A note on the maximum value of the Kakwani index

The Kakwani index computes the departure from proportionality of a progressive income tax by measuring the difference between the concentration coefficient for tax liabilities and the Gini coefficient for pre-tax incomes. In case of maximum progression, that is a situation in which only one taxpayer faces the overall tax burden, the index reaches its theoretical maximum value, given by 1 minus the Gini coefficient for pre-tax incomes. We argue that this phenomenon can happen in one special case that is not satisfied in real-world personal income taxes. As a matter of fact, the overall tax revenue of a real-world personal income tax cannot be eventually paid only by the richest taxpayer. Therefore, the maximum concentration coefficient for taxes cannot be equal to 1, and, consequently, the maximum value of the Kakwani index cannot be 1 minus the Gini coefficient for pre-tax incomes, as generally described in the related literature. According to different hypotheses, we give evidence of this phenomenon by employing the Italian personal income tax.


Introduction
In their seminal papers, Jakobsson (1976), Fellman (1976), Kakwani (1977) and Reynolds and Smolensky (1977) show how the degree of progression and the redistributive effect of a tax can be quantified. In particular, Kakwani (1977) proposes his famous index able to compute the departure from proportionality of a progressive income tax. This index measures the difference between the concentration coefficient for the tax liability distribution and the Gini coefficient for the pre-tax income one.
All the related tax literature (e.g. Lambert 2001) states that its maximum value is one minus the Gini coefficient for the pre-tax income distribution, and its minimum value − 1 minus the same Gini coefficient. We argue that these phenomena can happen in one special case that is not satisfied in real-world personal income taxes. As a consequence, the maximum (minimum) value of the Kakwani index is lower (greater) than its theoretical one. Focusing on the maximum value, we give evidence of its magnitude by two examples regarding the Italian personal income taxation.
The structure of the paper is as follows. Section 2 presents the basic inequality indexes. Section 3 focuses on the highest admittable value of the Kakwani index. Section 4 briefly introduces the data and the microsimulation model employed in this work (Sect. 4.1), and subsequently reports the results (Sect. 4.2). Section 5 concludes.

Basic notation
A population of N income earners, with i = 1, . . . , N , is considered. We denote by X = (x 1 , . . . , x N ) the gross income distribution ordered in non-decreasing order. Similarly, we call T = (t 1 , . . . , t N ) the tax liability distribution and Z = (z 1 , . . . , z N ) the post-tax income one. To evaluate the inequality within these distributions, we employ the Gini (1914) coefficient G = 2μ −1 cov ( , F( )) and the corresponding concentration one C |η = 2μ −1 cov ( , F(η)), where , η = (X , T , Z ), C η|η = G η = G = C | , μ is the average value of the considered distribution, cov represents the covariance, and F( ) is the cumulative distribution function (Kakwani 1980;Jenkins 1988 (Atkinson 1980;Plotnick 1981;Kakwani 1984). Similarly, the degree of tax progressivity can be computed by the Kakwani bounds are possible under the condition that the highest admittable value for C MAX T |X is 1 and the corresponding minimum value C MIN T |X is equal to −1. It has to be noted that the above-mentioned extreme values for K can be verified in one special case: the overall tax revenue (hereafter Υ ) is lower than the top (bottom) gross income x N (x 1 ) observed in the income distribution. 1 This is not what researchers observe in real-world taxation, since in general Υ > x N . As a consequence, the highest value of the tax liability concentration C MAX T |X is necessarily lower than N −1 N (1 for large samples) and the corresponding lowest value C MIN T |X is greater than 1−N N (−1 for large samples). In turn this implies that K MAX (K MIN ) depends on the distribution of X and the overall amount of the tax revenue to be collected Υ . In particular, ceteris paribus, K MAX (K MIN ) increases (decreases) as the overall amount of the tax revenue Υ decreases.

When re-ranking of post-tax incomes occurs
The empirical occurrence of the maximum theoretical value of C MAX T |X is even more unlikely if the condition of marginal tax rates not exceeding 100% holds. 2 Suppose initially that only the richest taxpayer has to face a positive tax liability. Until the tax revenue Υ is lower than or at most equal two or more taxpayers are needed for Υ to be paid, so that K MAX < N −1 N −G X monotonically decreases with Υ (also R E MAX decreases with Υ , whilst RS MAX increases): in these cases an unwanted and disproportionate re-ranking of post-tax income values is likely to occur, leading to situations in which R E MAX < 0 even if RS MAX > 0. Finally, K MAX becomes zero when Υ = N i=1 x i ; in this extreme case also R R MAX would be zero, whilst R E MAX = RS MAX would be equal to G X .

