with the same distributive weight) everyone's welfare—the traditional social welfare functions—as well as social evaluation indices that censor welfare at a poverty line and that are therefore not affected by changes in the welfare of the “rich”—in short, the much-used poverty indices. These classes of functions and indices are described in Section 3.

Verifying whether a tax reform is desirable then proceeds by checking whether it can command the unanimous approval of all those analysts who agree with some generally defined normative properties of the social evaluation indices. If so, the reform can be called “socially improving.” One well-known criterion to assess social improvement comes from the Pareto principle. Pareto improvement, however, is likely to provide a poor normative basis for the empirical analysis of tax reforms (for reasons that we discuss below in Section 6.3), and it is thus useful to consider “higher-order” ethical principles. We increase progressively the ethical content of our classes of social welfare and poverty indices by incorporating the anonymity—or symmetry—principle (for first-order improvement), the Pigou–Dalton principle of transfers (for second-order improvement), the principle of diminishing transfers (for third-order improvement), and subsequent “generalized” higher-order normative principles. These principles lead to the definition of what we term Pen-improving, Dalton-improving, Kolm-improving, and higher-order welfare-improving tax reforms. The definition of poverty-improving tax reforms proceeds in a similar way by censoring the social assessment at a poverty line. This is derived in Section 4.

The third difficulty lies in estimating the impact on tax revenues of changes in the structure of indirect taxation. It is well known that this impact is linked to the aggregate deadweight loss of taxation,⁵ and thus to the economic efficiency of a tax reform. Estimating the aggregate impact on tax revenues can be less difficult (in principle at least) than the estimation of household-specific changes in consumption, but the details of the estimation procedures can still lead to considerable disagreements among tax analysts. One solution is to carry out sensitivity analysis on the role of the unobserved economic efficiency parameter, in the manner of Ahmad and Stern (1984) for instance. We propose an alternative procedure that estimates the critical efficiency ratio up to which a tax reform can be said to be socially improving at a given ethical order. This leaves policy makers free to assess whether the actual efficiency ratio is likely (or can be safely estimated) to be below that critical value, and thus whether the tax reform can confidently be deemed socially improving. A similar device is constructed to handle, for poverty measurement, the role of poverty lines—whose estimation is also notoriously difficult and controversial. We show how to construct estimates of the critical poverty lines up to which a tax reform can be considered to be poverty improving. Policy makers and poverty analysts are then let free to assess or estimate whether those critical poverty lines are sufficiently high to encompass all plausible poverty line estimates. If that is so, the tax reform can be confidently described as poverty improving. These critical thresholds are derived in Section 5.

Social efficiency is checked in this article through the use of simple Consumption Dominance curves. curves display cumulative consumption shares when these are weighted by powers of poverty gaps. First-order curves show the share in the total consumption of a good of those at a given income level. Second-order curves indicate the combined share in the total consumption of a good of those whose income lies below a given threshold. Higher-order curves weight consumption shares by increasingly higher powers of poverty gaps. Increasing the tax on good j and decreasing the tax on good l is poverty improving at any given order of ethical dominance if the curve of that order for good l is higher than the curve for good j at every threshold under a maximum poverty line. When that maximum poverty line extends to infinity, the tax reform can be called welfare improving. The second-order welfare improvement condition we obtain is equivalent to the Dalton improvement condition of Yitzhaki and Slemrod (1991) and Mayshar and Yitzhaki (1995). Our social improvement conditions become less stringent when we consider poverty instead of welfare improvement, since we can then focus exclusively on those below a maximum admissible poverty line and do not need to consider the impact of the reform on the entire population. Increasing the order of dominance also facilitates the search for socially improving tax reforms since more ethical structure is then imposed on the properties of the admissible social evaluation indices. These links are described in some detail in Section 6.

curves were first used by Makdissi and Wodon (2002) in the context of poverty reduction. This article uses them for the broader analysis of social improvement. The current paper also introduces the concept of first-order social improvement and shows how to test it, and also discusses how it differs from the traditional and important concept of Pareto improvement. The concept of social improvement further leads to an interpretation of the ratio of curves as a natural indicator of the distributive benefit of a tax reform, a benefit that can be compared to the average deadweight loss or economic efficiency cost of the reform. We also see how the concepts of critical economic efficiency ratios and critical poverty lines may help circumvent the difficulties faced by tax analysts.

Tax policy analysis is clearly of practical and policy interest. It is thus essential to consider how the tools described above can be implemented empirically using survey data. A crucial part of this article is devoted to this through implementing estimators of curves, critical poverty lines, and critical economic efficiency thresholds. These tools are applied in Section 7 to the analysis of Mexican consumption data. Section 8 concludes the article, and the proofs of the various lemmas and theorems can be found in the Appendix.

2. notation and definitions

Consider a vector q of K consumer prices. For expositional simplicity, and as is customary in the partial equilibrium literature, we set the vector of producer prices to 1 and assume them to be constant and invariant to changes in t, the vector of K tax rates. We then have q= 1 +t and dq_k=dt_k, where q_k and t_k denote respectively the price of and the tax rate on good k.

