Document(s) 9 of 20

As is well-known, the assessment of tax and transfer systems draws mainly on two fundamental principles: efficiency and equity. The former relates to the presence of distortions in the economic behavior of agents, while the latter focuses on distributive justice. Vertical equity as a principle of distributive justice is rarely questioned as such, although the extent to which it must be precisely weighted against efficiency is a matter of intense disagreement among policy analysts. A principle of redistributive justice which gathers even greater support is that of horizontal equity, the equal treatment of equals. The HE principle is often seen as a consequence of the fundamental moral principle of the equal worth of human beings, and as a corollary of the equal sacrifice theories of taxation. This chapter and the next cover in turn the measurement of each of these principles.

7.1 Taxes and transfers

Let X and N represent respectively gross and net incomes, and let T be taxes net of transfers — the net tax for short. Gross income is pre-tax and/or pre-transfer income, and net income is post-tax and/or post-transfer income, that is, N = X - T. For expositional simplicity, we assume in this chapter that gross incomes are exogenous. This is a common assumption in the literature on the measurement of the impact of taxes and transfers, although it can fail to capture the true impact of tax and transfer policies on well-being when these taxes and transfers are non-marginal.

We can expect a part of the net tax to be a function of the value of gross income X. Otherwise, taxes would be lump sum and orthogonal to gross income. We denote this deterministic part by T(X). For several reasons, we also expect T to be stochastically linked to X. In real life, taxes and transfers depend on a number of variables other than gross incomes, such as family

size and composition, age, sex, area of residence, sources of income, type of consumption and savings behavior, and the ability to avoid taxes or claim transfers. Thus, we can think of T as being a stochastic function of X, with

where v is a stochastic tax determinant.

We denote by F_X,N(.,.) the joint cumulative distribution function (cdf) of gross and net incomes. Let Q_x(p), Q_N(p) and Q_T(p) be the p-quantile functions for gross incomes, net incomes and net taxes, respectively. Let F_{N \ x}(.) be the cdf of N conditional on gross income being equal to x. The q-quantile function for net incomes conditional on a p-quantile value for gross incomes is then technically defined as Q_N(q\p) = inf for q ∈ [0, 1], assuming that net incomes are non-negative. QN(q|p) thus gives the net income of the individual whose net income rank is q among all those with gross income equal to Q_x(p).

The expected net income of those with Q_x(p) is then given by¹

and the expected net tax of those with Q_x(p) is obtained as

7.2 Concentration curves

An important descriptive and normative tool for capturing the impact of tax and transfer policies is the concentration curve. As we will see, concentration curves can help capture the horizontal and vertical equity of existing tax and transfer systems. They can also serve to predict the impact of reforms to these systems.

E:18.8.11

The concentration curve for T is²:

where is average taxes across the population. C _T (P) shows the proportion of total taxes paid by the p bottom proportion of the population.

In practice, concentration curves are usually estimated by ordering a finite number n of sample observations (X₁, N₁),..., (X_n, N_n) in increasing values

¹DAD: Distribution|Non-Parametric Regression.

²DAD: Curves|Concentration.

of gross incomes, such that X₁ ≤ X₂ ≤.... ≤ X_n, with percentiles p_{i = i/n}, i = 1,....., n and with T_i = X_i- N_i. For i = 1,...n, the sample (or "empirical") concentration curve for taxes (T_i = X_i - N_i)is then defined as

As for the empirical Lorenz curves, other values of C_T(P) can be estimated by interpolation.

The concentration curve C_N(p) for net incomes is analogously defined as

and typically estimated as

where the N_j have been ordered in increasing values of the associated gross incomes X_j. Note that C_N(P) is different from the Lorenz curve of net incomes, L_N(p), which is defined as:

Empirically, the Lorenz curve for net income is typically estimated as

but where the observations have been re-ordered in increasing values of net incomes, with N₁ ≤ N₂ ≤...≤ N_n. Thus, C_N(p) sums up the expected value of net incomes up to gross income percentile _p. L_N(p) however, sums up net incomes up to a net income percentile _p.

