Designing and Displaying the Income Tax Scale under Progressivity

The personal income tax is usually progressive and based on a marginal tax rate (MTR) that increases in steps [Johansen, 1965, p. 213]. Such a tax scale implies rapid hikes of the MTR and thus break points on the average tax rate (ATR) curve. In some tax systems, tax privileges in the form of the basic allowance or a reduced tax rate are phased out and thus not provided for a higher income. This leads to an increased MTR within the interval of the expiring tax allowance (the so-called “bubble”). The purpose of this article is to present different approaches regarding the design of a progressive tax function (Section 1). Against this background, the German tax scale is outlined as an example of a linear increasing MTR (Section 2). Moreover, the Polish tax scale is analysed by taking into consideration the introduced MTR bubbles (Section 3). The following Sections 4 and 5 deepen this issue by presenting the bubbles of the British and the US tax scale, as well as raising the problem of the joint taxation under such a design of the tax function. Pobrane z czasopisma Annales H Oeconomia http://oeconomia.annales.umcs.pl Data: 05/12/2018 13:48:36


Introduction
The personal income tax is usually progressive and based on a marginal tax rate (MTR) that increases in steps [Johansen, 1965, p. 213].Such a tax scale implies rapid hikes of the MTR and thus break points on the average tax rate (ATR) curve.In some tax systems, tax privileges in the form of the basic allowance or a reduced tax rate are phased out and thus not provided for a higher income.This leads to an increased MTR within the interval of the expiring tax allowance (the so-called "bubble").
The purpose of this article is to present different approaches regarding the design of a progressive tax function (Section 1).Against this background, the German tax scale is outlined as an example of a linear increasing MTR (Section 2).Moreover, the Polish tax scale is analysed by taking into consideration the introduced MTR bubbles (Section 3).The following Sections 4 and 5 deepen this issue by presenting the bubbles of the British and the US tax scale, as well as raising the problem of the joint taxation under such a design of the tax function.

Design of a progressive tax
A tax is deemed to be progressive if the tax amount increases overproportionately in relation to income: , whereby denotes income and the tax amount for an income of .The progressivity implies that the increasing ATR is lower than the MTR: . This is usually fulfilled, since in most countries the tax scale is based on a MTR that increases in steps for the consecutive income tax brackets [Seidl, Pogorelskiy, Traub, 2013, p. 8;Endres, Spengel, 2015, p. 76;IBFD, 2016b].
This so-called traditional income tax formula, leads to rapid hikes of the MTR and thus break points of the ATR curve at the upper thresholds of the tax brackets.This drawback has attracted the attention of economists and mathematicians for more than a century now [Voigt, 1912;Seidl, Topritzhofer, Grafendorfer, 1970] and is presented (see Figure 1) using the following baseline tax scale: is the basic allowance, after which the is applied to an income .The income exceeding the upper limit of is taxed at a higher .In order to smooth the ATR function, further tax brackets could be introduced.The reduction of the lowest MTR ( ) on an income at the same time would smooth the rapid tax increase for an income exceeding the basic allowance ( ).The tax scale would be extended to: The amount of the new threshold as well as the level of the additional MTR ( ) can be varied for the purpose of the revenue neutral amendment (see Figure 2).Nevertheless, the gradual increase of the MTR -tax bracket by tax bracketcauses breakpoints of the ATR function.These can be avoided by means of a linear function.Let us assume that the ATR should increase linearly within the tax bracket , whereby .In order to avoid breakpoints at the extreme values of the given interval, the ATR linear function should connect the ATR values of the baseline tax scale (see Figure 1).Based upon these two points of intersection, the following equation is calculated The derived tax function has the form and the adjusted tax scale is as follows (see Figure 3):  The above presented adjustments of the tax scale provide an instrument for the legislator when designing a progressive tax scale.

German personal income tax scale
The German Tax Reform Act of 1990 introduced the linear increasing marginal tax rate to eliminate the rapid hikes of the MTR [Deutscher Bundestag, 1988, p. 10, 149] that appear under the traditional income tax formula (see Figure 1).The complexity of this tax scale has led to some simplification proposals towards the traditional income tax formula [Kirchhof, 2003;FDP, 2009, p. 6;Rose, 2011, p. 323].However, rather than the tax scale, the provisions on the calculation of the taxable income contribute to the complexity of the tax system.Furthermore, by means of the formulas included in the German tax scale, a continuous increase of the MTR is (partially) achieved (see Figure 5).
Figure 5 presents the MTR, ATR as well as the amount of the personal income tax in Germany for 2016 and 2017.There are no significant differences between these two consecutive years.The amendment aims to tackle the bracket creep that appears when the nominal increase of the taxable income leads to the application of a higher tax rate, yet does not result in a higher real income (adjusted for inflation).Therefore, the basic allowance as well as the remaining thresholds has been increased (see Table 1, 2 nd column).Source: own diagram based on Table 1.