Let y be nominal income, and denote consumers' preferences by θ. The indirect utility function is given by v(y, θ, q). Following King (1983), we will be using a vector of reference prices, q^R, to assess consumers' well-being in the presence of varying tax rates. Denote the real (or equivalent) income in the post-reform situation by y^R, where y^R is measured on the basis of the reference prices q^R . y^R is implicitly defined by v(y^R, θ, q^R) =v(y, θ, q), and explicitly by the real income function y^R=ρ(y, θ, q, q^R), where

(1)

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 consumer welfare is affected by a marginal change in tax rates. Let x_k(y, θ, q) be the consumption of good k of a consumer with income y, preferences θ and facing prices q. Using Roy's identity and setting reference prices to pre-reform prices, we find

(2)

Equation (2) says that observed pre-reform consumption of good k is a sufficient statistic to know the impact on consumer welfare of a marginal change in the price of good k.⁶

Assume that preferences θ and nominal income y are jointly distributed according to the distribution function F(y, θ). The conditional distribution of θ given y is denoted by F(θ | y), and the marginal distribution of nominal income is given by F(y). Continuity across y of the conditional distribution function F(θ | y) is not generally needed for the methodological results of this article, but empirical applicability of some of these results will sometimes require such a continuity assumption (see Duclos et al., 2004). Let preferences belong to the set Θ, and assume income to be distributed over [0, a]. Expected consumption of good k at income y is given by x_k(y, q), such that

(3)

Let X_k(q) then be the per capita consumption of the kth good, defined as By (2), X_k(q) is also the average cost in well-being of an increase in the price of good k. We will often normalize by X_k(q) some of the various measures that will be associated to a good k, and we will distinguish these normalized measures with a . Hence, as a proportion of per capita consumption, consumption of good k at income y is expressed as .

We now turn to the government budget constraint. Per capita commodity tax revenues, R(q), equal Assume that the government's tax reform increases the tax rate on the jth commodity and uses the proceeds to decrease the tax rate (or to increase the subsidy) on the lth commodity. As is conventional in the optimal taxation literature, total tax revenues are kept invariant to the tax reform. Revenue neutrality of the tax reform requires that

(4)

Define γ_lj as

(5)

where for expositional simplicity, the (q)'s have been omitted. The numerator in (5) gives the marginal tax revenue of a marginal increase in the price of good l, 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_l) from taxing l.⁷ The denominator gives exactly the same measures for an increase in the price of good j. We may thus interpret γ_lj as the economic efficiency cost of taxing j relative to that of taxing l: MECF_j/MECF_l. The higher the value of γ_lj, the less economically efficient is taxing good j. For expositional simplicity, we will sometimes refer to γ_lj simply as γ.

By simple algebraic manipulation, we can then rewrite Equation (4) as

(6)

which fixes dq_j as a revenue-neutral proportion of dq_l.

3. measuring poverty and social welfare

To assess the impact of a tax reform on poverty and social welfare, we follow the main custom in the measurement literature and focus for simplicity on additive poverty and social welfare indices. For first, second, and third order dominance, this is only to ease the necessity and sufficiency proofs of the various social improvement conditions: We could use some of the results of the previous literature to show that the relevant classes of indices also include subclasses of nonadditive indices as well.⁸ This additivity assumption is such that poverty indices can be expressed as

(7)

where P(z) is an additive poverty index, z is the poverty line defined in income space, and p(y, z) is the contribution to total poverty of a consumer with income y (with p(y, z) = 0 for all y > z).

It is best here to think of y and z as defined with respect to constant reference prices—or as “real” variables. Defining the poverty line in the real income space instead of in the nominal income space is convenient since the poverty line is then invariant to tax reforms. In this article, as mentioned above, we generally suppose that pre-reform nominal and real incomes are the same since we take reference prices as the pre-reform prices.⁹ Define the nominal income function, y=η(y^R, θ, q, q^R), as the inverse of ρ(y, θ, q, q^R):

(8)

The nominal income function gives the level of income that yields the same utility under q as y^R gives under q^R. The marginal distribution of real income, F_R(y^R), can always be computed from the joint distribution of nominal income and preferences F(y, θ):

(9)

For the poverty improvement results of this article, we consider the classes of poverty indices P(z) ∈Π^s defined such that

(10)

where is the set of functions that are s-time piecewise differentiable¹⁰ over [0, z], and where the superscript ^(s) stands for an sth-order derivative with respect to y. We will return very shortly to the interpretation of the derivative assumptions.

A particular subclass of additive poverty indices to which we will make repeated references below is found in Foster et al. (1984) and is defined for α≥ 0 by

(11)

FGT⁰(z) gives the most widely used index of poverty, the so-called poverty headcount, and FGT¹(z) yields the second most popular index, the (normalized) average poverty gap. Note that FGT^α(z) belongs to Π^s(z) for α≥s− 1. The FGT indices were in fact used by Besley and Kanbur (1988) for their analysis of price subsidies (thus in fact supposing an explicit form for p(y, z)). Other well-known additive indices that also belong to Π¹(z) and Π²(z) include the Watts (1968) index, the second class of indices proposed by Clark et al. (1981), and the class of indices proposed by Chakravarty (1983).¹¹

Turning now to social welfare, we consider utilitarian social welfare functions U such that

(12)

We focus on social welfare indices that belong to the classes Ω^s, s= 1, 2, … . They are defined as

(13)

It is useful at this point to provide a normative interpretation of the different classes of indices of poverty and social welfare. When s≥ 1, poverty indices weakly decrease (p⁽¹⁾(y, z) ≤ 0) whereas welfare indices weakly increase (u⁽¹⁾(y) ≥ 0) when an individual's income increases. These indices are thus Paretian but they also obey the well-known symmetry or anonymity axiom: Interchanging any two individuals' incomes leaves unchanged the poverty and social welfare indices. Ordering two distributions of living standards over the first-order classes of indices is equivalent to making the living standards “parade” simultaneously alongside each other, and verifying if one parade weakly dominates the other (this exercise was first suggested by Pen, 1971, chapter III). For poverty comparisons, the distributions of living standards are simply censored at z. For simplicity, we will refer to first-order welfare-improving tax reforms as “Pen-improving tax reforms.”