Denote as t the average tax as a proportion of average gross income, with t = μT/μX. When ≠0,we can show that

For a positive t, this indicates that the more concentrated are the taxes among the poor (the smaller the difference L_x(p) - C_T(P))the less concentrated among the poor will net incomes be. The reverse is true for transfers (negative t): the more concentrated they are among the poor, the more concentrated net income is among the poor. This link will prove useful later in defining indices of tax progressivity.

7.3 Concentration indices

As for the Lorenz curves and the S-Gini indices of inequality introduced earlier, we can aggregate the distance between p and the concentration curves C(p) to obtain summary indices of concentration. These indices of concentration are useful to compute aggregate indices of progressivity and vertical equity. More generally, they can also serve to decompose the inequality in total income or total consumption into a sum of the concentration of the components of that total income or consumption, such as different sources of income (different types of earnings, interests, dividends, capital gains, taxes, transfers, etc.) or different types of consumption (of food, clothing, housing, etc.).

To define indices of concentration, we can simply weight the distance p - C(p) by an ethical weight κ(P), of which a popular form is again given by κ(p; p) in equation (4.8). This gives the following class of S-Gini indices of concentration, IC(p) ³:

7.4 Decomposition of inequality into income components

7.4.1 Using concentration curves and indices

An S-Gini inequality index for a variable can easily be decomposed as a sum of the concentration indices of the component variables that add up to that variable. This can be useful, for instance, for decomposing total income inequality as a sum of concentration indices for the different sources of income (employment, capital, transfers, etc.), or total expenditure inequality as a sum of concentration indices for food and non-food expenditures, say. For example, let X₍₁₎ and X₍₂₎ be two types of expenditures, and let X = X₍₁₎ + X₍₂₎ be total consumption. Let Cx₍₁₎ (p) and Cx₍₂₎ (p) be the concentration curves of each of the two types of consumption (using X as the ordering variable). The concentration indices for X_(c), IC_x(c) (p)), c = 1, 2, are as follows:

³DAD: Redistribution|Coefficient of Concentration.

Inequality in X can then be decomposed as a sum of the inequality in X₍₁₎ and in X₍₂₎. The Lorenz curve for total consumption is given by:

which is a simple weighted sum of the concentration curves for each of the two types of consumption. The index of inequality in total consumption is similarly a simple weighted sum of the concentration indices of each of the two types of consumption⁴:

For given μX₍₁₎ and μX₍₂₎, the higher the concentration indices IC_X(1)(P) and IC_X(2)(ρ), the larger the S-Gini index of inequality in total consumption. Moreover, the higher the share μX_(C)/μX of the more highly concentrated expenditure, the higher the inequality in total expenditures⁵.

E:18.8.32

One possible difficulty with the above is that a component which has the same value for all will be judged by the decompositions in (7.13) and (7.14) to have a zero contribution to total inequality. This is because C_X(c) (p) = p for all p and IC_X(c) = 0 if component c is equally distributed across all individuals. It may be argued, however, that in such a case contribution c should be seen as contributing negatively to total inequality. Being the same for all, component c indeed decreases the inequality introduced by other components. One way to capture this is to rewrite the decompositions (7.13) and (7.14) in reference to L_X(p) and I_X(ρ).This gives:

and

The two terms on the left of each of these last two expressions give respectively the contributions of components 1 and 2 to the Lorenz curve and the inequality index of total expenditure X. Those conditions must sum to zero.

⁴DAD: Decomposition|S-Gini: Decomposition by Sources.

⁵DAD: Decomposition|S-Gini: Decomposition by Sources.

7.4.2 Using the Shapley value

An alternative approach uses the Shapley value to express inequality in total income as a sum of the contributions of inequality in individual income components. For expositional simplicity, assume again that there are only two income components, X₍₁₎ and X₍₂₎. Total inequality is then given by I (X₍₁₎, X₍₂₎). Suppose that we replace the two income components X₍₁₎ and X₍₂₎ by their mean value μx1 and μx₍₂₎, to yield I(μX₍₁₎, μX₍₂₎). Clearly, inequality would be zero after such a substitution. Total inequality can then be expressed as:

An estimate of the contribution of component 1 to total inequality would be given by the second line, and the third line would indicate the contribution of component 2. These estimated contributions are in general dependent upon the order in which the components are replaced by their mean value. The contribution of component 1 could for instance be estimated alternatively as I (X₍₁₎,μx2). To solve this order dependency problem, we can use the Shapley value to define the contribution of a component c to total inequality as its expected contribution to inequality reduction when it is added randomly to anyone of the various subsets of components that one can choose from the set of all components. With two components, this gives ⁶:

7.5 Progressivity comparisons

7.5.1 Deterministic tax and benefit systems

Let us for a moment assume that the tax system is non-stochastic (or deterministic), namely, that v equals a constant zero. Suppose also for now that this deterministic tax system does not rerank individuals, or equivalently that

⁶DAD: Decomposition|S-Gini: Decomposition by Sources.

