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In this chapter, we examine in more detail a more neglected aspect of the notion of redistributive justice: horizontal equity (HE) in taxation (including negative taxation).¹ Two main approaches to the measurement of HE are found in the literature, which has evolved substantially in the last thirty years. The classical formulation of the HE principle prescribes the equal treatment of individuals who share the same level of welfare before government intervention. HE may also be viewed as implying the absence of reranking: for a tax to be horizontally equitable, the ranking of individuals on the basis of pre-tax welfare should not be altered by a fiscal system. Most of the analysis below will involve ethical indices. We will see that, depending on the choice of the underlying social welfare function or inequality index, horizontal inequity will be captured either by a "classical" horizontal inequity index or by a "reranking" one.

8.1 Ethical and other foundations

Why should concerns for horizontal equity influence the design of an optimal tax and transfer system? Several answers have been provided, using either of two approaches. The traditional or "classical" approach defines HE as the equal treatment of equals (see Musgrave (1959)). While this principle is generally well accepted, different rationales are advanced to support it. First, a tax which discriminates between comparable individuals is liable to create resentment and a sense of insecurity, possibly also leading to social unrest.

Second, the principles of progressivity and income redistribution, which are key elements of most tax and transfer systems, are generally undermined by horizontal inequity (HI) - as we shall see in our own treatment below. This has indeed been one of the main themes in the development of the reranking approach in the last decades. Hence, a desire for HE may simply derive from a general aversion to inequality, without any further appeal to other normative criteria. HI may moreover suggest the presence of imperfections in the operation of the tax and transfer system, such as an imperfect delivery of social welfare benefits, attributable to poor targeting or to incomplete take-up. It can also signal tax evasion, which can inter alia cost the government significant losses of tax revenue.

¹This chapter draws extensively from Duclos, Jalbert, and Araar (2003), where more details can be found.

Third, HE can be argued to be an ethically more robust principle than VE. VE asks for the reduction of welfare gaps between unequal individuals. Depending on the retained specification of distributive fairness, the strength of the requirements of vertical justice can vary considerably, while the integrity of the principle of horizontal equity remains essentially invariant. This has led several authors to advocate that HE be treated as a separate principle from VE, and thus that HE be one of the objectives over which optimal trade-offs are assessed for the setting of tax policy.

The theory of relative deprivation also suggests that people often specifically compare their relative individual fortune with that of others in similar or close circumstances. The first to formalize the theory of relative deprivation, Davis (1959), expressly allowed for this by suggesting how comparisons with similar vs dissimilar others lead to different kinds of emotional reactions; he used the expression "relative deprivation" for "in-group" comparisons (i.e., for HI), and "relative subordination" for "out-group" comparisons (i.e., for VE) (Davis 1959, p.283). Moreover, in the words of Runciman (1966), another important contributor to that theory, "people often choose reference groups closer to their actual circumstances than those which might be forced on them if their opportunities were better than they are" (p.29).

In a discussion of the post-war British welfare state, Runciman also notes that "the reference groups of the recipients of welfare were virtually bound to remain within the broadly delimited area of potential fellow-beneficiaries. It was anomalies within this area which were the focus of successive grievances, not the relative prosperity of people not obviously comparable" (p.71). Finally, in his theory of social comparison processes, Festinger (1954) also argues that "given a range of possible persons for comparison, someone close to one's own ability or opinion will be chosen for comparison" (p.121). In an income redistribution context, it is thus plausible to assume that comparative reference groups are established on the basis of similar gross incomes and proximate pre-tax ranks, and that individuals subsequently make comparisons of post-tax outcomes across these groups. Individuals would then assess their relative redistributive ill-fortune in reference groups of comparables by monitoring inter alia how they fare compared to similar others, and by assessing whether they

are overtaken by or overtake these comparables in income status, thus providing a plausible "micro-foundation" for the use of HE as a normative criterion.

This suggests that comparisons with close individuals (but not necessarily exact equals) would be at least as important in terms of social and psychological reactions as comparisons with dissimilar individuals, and thus that analysis of HI and reranking in that context should be at least as important as considerations of VE. It also says that, although classical HI and reranking are both necessary and sufficient signs of HI, they are (and will be perceived as) different manifestations of violations of the HE principle.