When s≥ 2, poverty indices are convex and welfare indices are concave. They must thus respect the Pigou–Dalton principle of transfers,¹² which postulates that a mean-preserving transfer of income from a higher-income person to a lower-income person constitutes a social improvement, in the form of increasing social welfare or decreasing poverty. We follow the lead of Mayshar and Yitzhaki (1995) by denoting as “Dalton-improving tax reforms” those reforms that will be found to be second-order welfare improving.

By their third-order derivative, the poverty and social welfare indices that belong to Π³ and Ω³ must also be sensitive to favorable composite transfers. These transfers are such that a beneficial Pigou–Dalton transfer within the lower part of the distribution, accompanied by an adverse Pigou–Dalton transfer within the upper part of the distribution, will add to social welfare, provided that the variance of the distribution is not increased. Kolm (1976) was the first to introduce this condition into the inequality literature, and we therefore refer for simplicity to third-order welfare improving tax reforms as “Kolm-improving tax reforms.”Kakwani (1980) subsequently adapted this principle to poverty measurement.¹³

For the interpretation of higher orders of dominance, we can use the generalized transfer principles of Fishburn and Willig (1984). For s≥ 4, for instance, consider a combination of two exactly opposite and symmetric composite transfers, the first one being favorable and occurring within the lower part of the distribution, and the second one being unfavorable and occurring within the higher part of the distribution. Because the favorable composite transfer occurs lower down in the income distribution, indices that are members of the Π⁴ and Ω⁴ classes will respond favorably to this combination of composite transfers. Generalized higher-order transfer principles essentially postulate that, as s increases, the weight assigned to the effect of transfers occurring at the bottom of the distribution also increases. Blackorby and Donaldson (1978) describe these indices as becoming more Rawlsian. Thus, as shown in Davidson and Duclos (2000) for poverty indices, when s→∞ only the lowest income counts, although this result does not generalize to welfare dominance, as shown in Duclos and Makdissi (2004) and as will also appear later.

4. identifying socially improving tax reforms

To derive the conditions by which the social improvement of tax reforms can be checked, it is handy first to refer to stochastic dominance curves. When q=q^R, these are defined for orders of dominance s= 1, 2, … as (see, for instance, Davidson and Duclos, 2000)

(14)

Note that this expression can also be obtained by a process of sequential integration with  , and with D¹(z) =F(z). More generally, in real income space we have

(15)

which reduces to (14) since we have that z=η (z, θ, q, q^R) =ρ (z, θ, q, q^R) when q=q^R.

Dominance curves are therefore just sums of powers of poverty gaps. They can thus be interpreted as ethically weighted sums of individual deprivation. For s= 1, we have what is often called the “poverty incidence curve”; D²(z) gives the “poverty deficit” or the “poverty intensity” curve. The larger the value of s, the larger the weights on the largest poverty gaps. Clearly, as can be seen by comparing (11) and (14), dominance curves have a convenient link with the FGT indices since FGT^α(z) = (α)!z^−α D^α+1(z).

It is thus useful to consider how the dominance curves are affected by changes in prices. By (2) and (15), we can show that

(16)

where f(z) is the density of income at z. (More details on the derivation of (16) can be found at the end of the Proof of Theorem 1 in the Appendix.) For reasons that will become clear later, these derivatives can serve to define “consumption dominance” (CD) curves:

(17)

Because CD^s_k(z) curves describe changes in ethically weighted sums of deprivation, they can be interpreted as the ethically weighted cost of taxing k. Normalized CD curves, , are just the above CD curves for good k normalized by the average consumption of that good:

(18)

curves are thus the ethically weighted (or social) cost of taxing k as a proportion of the average welfare cost. Note that the social cost depends on s and z. By (16), only takes account of the consumption pattern of those precisely at z. The curve gives the share in the total consumption of k of those individuals with income less than z. For s= 3, 4, … , greater weight is given to the shares of those with higher poverty gaps. For z < a, the pattern of consumption and the welfare cost for some may be ignored in the computation of the social cost of taxing k. The social and average welfare cost coincide only when s= 2 and z≥a, since we have for all z≥a and all k.

Define a distribution function for the share of good k as

(19)

G_k(y) is the proportion of the total consumption of good k that is consumed by those with incomes less than y. Note that G_k(a) = 1. For s≥ 2, it follows from (16) that

(20)

Hence, the curves are FGT^α indices (multiplied by a factor of z^s−2/(s− 2)!), using α=s− 2 and a transformed income distribution G_k that weights individuals by their share in the total consumption of the good k. When multiplied by (s− 1)!/z^s−1, the CD curves for s=α+ 1 have the convenient feature of providing the impact on the FGT^α(z) indices of a marginal increase in the price of good k. This is important in its own right given the popularity of the FGT class of poverty indices.