T^(1)(X) ≤ 1. Furthermore, denote the average rate of taxation at gross income X by t(X) with t(X) = T(X)/X⁷. Assuming no reranking, a net tax

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(possibly including a transfer or subsidy) T(X) is said to be

locally progressive at X = x if the average rate of taxation increases with X, that is, if t⁽¹⁾(x) >0;

locally proportional at X = x if the average rate of taxation stays constant with X, that is, if t⁽¹⁾(x)=0;

and locally regressive at X = x if the average rate of taxation decreases with X, that is, if t⁽¹⁾(x) < 0.

E: 18.8.10

There are two popular "local" measures to capture the change in taxes and net income as gross income increases. One is the elasticity of taxes with respect to X, also called Liability Progression, LP(X):

LP(X) is simply the ratio of the marginal tax rate over the average tax rate at X. It is possible to show that a tax system is everywhere progressive (namely, t⁽¹⁾(X) >0 everywhere) if LP(X) > 1 everywhere. The larger this measure at every X, the more concentrated among the richer are the taxes.

One problem with LP(X) is that it is not defined when T(X) = 0, and that it is awkward to interpret when a net tax is sometimes negative and sometimes positive across gross income. Another problem is that it is linked to the relative distribution of taxes, not with the relative distribution of the associated net incomes.

These problems are avoided by the use of a second local measure of progression, called Residual Progression (RP(X)), which is the elasticity of net income with respect to gross income:

Unlike LP(X), RP(X) is well defined and easily interpretable even when taxes are sometimes negative, positive or zero, so long as gross and net incomes are strictly positive. It is then possible to show that a tax system is everywhere progressive (again, this means that t⁽¹⁾(X)> 0 everywhere) if RP(X) <1 everywhere.

There is a nice link between these measures of progressivity and the redistributive impact of taxes.

⁷DAD: Distribution|Non-Parametric Regression.

Progressivity and inequality reduction

Assuming no reranking, the following conditions are equivalent:

1. t⁽¹⁾(x)> 0 for all X;
   
   
2. LP(X) > 1 for all X (assuming T(X) > 0);
   
   
3. RP(X)< 1 for all x;
   
   
4. Lx(p) > C_T(P) for all p and for any distribution F_X of gross income (assuming μT > 0);
   
   
5. L_N(p) > L_X(P) for all p and for any distribution F_X of gross income.

Progressive taxation will thus make the distribution of net incomes unambiguously more equal than the distribution of gross incomes, regardless of that actual distribution of gross incomes. Moreover, if the residual progression for a tax system A is always lower than that of a tax system B, whatever the value of X, then the tax system A is said to be everywhere more residual -progressive than the tax system B, and the distribution of net incomes will always be more equal under A than under B, again regardless of the distribution of gross incomes.

Hence, an important distributive consequence of progressive taxation is to make the inequality of net incomes lower than that of gross income. Analogously, proportional taxation will not change inequality, and regressive taxation will increase inequality. The more progressive the tax system, the more inequality-reducing it is. To check whether a deterministic tax system is progressive, proportional or regressive, we may thus simply plot the average tax rate as a function of X and observe its slope. Alternatively, we may estimate and graph its Liability progression or its residual progression at various values of X. To check whether a tax system is more residual -progressive (and thus more redistributive) than another one, we simply plot and compare the elasticity of net incomes with respect to gross incomes. All of this can be done using non-parametric regressions of T(X) and N against X.⁸

Another informative descriptive approach is to compare the share in taxes and benefits to the share in the population of individuals at various ranks in the distribution of gross income. This is most easily done by plotting on a graph the ratios T(X)/μT or for various values of X or p. If these ratios exceed 1, then those individuals with those incomes or ranks pay a greater share of total taxes than their population share. A similar intuition applies when T(·) is a benefit: a ratio T(X)/nr or that exceeds 1 indicates that the benefit share exceeds the population share. If T(X) or increases proportionately faster than X or Qx(p), then the tax system is everywhere locally progressive.