The value of studying classical HI has nonetheless been questioned by a few authors, who reject the premise that the initial distribution is necessarily just, or who point out that utilitarianism and the Pareto principle may justify the unequal treatment of equals (as discussed above). A number of authors have also expressed dissatisfaction with the classical approach to HE because of the implementation difficulties it was seen to present. Indeed, since no two individuals are ever exactly alike in a finite sample, it was argued that analysis of equals had to proceed on the basis of groupings of unequals which were ultimately arbitrary. The proposed alternative was then to link HI and reranking and to note that the absence of reranking implies the classical requirement of HE. For instance, Feldstein (1976), p.94, argues that

the tax system should preserve the utility order, implying that if two individuals would have the same utility level in the absence of taxation, they should also have the same utility level if there is a tax.

Various other ethical justifications have also been suggested for the requirement of no-reranking. For instance, King (1983) argues in favor of adding (for normative consistency) the qualification "and treating unequals accordingly" to the classical definition of HE. It then becomes clear that classical HE also implies the absence of reranking. Indeed, if two unequals are reranked by some redistribution, then it could be argued at a conceptual level that at a particular point in that process of redistribution, these two unequals became equals and were then made unequal (and reranked), thus violating classical HE. Hence, from the above, it would seem that (quoting again from King 1983, p. 102) "a necessary and sufficient condition for the existence of horizontal inequity is a change in ranking between the ex ante and the ex post distributions". We thus follow each of the approaches in turn, starting with reranking.

8.2 Measuring reranking and redistribution

We first show how to decompose the net redistributive effect of taxes and transfers into vertical equity (VE) and reranking (RR) components. The VE effect measures the tendency of a tax system to "compress" the distribution of net incomes, which is linked to the progressivity of the tax system. The RR term contributes negatively to the net redistributive effect of the tax system.

The use of Lorenz and concentration curves and of the associated S-Gini indices of inequality and redistribution will enable this integration of reranking and horizontal inequity.

8.2.1 Reranking

Recall first the definition of a concentration curve for net income in (7.6). We can show that C_N(p) will never be lower than the Lorenz curve L_N(P), and will be strictly greater than L_N(p) for at least one value of p if there is "reranking" in the redistribution of incomes. (In a continuous distribution, a sufficient condition for reranking is that v in (7.1) is not degenerate, namely, that it is not a constant.) Intuitively, C_N(p) cumulates some net incomes whose percentiles in the net income distribution exceed p. These are net incomes that exceed and QN(P) Such high incomes are nevertheless possible, however, due to the stochastic term v in (7.1). L_N(p) only cumulates the net incomes which equal Q_N(P) or less. Hence, C_N(P) ≥ L_N(p). This can also be seen by comparing the estimators in equations (7.7) and (7.9). In (7.9), the observations of Nj are cumulated in increasing values of Nj, but in (7.7), the observations of Nj ate cumulated in increasing values of Xj, which means that some higher values of Nj may be cumulated before some lower ones.

It is therefore straightforward to conclude that a net tax T will cause reranking (and hence horizontal inequity) if and only if C_N(p) > L_N(P) for at least one value of p ε]0, 1[. The distance C_N(p) - L_N(P) can therefore be used

E: 18.8.5

as an indicator of reranking². A natural S-Gini index of rerankingind is then obtained as a weighted distance between the two curves:

Denoting IC_N(p) as the index of concentration of net incomes (recall (7.11)), this index of reranking can also be obtained as

8.2.2 S-Gini indices of equity and redistribution

As for comparisons of inequality and concentration, it is often useful to summarize the progressivity, vertical equity, horizontal inequity as well as the redistributive effect of taxes and transfers into summary indices. We can do this by weighting the differences expressed above by the weights k(p; p) of the S-Gini indices to obtain S-Gini indices of TR- progressivity (IT(p)), IR- progressivity and vertical equity (IV(p)), reranking (RR(p)), and redistribution (IR(p)):

²DAD: Curves|Lorenz and DAD: Curves|Concentration.

These indices can also be computed as differences between S-Gini indices of inequality and concentration:

Many of these indices have first been proposed with p = 2, which corresponds to the case of the standard Gini index. IT(p = 2) is known as the Kakwani index of TR progressivity³, IV(p = 2) is known as the Reynolds-

E:18.8.4

Smolensky index of IR progressivity and vertical equity, and RR(p = 2) is known as the Atkinson-Plotnick index of reranking.