Most importantly it would seem, CD curves can be used to test whether tax reforms are poverty or welfare improving. This is shown in Theorems 1 and 2, where γ is defined as in (5).

Theorem 1 A necessary and sufficient condition for a marginal tax reform, to be s-order poverty improving, that is, to decrease poverty weakly for all P(z) ∈Π^s(z), for all z∈[0, z⁺] and for a given s∈{1, 2, 3, … }, is that

(21)

Proof See the Appendix.

Theorem 1 is similar to the result obtained in Makdissi and Wodon (2002). The most important difference is in the formulation of first-order poverty improvement, which requires a specification of preference heterogeneity in the form of (3) and which is not immediately implementable using the framework of Makdissi and Wodon (2002). Note that (21) can be rewritten (for s > 1) as

(22)

which shows that checking the sign of the impact of the tax reform reduces to comparing consumption patterns weighted by powers of poverty gaps at different thresholds z.

Note that for s larger than 2, the curves in (21) use poverty gaps (z−y) as weights on consumption patterns. But this procedure does not require more sampling information than for the construction of tests with lower values of s since the sample values of y are also needed for s= 1 and s= 2. Note also that for s larger than 2, one must use in (20) the value of z both in the upper bound of the integral and in the integrand itself. Further, and as is clear from (10), the ethical conditions imposed on Π^s(z) are more demanding as s increases.

A more general result can also be derived for reforms involving changes in more than two tax rates. In a context of a n-good tax reform, for instance, condition (21) becomes

(23)

where any good of the n goods involved in the tax reform may be chosen as good 1. Note that (23) adds additional “degrees of freedom” to condition (21), in that one now has the freedom to check over different combinations of dq₂, dq₃, … , dq_n to verify (23). dq₁ is then obtained as .

Yitzhaki and Lewis (1996) also hint to an adaptation of their framework to poverty analysis, but it is less straightforward to do so than with CD curves. This is because poverty measurement censors well-being at poverty lines, whereas the tools used by Yitzhaki and Lewis (1996) censor the population at percentiles. One advantage of the CD curves over the concentration ones is indeed to focus on the consumption patterns of those under monetary thresholds, not on those who belong to certain percentiles of the population, as is more common in the inequality literature for instance. From a statistical perspective, censoring over percentiles has the additional disadvantage of introducing sampling variability over the value of the threshold—the value of p=F(z).

For welfare dominance, a theorem similar to Theorem 1 also applies.

Theorem 2 A sufficient condition for a marginal tax reform, to be s-order welfare improving, that is, to increase social welfare weakly for all W∈Ω^s and for a given s∈{1, 2, 3, … }, is that

(24)

5. critical poverty lines and efficiency ratios

The ratio of normalized consumption dominance curves can be interpreted as the distributive benefit of taxing j instead of l. To see why, assume again an increase in the tax on good j and thus a fall in the price of good l. The socially weighted gain from this fall in q_l, as a proportion of the average welfare gain, is given by , which is therefore an indicator of the distributive benefit of taxing j. The same naturally holds for the distributive benefit of taxing l, which is given by . Denote this distributive benefit ratio as . If , the relative distributive benefit of taxing j is then effectively infinite, but for tractability we will then define it as γ⁺⁺, which we may set to as large a finite value as we wish. δ^s(z) is thus given by

(25)

Note that δ¹(z) is a normalized ratio of Engel curves at income z. More precisely, it is the ratio, at z, of the share of good l over the share of good j, divided by the ratio of the average shares over the entire population.

Thus, δ^s(z) is the distributive benefit of taxing j relative to that of taxing l. Recall that γ is the economic cost of taxing j relative to that of taxing l. Comparing distributive benefit δ^s(z) to economic cost γ is crucial in determining whether a tax reform that increases t_j and decreases t_l is socially improving. We can indeed re-write the conditions of Theorems 1 and 2 by checking whether δ^s(z) ≥γ for all z∈[0, z⁺] and for all z∈[0, ∞) (Equations (21) and (24)), respectively. In words, a tax reform is s-order socially improving if its distributive benefit exceeds its economic cost over a range of alternative poverty lines. Note that when preferences are identical and homothetic, then by (19)G_j(y) =G_l(y) for all y, and therefore δ^s(z) = 1 whatever the values of z and s. There is then no distributive benefit to the tax reform whatever the social welfare or poverty objectives. The optimal tax system is then one in which the marginal deadweight loss of taxation is the same across the two goods.

For a given value of γ=γ⁺, we assume from now onwards that—for some z⁺ and without loss of generality—we have that

(26)

with strict inequality over at least part of the range. Such curves and are shown on Figure 1. Condition (26) is clearly also valid for all other values of γ < γ⁺. Hence, either (26) also holds for all superior values of γ > γ⁺, in which case we must have that δ^s(z) =γ⁺⁺ for all z∈[0, z⁺], or there exists a critical value of γ beyond which (26) does not hold everywhere over [0, z⁺], and this is defined by . Denoting this critical economic efficiency threshold as γ_s(z⁺), we have

(27)

Such a value of γ_s(z⁺) is shown in Figure 1 as that precise value of γ that makes and cross at z=z⁺.