⁸DAD: Distribution|Non-Parametric Regression.

A competing descriptive tool is to plot the ratio of taxes over gross income, that is, T(X)/X, perhaps assessed at some rank p to give Such a graph shows how the average tax rate evolves with gross income or ranks. When these ratios increase everywhere with X, the tax is everywhere locally progressive.

7.5.2 General tax and benefit systems

Although graphically informative, the above simple descriptive approaches present three main problems. First, if T⁽¹⁾ (X) > 1, the tax system will induce reranking, even if it is a deterministic function of X. As we will see below, reranking (and, more generally, horizontal inequity) decreases the redistributive effect of taxation, besides being of significant ethical concern in its own right.

Second, and more importantly in empirical applications, taxes are typically not a deterministic function of gross income, and randomness in taxes will introduce greater variability and inequality in net incomes than the above deterministic approach would predict. X - T(X) may then be an unreliable guide to the distribution of net incomes, and the above theorems relating local progression measures to global redistributive impact lose a great part of their practical usefulness. Randomness in taxes will also introduce further reranking. These features will reduce the redistributive effect of the tax, and may even in the most extreme cases increase inequality even when the "deterministic trend" of the tax is progressive - even when t⁽¹⁾ (X) > 0.

Third, the actual redistribution effected by taxes depends on the distribution of gross incomes, and not only on the shape of the tax function T. Said differently, the actual redistributive effect of Liability or residual progression will depend on the actual distribution of gross incomes. Arguably, the actual redistribution operated by a tax system is probably of greater interest than its potential impact. A tax may be very locally progressive over some ranges of gross income, but the actual redistributive impact will depend on the interaction of this local progression with the distribution of gross incomes.

7.6 Tax and income redistribution

To deal with these difficulties, we can use the actual distribution of taxes T and net incomes N (instead of their predicted values T(X) and X - T(X)) to determine whether the actual tax system is really progressive and inequality-reducing. This amounts to combining the local measures of progressivity with the distribution of gross incomes to generate global measures of progressivity.

There are two leading approaches for this exercise. The first is the Tax-redistribution (TR) approach, and the second is the Income- redistribution (IR) approach. The global definitions of tax progressivity associated to each of these approaches are as follows.

E: 18.8.2

1 For TR progressivity:

(a) A tax T is TR-progressive if⁹

(b) A benefit B is TR-progressive if ¹⁰

(c) A tax T₍₁₎ is more TR-progressive than a tax T₍₂₎ if ¹¹

(d) A benefit B₍₁₎is more TR-progressive than a benefit B₍₂₎ if ¹²

(e) A tax T is more TR -progressive than a benefit B if ¹³

E:18.8.3

2 For IR progressivity:

(a) A net tax T is IR-progressive if ¹⁴

(b) A net tax T₍₁₎ is more IR-progressive than a tax (and/or a transfer) T₍₂₎ if ¹⁵

These two TR and IR approaches are consistent with the use above of Liability and residual progression in a deterministic tax system. If v = 0 in (7.1), and if t⁽¹⁾ (X) > 0 and T⁽¹⁾ (X) ≤ 1 (namely, no reranking), then, whatever the actual distribution of gross incomes, T(X) is both TR- and IR-progressive. Furthermore, if LP₍₁₎(X) > LP₍₂₎(X) at all values of X, then the tax system 1 is necessarily more TR progressive than the tax system 2. And if RP₍₁₎(X) < RP₍₂₎(X) at all values of X, then the tax system 1 is necessarily more IR progressive than the tax system 2.

⁹DAD: Redistribution|Tax or Transfer.

¹⁰DAD: Redistribution|Tax or Transfer.

¹¹DAD: Redistribution Tax/Transfer vs Tax/Transfer.

¹²DAD: Redistribution Tax/Transfer vs Tax/Transfer.

¹³DAD: Redistribution Transfer vs Tax.

¹⁴DAD: Redistribution|Tax or Transfer.

¹⁵DAD: Redistribution|Tax/Transfer vs Tax/Transfer.

