Abstract
This is the core chapter of the theoretical part of the book. First, uniform measures of tax progression for identical incomes distributions are introduced. Next, the concept of uniform tax progression is extended to comparisons that involve different income distributions in order to facilitate international and intertemporal comparisons. It is also shown that necessary and sufficient conditions do not exist for uniform tax progression, whenever different income distributions for the situations to be compared are admitted. A discrete formulation of the extended concept, being indispensable for empirical purposes, is given as well. Six definitions of the relation “is more progressive than” are given. They are based either on Lorenz or Suits curve differences, and refer to taxes, net incomes, or second order differences. The rest of the chapter deals with the economic intuition behind the different definitions and their interrelationships.
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- 1.
- 2.
Elasticities raise problems for intervals of income for which, due to tax allowances, no tax is levied. In this book, tax allowances will be largely neglected. For a thorough analysis within the framework of this section see Keen et al. (2000). Using the Kakwani index of impost progression, Wagstaff and van Doorslaer (2001) studied the dominant factors of progression in 15 OECD countries. They identified three clusters of the main drivers of progression, viz. (a) the rate structure countries, where the rate effect is the dominant source of progression, (b) the allowance countries, where allowances are the dominant source of progression, and (c) the mixed structure countries, where roughly half of the progression is attributable to the rate structure and half to tax allowances. In this book, we do not differentiate between sources of impost progression. Rather we measure progression irrespective of its sources.
- 3.
Since Jakobsson’s result is somewhat vaguely stated, it provoked misunderstandings. For an unambiguous analysis see Thistle (1988).
- 4.
The sufficiency part of this theorem for Lorenz dominance was proven by Fellman (1976).
- 5.
For clarifications of the relationship between the single-crossing condition and Jakobsson’s theorem see Thistle (1988).
- 6.
On the other hand, Kakwani’s (1977a) sufficiency conditions extend to necessary conditions if they should apply to the universe of income distributions. Suppose \({\epsilon }^{1}(Y ) \geq {\epsilon }^{2}(Y )\) on \([{Y }_{{_\ast}},{Y }_{{_\ast}{_\ast}}]\) except on the subinterval \(\tilde{Y } \subset [{Y }_{{_\ast}},{Y }_{{_\ast}{_\ast}}]\) on which \({\epsilon }^{1}(Y ) < {\epsilon }^{2}(Y )\) holds. Then for all income distributions defined on \(\tilde{Y }\), T 2 is more progressive than T 1. Hence, \({\epsilon }^{1}(Y ) \geq {\epsilon }^{2}(Y )\) on the whole support \([{Y }_{{_\ast}},{Y }_{{_\ast}{_\ast}}]\) is a necessary and sufficient condition for T 1 to be more progressive than T 2 if all income distributions are admissible. The extension to \(\eta (Y )\) is immediate.
- 7.
Note the this approach has also the advantage that the effects of inflation for intertemporal comparisons of tax progression are normalized by population or income shares. At the same time, different currencies are also calibrated and unified by this approach, which renders international comparisons of tax progression viable.
- 8.
Equations (4.9) and (4.38) (the latter for the discrete case) are the so-called Suits (1977) curves. Equation (4.10) is the Suits-curve equivalent for net incomes. Recall from Sect. 3.2 that Suits used the tax curve to construct a global measure of tax progression similarly to the Gini coefficient for measuring income inequality. Note that \({F}_{Y }^{Y }(p)\) is just the diagonal through the unit square. Therefore, this formulation is omitted.
- 9.
Note that such transformations of variables work only for monotonous functions for which inverse functions exist. Basically these transformations hypothesize a functional relationship between the ranges of two functions. Consider, for instance, F Y (q). It is composed of F(Y ) = q and F Y (Y ) = p, i.e., of two functions of Y. Then a function between the two images of these functions for equal values of the common domain variable Y is hypothesized, such that q forms the domain and p the image of this function: \(p = {F}_{Y }(Y ) = {F}_{Y }[{F}^{-1}(q)] = {F}_{Y }(q)\). This holds mutatis mutandis also for (4.7)–(4.10). This phenomenon becomes more visible for the discrete transformations (4.35)–(4.39).
- 10.
This means that we assume awayre-ranking in our theoretical analysis. We will come back to this point in the next section.
- 11.
Stroup (2005, p. 205–6) observes that “if the richest 10% of society pays 25% of the total income tax burden, this figure means something different whether the same percentile earned 20% of total income in society or earned 30%—the former indicates a progressive tax system while the latter implies a regressive tax system.” This represents a concise intuition of the difference between \({F}_{T}(q)\) and \({F}_{T}^{Y }(p)\). Note that Stroup (2005) based his proposal of a measure of tax progression on the difference between F Y (q) and \({F}_{T}^{Y }(p)\). With respect to our definitions of greater tax progression as developed in Sect. 4.3, Box 1 readily shows us that the uniform equivalent of Stroup’s measure of tax progression is: Definition 5 − Definition 1 + Definition 2.