A similar exercise leads to a definition of a critical upper poverty threshold z_s(γ⁺). To see this, assume again that, for a given value of γ=γ⁺, (26) holds for all values of z within some bottom range of z∈[0, z⁺], with strict inequality over at least part of the range. Then, either (26) holds for any larger value of z⁺, in which case we can set z_s(γ⁺) to as large a value as we wish and denote it by z⁺⁺, or there exists a critical value of z⁺ beyond which (26) does not hold anymore, and this is given by .¹⁴ We may then define z_s(γ⁺) as

(28)

Figure 1 shows such a z_s(γ) as the first crossing point of and . Equations (27) and (28) say that increasing t_j and decreasing t_l is socially improving so long as γ and z are not allowed to exceed certain critical thresholds, which depend on the order of ethical dominance s. For a given γ⁺ and z⁺, z_s(γ⁺) and γ_s(z⁺) give respectively the critical upper poverty line and the critical economic efficiency ratio up to which the tax reform is necessarily s-order poverty improving.

z _s (γ⁺) also gives the first poverty line (starting from z= 0) at which a poverty analyst using an FGT^s−1(z) poverty index would be exactly indifferent to the tax reform when the MECF ratio is given by γ⁺. At all lower poverty lines, the FGT poverty analyst would choose increasing t_j, and at poverty lines just higher than z_s(γ⁺) he would prefer decreasing t_j. Alternatively, γ_s(z⁺) gives the MECF ratio for which a FGT^s−1 (z⁺) analyst would be exactly indifferent between reallocating or not the burden of indirect taxation across the goods j and l. At lower values of γ, the FGT analyst prefers taxing j further; at higher values, he prefers taxing l further.

6. discussion

Two points are worth mentioning at the outset. First, although notationally and expositionally more complicated, it would not be analytically much more difficult to allow for general equilibrium effects from changes in t. This would involve considering changes in producer prices, with effects on the welfare of producers, which would typically feed into changes in the consumer and the producer prices of goods other than those whose tax rates are changed by the government. The analysis would then take into account the sum of the welfare effects of the marginal changes in the various consumer and producer prices. We hope to consider such a generalization in future work.

Second, the tools developed above are in principle strictly limited to the analysis of infinitesimal price changes. For changes in tax rates that are not infinitely small, the impact of a change in t_k on an exact measure of individual welfare is not precisely given by the pre-reform observed demand of good k times the change in t_k. The proportional difference between the exact measure and the inexact measure that we use is approximately equal to one half the compensated price elasticity of good k times dt_k/dq_k. Hence, for a compensated price elasticity of 1, a 5% increase in the price of a good k leads approximately to a 2.5% error in the estimate of consumer welfare change when one uses the pre-reform consumption of k to compute that estimate. For many purposes, this error would seem relatively small.

6.1. Efficiency and Poverty and Welfare Improvements A tax reform cannot be Pen improving when γ≥ 1, as Lemma A.1 shows in the Appendix. Pen-improving tax reforms are, however, theoretically possible so long as γ < 1. Yitzhaki and Thirsk (1990) and Yitzhaki and Slemrod (1991) find that if γ is larger than one, a tax reform cannot be Dalton improving, a result that is also shown in Lemma A.2 in the Appendix. The proof of Proposition 2 also shows that γ_s(∞) ≤ 1 for any s= 3, 4, … Thus, for a reform to be welfare improving at any ethical order requires that it also be economically efficient, namely, that it weakly decreases the average deadweight loss.

A tax reform may, however, be poverty improving at any order—even at the first—even when γ > 1. For first- and second-order poverty improvement, the excess burden or economic efficiency cost of the reform then has to be paid by those households whose income is above the maximum poverty line z⁺, and to whose change in well-being poverty analysts are ethically indifferent. More surprisingly perhaps, for ethical orders s= 3, 4, … , a tax reform may be poverty improving when γ > 1 even when z⁺ > a (which says that all can then be considered poor, though not equally poor). We sum up these results in the following remark.

Remark 1 Welfare improvement of any order requires economic efficiency. Economic efficiency is not, however, needed for poverty improvement; for s > 2 this is true even in cases in which everyone may be considered poor.

If a tax reform is s-order poverty improving up to some z⁺, then it is also poverty improving up to z⁺ at the s+ 1 order (Lemma A.3). This is because a CD curve of order s can be obtained as the integral of the CD curve of order s− 1, as in the usual stochastic dominance setting (see Makdissi and Wodon, 2002, p. 230) and as is shown in Equation (A.5) in the proof of Lemma A.3. Lemma A.3 also says that if γ_s(z⁺) < γ⁺⁺ and if there is strict dominance over a bottom range of z, then γ_s+1(z⁺) > γ_s(z⁺), meaning that, as the order of ethical dominance increases, economic efficiency becomes less constraining. Increasing the order of dominance also increases the range of poverty lines over which a reform is poverty improving. Indeed, if z_s(γ⁺) < z⁺⁺ and if there is strict dominance over a bottom range of z, then z_s(γ⁺) < z_s+1(γ⁺) (see Lemma A.4). Finally, if there is poverty improvement over a bottom range that extends to z⁺, with strict dominance over at least part of that range, then for any other finite threshold z⁺, there will also be poverty improvement for a sufficiently large order s (this is by lemma 1 of Davidson and Duclos, 2000).