Note that these progressivity comparisons have as a reference point the initial Lorenz curve. In other words, a tax is progressive if the poorest individuals bear a share of the total tax burden that is less than their share in total gross income. As mentioned above, an alternative reference point would be the cumulative shares in the population. This is often argued in the context of state support — the reference point to assess the equity of public expenditures is population share. The analytical framework above can easily allow for this alternative view — for instance, simply by replacing LX (p) by p in the above definitions of TR progressivity. This will make more stringent the conditions to declare a benefit to be progressive, but it will also make it easier for a tax to be declared progressive — to see this, compare (7.21) and (7.22).

7.7 References

Many of the classical texts on the concept, the role and the measurement of tax progressivity date from the 1950's but they are still very relevant today — they include Blum and Kahen Jr. (1963), Musgrave and Thin (1948), Slitor (1948) and Vickrey (1972). See also Okun (1975) for an influential discussion of the interaction between efficiency and equity issues, as well as Pechman (1985) on incidence analysis.

The measurement of progressivity and vertical equity moved forward significantly in the middle of the 1970's following the slightly earlier advances on the measurement of inequality — see, for instance, Fellman (1976), Jakobsson (1976) and Kakwani (1977a) for the link between progressivity and inequality reduction, and Kakwani (1977b), Suits (1977) and Reynolds and Smolensky (1977) for influential indices of tax progressivity and vertical equity. Reviews of the literature can be found in Lambert (1993) and Lambert (2001).

For papers that address general links between progressivity and inequality, see Davies and Hoy (2002) (for the inequality-reducing properties of "flat taxes"), Latham (1993) (for how to assess whether one tax is more progressive than another), Liu (1985) (for tax progressivity and Lorenz dominance), Moyes and Shorrocks (1998) (on the difficulties that arise for the measurement of progressivity when households differ in needs), and Thistle (1988) (for residual progression and progressivity).

Numerous versions of other specific tax progressivity indices have been discussed and presented over the years. These include, for example, Baum (1987), for "relative share adjustment" indices; Blackorby and Donaldson (1984) and

Kiefer (1984), for normatively-based indices of progressivity; Duclos (1995a), Duclos and Tabi (1996) and Duclos (1997b), for indices of the "social performance" of tax progressivity; Duclos (1998), for normative foundations for the Suits progressivity index; Hayes, Slottje, and Lambert (1992), for effective tax progression across percentiles; and Zandvakili (1994), Zandvakili (1995) and Zandvakili and Mills (2001) for the use of progressivity indices derived from Generalized entropy and Atkinson inequality indices.

Linear indices of progressivity derive from the class of linear inequality indices introduced in Mehran (1976). They are discussed inter alia Duclos (2000), Kakwani (1987), Pfahler (1983) and Pfahler (1987).

Some of the literature has also tended to focus on the tension and on the links between local and global progressivity. See, for instance, Baum (1998), Cassady, Ruggeri, and Van Wart (1996), Formby, Seaks, and Smith (1984), Formby, Smith, and Thistle (1987), Formby, Smith, and Thistle (1990) and Formby, Smith, and Sykes (1986). See also Duclos (1995a) for a method for estimating the average residual progression of unevenly progressive tax and benefit systems, and Keen, Papapanagos, and Shorrocks (2000) and Le Breton, Moyes, and Trannoy (1996) for the impact of changes in tax components (such as sizes of allowances) on the progressivity of the overall tax system.

The influence of the "initial distribution" (that of gross incomes) on progressivity measurement is studied in Dardanoni and Lambert (2002) (for a "transplant-and-compare" procedure) and in Lambert and Pfahler (1992) - see also the comment by Milanovic (1994a). Yardsticks for assessing the effectiveness of tax and benefit policies in reducing initial inequality are proposed in Fellman, Jäntti, and Lambert (1999) and Fellman (2001).