When re-ranking of post-tax incomes is avoided
Whilst focusing on the highest value for the Kakwani index, in order to avoid the unpleasant outcomes underlined in Sect. 3.1, a different strategy has to be employed: the tax structure associated to K MAX needs to avoid post-tax incomes re-ranking.
Following the methodology described in Mantovani (2017), this distribution can be obtained by imposing a deduction D, equal for all taxpayers, to their pre-tax income x i whilst applying a 100% statutory marginal tax rate and avoiding the negative income tax (Keen et al. 2000).
Given a pre-tax income distribution (and its G X value) and a specific amount for Υ , R E MAX can be obtained through the minimum value for G MIN Z . Focusing on a tax, and avoiding the negative income taxation, z i ≤ x 1 for all incomes. G MIN Z can then be obtained by levelling top pre-tax incomes in order to obtain Z = ( x 1 , x 2 , . . . , x n , D, D, . . . , D). This is possible by applying T = (0, 0, . . . , 0, . . , x n ) and Z sup = (D, D, . . . , D).
For a given Υ and excluding negative taxes, three situations are possible: (a) were a redistributive transfer from Z inf to Z sup considered, G Z would increase and R E decrease; (b) either a redistributive transfer within Z inf or an egalitarian redistributive transfer from Z sup towards Z inf is not possible to occur, because some pre-tax incomes within Z inf would increase; (c) a redistributive transfer taking place within Z sup would determine an increase of G Z . As a consequence, the maximum value of the Kakwani index compatible with the maximisation of R E MAX is obtainable by considering

The data and the microsimulation model
We make use of a static microsimulation model concerning the Italian personal income tax (Pellegrino 2007)  According to the microsimulation model, the 2014 overall amount of pre-tax incomes N i=1 x i is 807.85 billion euros, whilst the overall tax revenue N i=1 t i is 151.67 billion euros. As a consequence, θ = 0.18774. The Gini coefficient for the gross income distribution G X is 0.45253, whilst that for the net income distribution G Z is 0.40248, and the one for the tax liability distribution is G T = 0.68484. The overall redistributive effect R E is then 0.05005. The concentration coefficient for the net income distribution C Z |X is 0.40160, whilst the one on the net tax liability distribution C T |X is 0.67289; therefore, the Reynolds-Smolensky RS index is equal to 0.05093 and the Kakwani index K is 0.22035. Finally, the Atkinson-Plotnick-Kakwani index R R is equal to 0.00088 (Table 1, column Present tax).

Results
Having ranked pre-tax values in non-decreasing order and considered sample weights, the top 1.46 million taxpayers (3.7% of all) earn a pre-tax income equal to N i=1 t i = Υ . Supposing all these taxpayers face a tax liability equal to their pre-tax income, and the remaining ones a zero tax liability (Table 1, column Maximum with re-ranking), G MAX T = C MAX T |X = 0.97379 and K MAX = 0.52116. Note that the empirical maximum value of the Kakwani index is lower than its theoretical one (1 − G X = 0.54747). 3 R E MAX would be remarkably lower than the one observed according to the present tax structure (0.04734) since this hypothetical tax liability distribution would generate a huge re-ranking of post-tax incomes (R R MAX = 0.07312, 83 times greater than the one observed today). As a consequence, RS MAX would be 2.36 times the one registered according to the present tax structure (0.12046) even if R E MAX is lower.
From the empirical point of view, it can be interesting to determine an hypothetical tax liability distribution able both to guarantee the total tax revenue observed according to the present tax structure and no re-ranking of post-tax incomes (see Sect. 3.2).
In order to obtain a total tax revenue equal to 151.67 billion euros, the value of D in the 2014 Italian case should be 29,763.83 euros (Table 1, column Maximum without re-ranking). This tax liability distribution would be able to maximise R E MAX , whilst guaranteeing no re-ranking of both tax liability and post-tax income distributions. In particular, G MAX

Conclusions
In this paper, we stress that, even if desired, the overall tax revenue of a personal income tax cannot be concentrated only on the richest income earner, simply because the overall tax revenue is remarkably greater than the top gross income observed in real-world income distributions.
From this simple observation follows that the maximum concentration coefficient for taxes cannot be 1, and, consequently, the maximum value of the Kakwani index cannot be equal to 1 minus the Gini coefficient for pre-tax incomes as generally described in the related literature. We give evidence of this phenomenon by illustrating empirical examples when considering a real-world tax.