These relationships are shown in Figure 2 using the distributive benefit ratios of Equation (25). When a reform's distributive benefit δ^s(z) over [0, z⁺] exceeds its economic cost γ, the reform is deemed poverty improving. At order s= 1, this is the case in Figure 2 for all γ up to γ₁(z⁺). It is also necessarily true for all higher orders s= 2, 3, … . It can also be seen that the distributive benefit increases with s over [0, z⁺], which then graphically means that γ_s(z⁺) also rises with s ({e.g.}, γ₃(z⁺) > γ₁(z⁺) in Figure 2). Alternatively, for a given γ⁺, the critical poverty line z_s(γ⁺) (up to which the distributive benefit is higher than the economic cost) increases with s; in Figure 2, this is shown by z₃(γ⁺) > z₂(γ⁺) > z₁(γ⁺).

6.2. CD and Concentration Curves The Dalton improvement condition of Theorem 2 is equivalent for s= 2 to a condition derived in Yitzhaki and Slemrod (1991) and Yitzhaki and Thirsk (1990). Their approach, however, is different. Their condition must be tested over the range of percentiles [0, 1], whereas ours is defined over an income range. To see this difference more clearly, define the concentration curve for good k as :

(29)

where F⁻¹(p) is the (left) inverse of the marginal income distribution function of incomes, is often called the p-quantile, that is, roughly speaking, the income of the individual whose rank is p. Hence, .

There thus exists a parallel between the and the usual concentration curves. The curve gives the share in good k of those below y, whereas yields the share in k of those with rank p or below. This is illustrated in Figure 3. The northeast quadrant shows the distribution function p=F(z), the northwest quadrant shows the concentration curve, and the southeast quadrant shows the consumption dominance curve of order 2. The concentration and consumption curves are linked by the 45-degree line of the southwest quadrant. This is shown by the two rectangles that are also drawn, by which and .

Concentration curves thus act as dual curves, just as it is well known that Generalized Lorenz curves¹⁵ act as dual D²(z) dominance curves. With the dual curves, Dalton improvement is tested by checking . If that condition is met, then condition (24) is also met for s= 2 and the reform then leads to a distribution that “generalized-Lorenz” dominates the status quo.

We may also define a dual first-order CD curve, denoted as :

(30)

These curves simply show the expected consumption of good k (relative to average consumption) along Pen's income parade. Pen improvement can be tested by verifying whether . As can be easily seen, this is equivalent to checking Similarly, first-order poverty improvement can be tested by checking whether . Using primal as opposed to dual curves has, however, the advantage of simplifying testing procedures for dominance orders higher than 2.¹⁶ Apart from this, the empirical application of these two approaches makes similar demands on the data, although their comparative statistical efficiency is a topic better left for future investigation.

6.3. Pareto Improving Tax Reforms We mentioned above that Pen improvement implies robustness of social welfare increase over all social welfare functions that are increasing and anonymous (or symmetric) in real incomes. Pareto improvement does not, however, impose anonymity as a property of the social evaluation exercise. This difference between these two improvement concepts may be subtle, but it is fundamental to the usefulness of practical searches for socially improving tax reforms.

A tax reform is Pareto improving if it decreases no one's real income. Under a marginal tax reform for which dq_j=−γ (X_l/X_j) dq_l, the impact on one's real income (with nominal income y and preferences θ) is given by (consider (2))

(31)

This is nonnegative for dq_j > 0 if and only if Thus:

Definition 1 A tax reform is Pareto improving if and only if

(32)

Define now the “distributive benefit” of the tax reform for someone at z and with preferences θ as

(33)

Denote by γ₀(z⁺) the maximum ratio of the MECF for which the tax reform is Pareto improving. This is formally defined by

(34)

Let also z₀(γ⁺) be the critical poverty line up to which a tax reform is Pareto improving:

(35)

Lemma A.5 in the Appendix shows that γ₀(z⁺) ≤γ₁(z⁺), with strict inequality if there is heterogeneity in the ratio of Engel curves for goods l and j at each y∈[0, z⁺]. Lemma A.6 further shows that z₀(γ⁺) ≤z₁(γ⁺), with strict inequality if z₁(γ⁺) < z⁺⁺ and if there is heterogeneity in the ratio of Engel curves at z₁(γ⁺). These are just formal ways of saying that Pareto-improving reforms are theoretically more difficult to identify than first-order socially improving ones. It also implies that the implementation of a first-order improving tax reform with a MECF ratio equal to γ₁(z⁺) will usually generate losers among those whose income is below z⁺.

An arguably more important issue is whether in practice γ₀(z) is significantly lower than γ₁(z⁺), or whether z₀(γ⁺) is substantially lower than z₁(γ⁺). We can expect condition (32) to be very restrictive empirically (as has been argued before¹⁷). In fact, it would not be misleading to suggest that a search for Pareto-improving tax reforms will normally be doomed to failure. This is due to the considerable heterogeneity of preferences typically found in the observed consumption patterns of a real population of individuals. With Pareto improvement as a social welfare constraint, it will be very difficult to identify any socially improving movement away from the current tax system, thus giving a sort of “vetoing status” to existing tax systems, whatever the defects of these systems may be. Furthermore, the identification of Pareto-improving tax reforms requires the observation of consumption patterns that are free of measurement errors.¹⁸ This is less of a problem for Pen improvement since it is the expected value of consumption patterns that matters, and in computing those expected values the measurement errors are (at least partially) averaged out. Whether the concept of Pen improvement eases the search for socially improving reforms is ultimately, of course, an empirical issue, to which we will revert later.