8.2.3 Redistribution and vertical and horizontal equity

The difference between the Lorenz curve of net and gross incomes is given by:

The larger this difference, the more redistributive is the tax and benefit system. Alternatively, the net redistribution can be expressed in terms of S-indices⁴:

E:18.8.6

³DAD: Inequality|Gini/S-Gini Index and DAD: Redistribution|Coefficient of Concentration.

⁴DAD: Inequality|Gini/S-Gini Index and DAD: Redistribution|Goefficient of Concentration.

The first term VE in each of the above two expressions is clearly linked to the definition of IR- progressivity in equation (7.26). As shown in equation (7.10), it can also be expressed in terms of TR- progressivity when t ≠ 0:

and, using S-indices,

Furthermore, if there is more than one tax and/or benefit that make up T, we can decompose total VE as a sum of the IR and TR progressivity of each tax and transfer. Say that there are J such taxes or benefits. Let t_(j) be the (overall) average tax rate of the tax T_(j) with j = 1,..., J, such that and let C_T(j) (p) and C_N(j)  (p) be the concentration curves of net income and taxes corresponding to tax T_(j), with N_(j) = X — T_(j). Then, we have

and

C_N(j) (p)— LX(P) and IX(P)—IC_N(p) capture the vertical equity of tax or transfer j at percentile p, and again can be easily seen to be an element of the definition of IR- progressivity. Each of these VE contributions can also be expressed as a function of TR progressivity at p (when t_(j) ≠ 0):

or, using S-Gini indices of IR progressivity, as a function of S-Gini indices of TR progressivity:

The second term on the right-hand side of (8.11) and (8.12) is the redistribution-reducing reranking effect. As is well known from the literature on reranking (see Atkinson, 1979, and Plotnick, 1981, for instance), taking into account reranking when using rank-dependent inequality indices increases measured inequality and decreases the redistributive effect of taxation,

and this explains why I_N generally exceeds IC_N,and also why the difference can be interpreted as the impact of reranking on the net redistributive effect of taxation.

To interpret that second term, we may also think of individuals resenting being outranked by others, but enjoying outranking others, and then assess their net feeling of resentment by the amount by which the net income of the richer (than themselves) actually exceeds what the net income of the richer class would have been had no "new rich" displaced "old rich" in the distribution of net incomes. We can then show that μ_N(I_N(P) — IC_N(p)) is the expected net income resentment of the poorest person in samples of p — 1 randomly selected individuals, and thus that RR(p) is an ethically-weighted indicator of such net resentment in the population.

8.3 Measuring classical horizontal inequity and redistribution

We now turn to the measurement of classical horizontal equity, defined again as "the equal treatment of equals".

8.3.1 Horizontally-equitable net incomes

One natural avenue for measuring whether equals are treated equally is to estimate the variability of taxes and net incomes conditional on some initial value of gross income. We may, for instance, wish to estimate the conditional variability of T at some value of X. Alternatively, and perhaps better for expositional purposes, we may want to show that conditional variability over a range of percentiles p of gross income X, and we may thus want to estimate for example the conditional variance of T at gross income:⁵

E:18.8.7

Recent work has, however, attempted to make the measurement of classical HI flow from ethical (as opposed to descriptive or statistical) foundations. We show how this can be done using the popular Atkinson social welfareatk function W(t) introduced in (4.37). For the distribution of net incomes, this social welfare function equals:

Recall that the expected net income of those at rank p in the distribution of gross income is given by Hence, if the tax system were horizontally

⁵DAD: Distribution|Conditional Standard Deviation.

equitable and if all individuals at rank p in the distribution of gross income were granted in net incomes, the local level of utility would be U (N(p); e) and net-income social welfare would equal

The expected net income utility of those at rank p in the distribution of gross income is, however, equal to

If, instead of U(N(p); e), we assigned individuals at rank p their expected net income utility U(p; e), social welfare would equal

is social welfare using ex ante expected net income; is social welfare using ex ante expected net income utility. By the concavity of the utility function, we have that and this difference captures the local utility cost of net income uncertainty at p. Hence, we also have that a feature which we can use to capture the global social welfare cost of HI and its impact on redistribution.