How income is measured is also of importance for the measurement of progressivity and redistribution. See in particular Altshuler and Schwartz (1996) (for the annual vs a "time-exposure" incidence of the US child care tax credit), Caspersen and Metcalf (1994) (for the annual vs lifetime incidence of value-added taxes), Creedy and van de Ven (2001) (for the annual vs lifetime incidence of the Australian tax and benefit system), Lyon and Schwab (1995) (for the annual vs lifetime incidence of taxes on cigarettes and alcohol), Metcalf (1994) (for the lifetime incidence of US state and local taxes), Nelissen (1998) (for the lifetime incidence of Dutch social security),

Empirical studies of progressivity and redistribution have been very numerous over the last three decades. They include Bishop, Chow, and Formby (1995a) (redistribution in six LIS countries), Borg, Mason, and Shapiro (1991) (regressivity of taxes on casino gambling), Davidson and Duclos (1997) (progressivity in Canada), Decoster and Van Camp (2001) (the redistributive effect of a shift from direct to indirect taxation in Belgium), Dilnot, Kay, and Norris (1984) (progressivity in the UK between 1948 and 1982), Duclos and Tabi (1999) (redistribution in Canada), Giles and Johnson (1994) (redistri­bution in the UK), Gravelle (1992) (the redistributive effect of the 1986 US tax reform), Hanratty and Blank (1992) (the comparative poverty effect of redistributive policies in the US and in Canada), Heady, Mitrakos, and Tsak-loglou (2001) (the redistributive effect of social transfers in the European Union), Hills (1991) (the redistributive effect of British housing subsidies), Howard, Ruggeri, and Van Wart (1994) (the redistributive effect of taxes in Canada), Khetan and Poddar (1976) (redistribution in Canada), Loomis and Revier (1988) (the redistributive effect of excise taxes), Mercader Prats (1997) (redistribution in Spain, 1980-1994), Milanovic (1995) (the redistributive effect of transfers in Eastern Europe and in Russia), Morris and Preston (1986) (redistribution in the UK), Norregaard (1990) (tax progressivity in the OECD countries), O'higgins and Ruggles (1981) (redistribution in the UK), O'higgins, Schmaus, and Stephenson (1989) (comparative redistribution of taxes and transfers in seven countries), Persson and Wissen (1984) (the impact of tax evasion on redistribution), Price and Novak (1999) (the regressivity of implicit taxes on lottery games), Ruggeri, Van Wart, and Howard (1994) (the redistributive impact of government spending in Canada), Ruggles and O'higgins (1981) (the redistributive impact of government spending in the US), Schwarz and Gustafsson (1991) (redistribution in Sweden), Smeeding and Coder (1995) (redistribution in 6 LIS countries), van Doorslaer, Wagstaff, van der Burg, Christiansen, Citoni, Di Biase, Gerdtham, Gerfin, Gross, and Hakinnen (1999) (the redistributive impact of health care financing in 12 OECD countries), Vermaeten, Gillespie, and Vermaeten (1995) (the redistributive impact of taxes in Canada, 1951-1988), Wagstaff and van Doorslaer (1997) (the redistributive impact of health care financing in the Netherlands), Wagstaff, van Doorslaer, Hattem, Calonge, Christiansen, Citoni, Gerdtham, Gerfin, Gross, and Hakinnen (1999) (the redistributive impact of personal income taxation in 12 OECD countries), Younger, Sahn, Haggblade, and Dorosh (1999) (tax incidence in Madagascar).

Benefit incidence analysis is also regularly carried out in less developed economies - see, for instance, Lanjouw and Ravallion (1999) for the role of differentiated "program capture" in explaining the evolution of the incidence of benefits, Sahn, Younger, and Simler (2000) for a dominance analysis of benefit incidence in Romania, van de Walle (1998a) for a discussion of general issues, and Wodon and Yitzhaki (2002) for the role of program allocation rules in the study of benefit incidence.

There have been numerous papers decomposing the Gini indices into sums of contributions of income sources. These include Aaberge, Bjorklund, Jantti, Pedersen, Smith, and Wennemo (2000), Achdut (1996), Cancian and Reed (1998), Gustafsson and Shi (2001), Keeney (2000), Leibbrandt, Woolard, and Woolard (2000), Lerman (1999), Lerman and Yitzhaki (1985), Morduch and Sicular (2002), Podder (1993), Podder and Mukhopadhaya (2001), Podder and Chatterjee (2002), Reed and Cancian (2001), Shorrocks (1982), Silber (1989), Silber (1993), Silber (1989), Sotomayor (1996), Wodon (1999), and Yao (1997).

Document(s) 9 of 20