7. empirical illustration

This section briefly illustrates the above normative and statistical tools to household-level data from Mexico's 1996 ENIGH, a nationally representative survey with detailed income and consumption modules. In 1996, there was a VAT exemption on food expenditures in Mexico. The rationale for the exemption is apparent in Figure 4, which provides (normalized) expected food and nonfood expenditures and at different incomes z. Per capita incomes (on the horizontal axis) have been normalized by cost of living indices (the regional poverty lines used in the latest poverty assessment for Mexico completed at the World Bank), so that cost-of-living differences between urban and rural areas are taken into account. A value of one indicates that a household is at the level of the urban/rural poverty line. With these poverty lines, 60.9% of the population is poor (those with per capita income below z= 1). The estimators and associated sampling distributions are those developed in Duclos et al. (2004). Note that the illustration does not address issues related to administrative distortions; the policy recommendations may thus not be immediately implementable.

The two curves cross at , which is also the critical first-order poverty line z₁(γ) shown in the bottom panel of Table 1 for γ= 1. Note that the standard error of the sampling distribution of is estimated to be 0.099, which implies that a 95% confidence interval for the true value z₁(γ) would be [1.60, 1.98]. Simply stated, this means that if γ= 1, we can be 95% certain that for any poverty line below 1.60 it is first-order poverty improving to implement a balanced-budget indirect tax reform by reducing at the margin taxes (or providing a subsidy for) on food expenditures and increasing taxes on nonfood expenditures.

Table 1.
indirect taxation for food versus nonfood expenditures, Mexico 1996

	Value of second-order CD curves at various poverty lines z
z= 0.5	0.141	0.070	0.071
	(0.004)	(0.002)	(0.002)
z= 1.0	0.410	0.247	0.163
	(0.007)	(0.006)	(0.004)
z= 2.0	0.700	0.504	0.196
	(0.007)	(0.010)	(0.006)
	Critical efficiency ratios γs(z+) for different maximum poverty lines z+and for different orders of dominance s
	z += 0.5 (28.5% of population covered)	z += 1 (60.9% of population covered)	z += 2 (85.0% of population covered)
γ1(z+)	1.782	1.354	0.947
(0.032)		(0.027)	(0.029)
γ2(z+)	2.021	1.657	1.390
	(0.041)	(0.025)	(0.018)
γ3(z+)	2.140	1.822	1.551
	(0.049)	(0.028)	(0.020)
	Critical poverty lines zs(γ) for different ratios of economic efficiency costs γ and for different orders of dominance s(*)
	γ= 0.5	γ= 1.0	γ= 1.5
z 1(γ)	4.272	1.793	0.752
	(0.127)	(0.099)	(0.030)
z 2(γ)	–	–	1.483
	–	–	(0.156)
z 3(γ)	–	–	2.347
	–	–	(0.145)

• Source: Authors' estimation using 1996 ENIGH. Sample size is 14,022 observations.
• Standard errors in parentheses. (*) Only poverty lines higher than 0.25 are considered.

However, and as expected, this reform is not Pen improving for γ⁺= 1 since we do find a critical poverty line at which the two curves intersect. Furthermore, although a first-order poverty improving (including headcount-reducing) food/nonfood tax reform is feasible for poverty lines below z₁(1), a substantial share of the poor would lose from such a reform. Given the average food and nonfood expenditure shares, with γ= 1 a one percentage point tax reduction for food must be compensated by a 0.340 percentage point tax increase on nonfood expenditures for budget neutrality. Hence, any poor household whose food expenditure share is below one fourth will lose from such a reform. The relative importance of these losers appears in Figure 5, which gives the cumulative population share and the cumulative share of losers as functions of the poverty line. The ratio of the two cumulative shares represents the share of the population below a given poverty line that loses from the reform. It turns out that 19.0% of the population below z= 1 and 14.3% of the population below z= 0.5 would lose from this hypothetical tax reform. Hence, this reform is clearly not Pareto improving, and would not be so even if we were to censor the assessment of its impact at a poverty line. In fact—and as anticipated in Section 6.3—Pareto-improving food and nonfood tax reforms are virtually impossible, whatever the MECF economic efficiency parameter. This is due to the fact that for both food and nonfood expenditures, there are households with either zero or very low expenditure levels across the income distribution, so that is zero for virtually any z⁺.

Taxing nonfood expenditures and reducing taxes (or providing subsidies) for food expenditures is Dalton improving for a wide range of values of γ. This is demonstrated in Figures 6 and 7— is everywhere above for s= 2 and 3. The top panel of Table 1 gives the values of the curves for z= 0.5, 1, and 2, as well as the difference between the curves (all differences are statistically greater than zero). For s= 2, these curves represent the cumulative shares of food and nonfood expenditures accounted for by those with per capita income below a certain level. For example, the population below z= 1 accounts for 41.0% of total food expenditures and 24.7% of total nonfood expenditures. Figures 6 and 7 also provide the distributive benefit ratio δ²(z) and δ³(z). As long as the economic cost γ of taxing food items relative to nonfood items is below δ²(z), taxing nonfood items to give relief to food items is beneficial.