To show the social welfare cost of HI and its impact on redistribution, we can follow either of two approaches. Recall that we have just provided two locally horizontally-equitable tax systems:

one in which each individual at rank p in the distribution of gross incomes receives JV(p) and utility U(N(p); e),

and one in which each of these individuals receives U(p; e).

In the first case, but mean income is the same under the two distributions N(p) and since Hence, a consequence of HI is to increase inequality and to decrease the redistributive fall in inequality brought about by tax and benefit systems. This is further developed in Section 8.3.2.

The second case imposes a horizontally-equitable local distribution of utility U(p) that equals the ex ante expected local utility. Compared to the actual distribution of net incomes, this reduces inequality but maintain the overall level of social welfare. Hence, it must be that average income under U(p) is lower than under N(p). It also implies that the cost of inequality is lower with U(p). This is further developed in Section 8.3.3.

8.3.2 Change-in-inequality approach

Let the equally distributed equivalent (EDE) incomes for WN(∈), and be , respectively. As before, inequality can be measured by the differences between those ξ and the corresponding μ, as a proportion of μ Now observe that

since and . Hence, .HI increases inequality. The overall redistributive change in inequality that results from the effect of taxes and transfers can then be expressed as

Note also that, by (4.35), (8.25) is equivalent to when the means of X and N are the same.

Hence, using (8.25) we obtain the following decomposition of the net redistributive change iri inequality⁶:

VE represents the decrease in inequality yielded by a tax which treats equals equally. Thus, VE can be interpreted as a measure of the underlying vertical equity of horizontally-equitable net taxes measures the fall in redistribution attributable to the unequal post-tax treatment of pre-tax equals. The excess of over is due to the appearance of post-tax income inequality within groups of pre-tax equals.

8.3.3 Cost-of-inequality approach

In the above change-in-inequality approach, average income is kept the same while comparing distributions of actual and horizontally equitable net incomes. Social welfare and inequality do, however, vary across the distributions of N(p) and In the second approach, the cost-of-inequality approach, social welfare is kept the same across the distributions being compared but the mean income required to attain this level of welfare varies. Each element of the decomposition in this section thus corresponds to a difference in means at equal social welfare

⁶DAD: Redistribution|Duclos & Lambert (1999) and DAD: Redistribution|Duclos, Jalbert & Araar (2003).

The cost of inequality in the distribution of net income can be expressed as:

Recall that represents the level of per capita net income that society could use for the elimination of inequality with no loss of social welfare.

Let represent the cost of inequality subsequent to a flat (or proportional, and thus inequality neutral) tax on gross incomes that generates the same level of social welfare as the distribution of net incomes. Denote the average income under this welfare-neutral flat tax by μF The net effect of redistribution on the cost of inequality then becomes:

Since and since we also have

which is positive if The more progressive the net tax system, the greater the value of . If the net tax system is progressive, the greater the value of e, the greater the redistributive fall in the cost of inequality.

We then write the decomposition of the total variation in the cost of inequality as ⁷:

The redistributive fall in the cost of inequality then decomposes into two effects.

First, is the cost of inequality under a (horizontally-equitable) certainty-equivalent level of net income at all ranks p. This certainty-equivalent net income is given by at rank p. Hence, for constant social welfare, an horizontally-equitable tax system corresponds to a distribution of to each individual at pre-tax percentile p.

Second, in (8.30) measures the difference in the cost of inequality of two horizontally equitable tax systems, the first being a flat tax system, and the second granting everyone his certainty equivalent level of net income, with both systems yielding the same level of social welfare W_N. is positive if the tax system is progressive in an ex ante, certainty-equivalent, sense. In such a case, the distribution across percentiles of the certainty-equivalent net incomes is less inequality costly than the distribution of gross incomes.

⁷ DAD: Redistribution|Duclos & Lambert (1999) and DAD: Redistribution|Duclos, Jalbert & Araar (2003).