- 12.
It can also be performed in terms of Y but this would require that both income distributions involved have equal support; see Seidl (1994, pp. 347–9).
- 13.
Equivalently, one can require that the slope of a relative concentration curve be less than one below a unique value of its argument and greater than one thereafter. Whereas this is equivalent to strict convexity for the case of relative concentration curves, strict convexity produces the more intuitive and precise formulations of the sufficient conditions.
- 14.
Dardanoni and Lambert (2002, Theorem 4, p. 111) showed that the sufficient condition is at the same time the necessary condition iff all income distributions involved are related by an isoelastic transformation. In terms of Fig. 4.3, concave sections of the relative concentration curve must not occur in this case. Note that Dardanoni and Lambert (2002) carried out their analysis forresidual income progression (which is characterized as relative concentration curves above the diagonal), but this analysis is equivalent to the analysis in terms ofliability progression (cf. Dardanoni and Lambert 2002, Footnote 4, p. 101).
- 15.
In accordance with Figs. 4.2 and 4.3, we assume in Theorems 2 and 4 that \(({Y }^{1},{T}^{1})\) is more progressive than \(({Y }^{2},{T}^{2})\). Hence, \({F}_{{T}^{1}}(q),\) \({F}_{{Y }^{1}-{T}^{1}}(q),\) \({F}_{{T}^{1}}^{{Y }^{1} }(p),\) and \({F}_{{Y }^{1}-{T}^{1}}^{{Y }^{1} }(p)\) are placed on the ordinate and \({F}_{{T}^{2}}(q),\;{F}_{{Y }^{2}-{T}^{2}}(q),\;{F}_{{T}^{2}}^{{Y }^{2} }(p),\) and \({F}_{{Y }^{2}-{T}^{2}}^{{Y }^{2} }(p)\), are placed on the abscissa of the coordinates of the unit square in which the relative concentration curves are depicted.
- 16.
Although having a necessary and sufficient condition for their analysis in terms of deformed income distributions, viz.isoelasticity of the deformation functions, to warrant independence of the baseline distribution, Dardanoni and Lambert (2002, p. 106) had to concede that isoelasticity is a rather demanding condition which will hardly be met in the real world. (For this approach see also Footnotes 2 in Chap. 1 and 14 in Chap. 4.) The possibility of contradictions resulting from the choice of different baseline distributions was already recognized by Bishop et al. (1990, p. 10), who, however, did not dispose of the theoretical result of avoiding contradictions later derived by Dardanoni and Lambert (2002). Hence, comparison of tax progression “becomes an empirical question,” which is another impetus for our present work.
- 17.
Since “effective progressivity is directly derived from collected revenues and existing income distribution, which makes the identification of the causal effect of the effective progressivity on the outcomes from which it is derived highly problematic,” Sabirianova Peter et al. (2010, p. 19) relied basically on the tax schedule which they applied to percentiles of per-capita GDP over a range from zero to fourfold per-capita GDP.
- 18.
Even recently the possibility of such analyses was denied. For instance, in their impressive study on tax progression and income distribution, Duncan and Sabirianova Peter (2008, p. 15) remark: “The inequality-based [progression] measures generally require information on pre- and post-tax inequality and the distribution of the tax burden. Information on these variables is either not available or not comparable across countries. The more serious problem, though, is the issue of simultaneity in determination of income inequality and inequality-based progressivity, which inhibits the identification of the direct effect of tax progressivity on inequality.”
- 19.
Like in our theoretical analyses we exclude negative incomes and taxes from our empirical analyses.
- 20.
- 21.
Note that for the discrete version we continue to use superscripts to refer to the two different vectors of taxes and incomes, each representing the situation as a whole, while subscripts are used for vector components.
- 22.
Strong arguments in favor of this approach were brought forward on theoretical grounds because concentration curves capture the effect of transfer and impost progression. Such effects of the whole impost system have to be allowed for to evaluate its full effects. The difference between the Lorenz curve of net incomes and its concentration curve rather captures the effect ofhorizontal equity (which is not the focus of this book). See Kakwani (1977b, pp. 72–3), Plotnik (1981), Jenkins (1988), and Bishop et al. (1990, pp. 5–6).
- 23.
- 24.
Buhmann et al. (1988, pp. 119–122) investigated 34 equivalence scales which were proposed by various researchers, and found that the Luxembourg equivalence formula fits them well for various values of \(\alpha \) for four representative groups of proposed equivalence scales. Buhmann et al. (1988, p. 128) also observed that income inequality first decreases and then increases as \(\alpha \) increases, viz. inequality is an U-shaped function of \(\alpha \); poverty decreases as \(\alpha \) increases (p. 132). For more elaborate work see Coulter et al. (1992), Banks and Johnson (1994), Jenkins and Cowell (1994), Faik (1995), and Cowell and Mercader-Prats (1999).