The critical efficiency ratios γ_s(z⁺) under which the tax reform is Dalton improving are provided in the middle panel of Table 1. For s= 2, γ₂(z⁺) is equal to, respectively, 2.021 for z⁺= 0.5, 1.657 for z⁺= 1, and 1.390 for z⁺= 2. If, following standard practice in Latin America, we consider those with per capita income below half the poverty line as extreme poor, the tax reform would reduce all distributive-sensitive poverty indices for the extreme poor even if the economic cost of nonfood taxation were 100% higher than that of food taxation.

The bottom panel of Table 1 also gives the critical poverty lines under which the tax reform would remain poverty improving with various MECF ratios. For example, with γ⁺= 1.5, the tax reform is second-order poverty reducing up to a poverty line of 0.752 (with a standard error of 0.03). For γ⁺= 1, there is no critical poverty line for s= 2 or higher. This is due to the fact that and do not intersect. Remember also that when multiplied by gives the impact on the FGT^s−1(z) poverty indices of a marginal change in the price of a good or a marginal increase in the tax on that good. Since the differences between the curves are statistically significant for the values of z considered in Table 1, a tax reform would lead to a statistically significant reduction in the poverty gap whatever these values of z. This does not necessarily mean that universal food subsidies are the best policy option for poverty reduction. In Mexico, universal subsidies for tortillas have recently been terminated, with the savings used for better-targeted social programs, including school stipends targeted to poor children living in poor rural areas. Still, within the options provided by indirect tax reforms, it may be Dalton improving to increase the tax on nonfood expenditures while providing subsidies for food expenditures.

4-7 are provided for broad food and nonfood expenditure aggregates, but the tools can equally well be applied to more specific items. In Figure 8, we compare the curves for a mixed bundle of food (including baby food, packaged food, deserts, drinks, and food consumed away from the home) and pasteurized milk. The estimated curves cross for a value of z slightly larger than the reference poverty line z= 1, but Table 2 (which is analogous to Table 1) indicates that the difference between the two curves is not statistically significant at z= 1 (and, similarly, that γ₂(z⁺= 1) is not statistically larger than one). Still, since the curve for milk is below that for the mixed bundle up to (allowing for a 95% confidence interval), it is certainly feasible with γ= 1 to reduce extreme poverty for all distributive-sensitive poverty indices by taxing milk and providing a subsidy for the items in the mixed-food bundle. Yet, since the second-order CD curves cross, this is not a Dalton-improving tax reform for γ= 1. It can be shown that this is also not a Kolm-improving tax reform for γ= 1 either, since the curves intersect as well, albeit at a higher value of z (as predicted by Lemma A.4).

Table 2.
indirect taxation for mixed food bundle and pasteurized milk, Mexico 1996

	Value of second-order CD curves at various poverty lines z
z= 0.5	0.075	0.037	0.038
	(0.005)	(0.003)	(0.005)
z= 1.0	0.263	0.256	0.007
	(0.009)	(0.008)	(0.010)
z= 2.0	0.537	0.627	−0.09
	(0.012)	(0.011)	(0.012)
	Critical efficiency ratios γs(z+) for different maximum poverty lines z+and for different orders of dominance s
	z += 0.5 (28.5% of population covered)	z += 1 (60.9% of population covered)	z += 2 (85.0% of population covered)
γ1(z+)	1.222	0.8	0.705
	(0.080)	(0.043)	(0.072)
γ2(z+)	1.997	1.028	0.856
	(0.181)	(0.039)	(0.019)
γ3(z+)	2.214	1.256	0.946
	(0.250)	(0.059)	(0.025)
	Critical poverty lines zs(γ) for different ratios of economic efficiency costs γ and for different orders of dominance s (*)
	γ= 0.5	γ= 1.0	γ= 1.5
z 1(γ)	11.39	0.6	0.421
	(0.065)	(0.033)	(0.028)
z 2(γ)	–	1.062	0.613
		(0.212)	(0.083)
z 3(γ)	–	1.623	0.795
		(0.140)	(0.052)

• Source: Authors' estimation using 1996 ENIGH. Sample size is 14,022 observations.
• Standard errors in parentheses. (*) Only poverty lines higher than 0.25 are considered.

As in Table 1, Table 2 provides the critical efficiency ratios γ₂(z⁺) and the critical poverty lines under which taxation of pasteurized milk and subsidies for the items in the mixed bundle is socially improving. As s increases, economic efficiency is clearly less of a constraint. This is apparent in the fact that γ_s(z⁺) increases with s (as anticipated by Lemma A.3). Again, building up the ethical content of the classes of poverty indices considered ({i.e.}, increasing s) increases the range of poverty lines and/or economic efficiency ratios over which the reform can confidently be deemed good for poverty reduction.

8. conclusion

This article shows how one can use simple Consumption Dominance curves to assess the social improvement of indirect marginal tax reforms. The methods are similar in spirit to checking for nonintersecting (second-order) concentration curves, but they are more general in that they enable the analyst to choose the order of ethical dominance in which he is interested and to censor individual welfare at some upper bound of poverty lines if so desired.

The proposed graphical tools have considerable normative appeal, in that they may be used to determine whether commodity-tax changes can be deemed to improve social welfare or decreasing poverty for large classes of social welfare and poverty indices and for broad ranges of poverty lines. They also provide detailed and useful descriptive information on the distribution of expenditures across the entire income distribution. The article further implements recently developed estimators of critical poverty lines and economic efficiency ratios that can be used to characterize socially improving tax reforms—as well as their sampling distribution. The methodology is illustrated using Mexican data.