8.3.4 Decomposition of classical horizontal inequity

We may also wish to know at which percentile or for which population group HI is more pronounced, and by how much it contributes to total classical HI. For this, define the local cost of classical violations of HE at p as:

This is the "risk-premium" of net income uncertainty at percentile p, and it is thus a money-metric cost of local classical HI at p. It is then possible to show that aggregating (8.31) using population weights yields the global index of total classical HI in (8.30):

8.4 References

The literature on horizontal inequity has evolved very significantly over the last 25 years. Recent literature surveys can be found in Jenkins and Lambert (1999), Lambert and Ramos (1997a) and Lambert (2001) (see also the comment by Plotnick (1999) and the earlier reviews of Musgrave (1990) and Plotnick (1985)). See also Balcer and Sadka (1986), Feldstein (1976), Hettich (1983), Lambert and Yitzhaki (1995) and Stiglitz (1982) for a treatment of horizontal equity as a separate principle from vertical equity, and Kaplow (1989), Kaplow (1995) and Kaplow (2000) for a critique of the principle of horizontal inequity.

The early reranking approach was much influenced by Atkinson (1979), Plotnick (1981) and Plotnick (1982) (for the RR(2) index), and King (1983) (for a normative link between inequality, mobility and reranking). See also Chakravarty (1985) for normative links between inequality and reranking, Dardanoni and Lambert (2001) for a statistically-based look at the association between gross and net incomes, Duclos (1993) for the general form of the IR(p) indices, Jenkins (1988a) for a "within-group" horizontal equity focus, Kakwani and Lambert (1999) for a Hi-related analysis of tax discrimination, Kakwani and Lambert (1998) for an axiomatic construction of equity measures, Rosen (1978) for a (rare) utility-based evaluation of horizontal inequity, and Lerman and Yitzhaki (1995) for reasons for which reranking may decrease inequality.

Classical horizontal equity has seen extensive developments particularly in the last 10 years: see, for instance, Aronson, Johnson, and Lambert (1994), Aronson and Lambert (1994), Aronson, Lambert, and Trippeer (1999) and van de Ven, Creedy, and Lambert (2001), for the use of the Gini for calculating both reranking and classical horizontal inequity; Duclos and Lambert (2000), for a cost-of-inequality approach; and Auerbach and Hassett (2002) and Lambert and Ramos (1997b), for a change-in-inequality approach.

Empirical enquiries into the extent of horizontal inequity have also been relatively numerous. They include inter alia Ankrom (1993) for comparative Swedish, British and American evidence, Berliant and Strauss (1985) for the US federal income tax system, Bishop, Formby, and Lambert (2000) for the effects of noncompliance and tax evasion, Creedy (2001) and Creedy (2002) for the impact of non-uniform indirect taxes on horizontal inequity in Australia, Creedy and van de Ven (2001) for the impact on measured horizontal inequity of using different equivalence scales and of using annual vs lifetime income, Decoster, Schokkaert, and Van Camp (1997) for indirect taxation and horizontal inequity in Belgium, Duclos (1995b) for the role of imperfections in poverty alleviation programs, Jenkins (1988b) and Nolan (1987) for the extent of reranking in the UK, Sa Aadu, Shilling, and Sirmans (1991) for whether the treatment of capital gains on owner-occupied housing matters for horizontal inequity, and Stranahan and Borg (1998) for whether an implicit "lottery tax" is a source of horizontal inequity.

The advances in the measurement of horizontal inequity have also led to a desire to decompose the overall measurement of redistribution as a function of progressivity, vertical equity, reranking and classical horizontal inequity. This is done inter alia in Duclos (1993) (with the S-Gini), Duclos (1995b) (with redistributive imperfections), Kakwani (1984) and Kakwani (1986) by using the Gini index but not attempting to measure classical horizontal inequity; and in Aronson, Johnson, and Lambert (1994), Aronson and Lambert (1994), van Doorslaer, Wagstaff, van der Burg, Christiansen, Citoni, Di Biase, Gerdtham, Gerfin, Gross, and Hakinnen (1999) (for health financing in 12 OECD countries), Wagstaff and van Doorslaer (1997) (for health financing in the Netherlands), Wagstaff, van Doorslaer, Hattem, Calonge, Christiansen, Citoni, Gerdtham, Gerfin, Gross, and Hakinnen (1999) (for personal income taxes in 12 OECD countries), all using the Gini index and incorporating both reranking and classical horizontal inequity. See also Wagstaff and van Doorslaer (2001) for a decomposition of total tax progressivity in components such as the progressivity of tax credits, marginal tax rates, allowances and deductions.

Document(s) 10 of 20