- 25.
Buhmann et al. (1988, p. 127) argue that equivalence scales have greater effect in case of different household structures associated with the actual income distributions to be compared; greater households, in particular, influence the results. Peichl et al. (2009a,b) observed that part of the increase in income inequality in Germany in terms of equivalized incomes is due to the trend in the direction of smaller households in the last decades.
- 26.
- 27.
The case of a relative concentration curve being below (or above) the diagonal in the interior of the unit square is equivalent to a positive (negative) difference of the generating curves within the unit interval. Recall that a concentration curve of \({F}_{{T}^{1}}(\cdot )\) relative to \({F}_{{T}^{2}}(\cdot )\) does not cross the diagonal iff \({F}_{{T}^{1}}(\cdot ) - {F}_{{T}^{2}}(\cdot )\) has the same sign for all \(q,p \in (0, 1)\). Note that this applies analogously also to net incomes.
- 28.
See also Kleiber and Kotz (2003, pp. 42–3).
- 29.
For the properties of this curve see Dancelli (1990).
- 30.
We use the formulation of Greselin et al. (2010, p. 2).
- 31.
See Zenga (1990, pp. 103 and 106).
- 32.
Note that this is merely a possibility. Taxes are levied on absolute rather than relative incomes. Therefore, considerations beyond proportional taxation are subject to speculation.
- 33.
Rothschild and Stiglitz (1970) introduced the concept of a mean-preserving spread to the measurement of risk. A mean-preserving spread moves probability mass from the center to the tails of the distribution, leaving the mean constant. Atkinson (1970, p. 247) pointed out the mathematical analogy between a mean-preserving spread and theprinciple of transfers: rank-preserving transfers from richer to poorer persons yield more equally distributed income distributions. Note that this is just a corollary of a time-honored theorem of Hardy, Littlewood and Pólya; see Berge (1963, p. 184).
- 34.
For the sufficient conditions, Theorems 4 and 5 may at first sight impart the impression of separate influences of the elasticities of the tax schedules or net incomes on the one hand, and the elasticities \(\chi (p)\) and χ(q) on the other. However, this impression is wrong because the elasticities \(\epsilon (\cdot )\) and \(\eta (\cdot )\) themselves depend in intricate ways on the income distributions (which applies also to Theorem 2; see, in particular, Corollary 3). [Note that the analysis in terms of income is a different case. For the sufficient conditions in terms of taxes and net incomes we are able to additively separate the effects of the tax schedule from those of the income distributions in the form of the sum of the elasticity of the income distribution density function on the one hand, and the tax elasticity or the residual income elasticity on the other; see Seidl (1994, pp. 347–8). However, this analysis applies only to cases of identical monetary units and identical supports of the income distributions involved.] The work of Dardanoni and Lambert (2002) may also be viewed under the aspect of separating tax schedules and income distributions. These authors employ deformation functions to mimic the income distribution of the other country to be compared. However, this possible way of decomposition works only for isoelastic deformation functions to secure independence of the baseline distribution (see also Footnotes 2 in Chap. 1 and 14, and 16 in Chap. 4).
- 35.
For instance, the top 10 percent of population quantiles may count for 25 percent of aggregate income, whereas the top 10 percent of income quantiles may represent the aggregate income of only 2 percent of the population. The tax on the former group is expected to be much higher than the tax on the latter.
- 36.
Table 6.14 will be explained in detail in Sect. 6.2.2.
- 37.
These tables will be explained in detail in Sect. 6.2.1.
- 38.
The population principle requires that income inequality in a collectivity remains unaffected if each income recipient is replaced by the same number of clones with exactly the same income. This applies, mutatis mutandis, also to taxes and net incomes.
- 39.
Switzerland’s population is about 2 percent of the population of the United States. To have equivalent income distributions in both countries in discrete terms requires therefore 50 income clones in the United States for each Swiss income recipient. Construct an income distribution by taking one representative out of the 50 US clones, then both income distributions are identical under the population principle.
- 40.
The second part of expression (4.44) may not seem immediate. Consider \({F}_{Y -{T}^{1}}^{Y }({p}_{k})\,\geq {F}_{Y -{T}^{2}}^{Y }({p}_{k})\). Then there corresponds a unique q k to each p k (recall that \({q}_{k} \geq {p}_{k}\)). Since the same set of taxpayers has higher relative aggregate net income at p k under T 1, it must also have higher relative aggregate net income at q k under T 1, as the distribution of gross incomes is the same.
- 41.
Note that for several reasons we exclude negative taxes for most of our analyses. We use this example here to compare equally distributed net incomes resulting from unequally distributed gross incomes.
- 42.
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Seidl, C., Pogorelskiy, K., Traub, S. (2013). Uniform Measures. In: Tax Progression in OECD Countries. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28317-8_4
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