Polish personal income tax scale
In contrast to its German counterpart, the Polish tax scale does not contain a linear increasing MTR.Its design follows the most commonly applied scale with constant MTR that rapidly increases for the consecutive tax bracket(s).The tax scale does not seem to include any relevant changes between 2016 and 2017, although the tax credit amount (see Table 2, the grey marked fields) differs among four intervals and partially depends on the exact amount of the income.
In order to analyse the Polish tax scale, I developed Table 2 by deriving the "hidden" tax brackets (see Table 3).For the purpose of transparency, I propose this form of the tax scale in the tax act (the first three columns of Table 3).The developed tax scale reveals the significant increase of the basic allowance from PLN 3,091 to PLN 6,600.This is due to the judgement of the Polish Constitutional Court [Trybunał Konstytucyjny, 2015] according to which the lack of a mechanism to guarantee that at least the minimum subsistence level is exempt from taxation is unconstitutional.However, the increase in the basic allowance has not been accompanied by an adjustment of the whole tax scale and thus the threshold of PLN 85,528 for the highest MTR of 32% has remained unchanged.It is noteworthy that the developed tax scale reveals more MTR intervals for 2017 (see Table 3, last column and Figure 6).Moreover, the introduced MTR for the second and fourth tax bracket are higher than the MTR for the consecutive income interval, respectively (see Figure 6, MTR-2017).A progressive tax does not imply a non-decreasing MTR function.Due to the increasing ATR, the amended tax scale fulfils the condition of progressivity (see Figure 6, ATR-2017).Nevertheless, it appears strange that the low income right above the basic allowance is subject to the relatively high MTR of 32.36%.
This phenomenon is caused by the decreasing tax credit amount for a higher income.While under the 2016 legal status the basic allowance is provided for every income, the introduced variable tax credit reduces the benefit of the basic allowance that effectively is not granted for individuals with a taxable income over PLN 127,000.This leads to a higher tax burden in 2017 for incomes above the threshold of PLN 85,528 (see Figure 6, T-2016 vs. T-2017).3.
A closer look at the tax rates for a lower income (see Figure 7) clarifies the objective of the Polish legislator to only provide the tax relief for low earners.In order to achieve the same tax burden for a taxable income from PLN 11,000 before and after the amendment, the tax amount as well as the ATR rapidly increase within the income interval between the basic allowance (of PLN 6,600) and PLN 11,000 (see Figure 7, T-2017 andATR-2017).This leads to the aforementioned unusual high MTR of 32.36%, which may have a negative impact on the incentive to work.

"Bubbles" in the British and the US tax system
The up and down of the MTR (commonly known as a "bubble") is not a new concept of the Polish legislator; rather, the amendment of the tax scale in Poland is based on the British tax system, which essentially provides a personal allowance of £ 11,000.However, this allowance is phased out for an income over £ 100,000, declining by £ 1 for every £ 2 of income above this threshold and thus within the interval .At such a level, the income is subject to the "higher rate" of 40% and the MTR can be derived as follows: [HM Revenue & Customs, 2016].This MTR exceeds the consecutive MTR (on a higher Pobrane z czasopisma Annales H -Oeconomia http://oeconomia.annales.umcs.plData: 09/09/2019 10:26:57 U M C S income) and thus creates a bubble (see Figure 8).However, this bubble applies for a significantly higher income than its Polish counterpart.The phenomenon of tax bubbles also appears in the US tax law.The baseline corporate tax rate of 35% is exceeded by the MTR of 39% and 38% for two income intervals and , respectively [PWC, 2016, p. 47].These two bubbles aim to eliminate the benefit of the reduced tax rates applied for the lower income.Figure 9 illustrates the approximation of the ATR and MTR within the bubble intervals.To conclude, the bubbles as an income interval with a higher MTR than the MTRs applied for a lower and higher income around the bubble are used to eliminate tax benefits for a higher income.However, the presented first bubble in the Polish tax scale applies for a relatively low income and thus may negatively affect the willingness to work.

"Bubbles" and the joint taxation
The joint taxation provides tax benefits that result from the progressivity of the tax scale.Under this taxation form, the tax burden for married couples is usually computed as the double tax amount calculated for half of the aggregate income of both spouses.Thereby, the tax benefit can be calculated by means of the formula , where denote the income of one spouse and the other, respectively.
is the tax amount for a taxable income of .Assuming that , the tax advantage is equal to the difference of the tax reduction regarding the higher income spouse and the tax increase for the other spouse .The joint taxation (or income splitting) can be interpreted as a transfer of the taxable from one spouse to the other (unit by unit) until the incomes are equalised .With this in mind, the tax benefit can be calculated by means of the MTR as follows: The joint taxation is advantageous if the MTR constitutes a non-decreasing function (see Figure 10), which implies a progressive tax scale.However, a progressive tax scale -in other words, an increasing ATR -does not imply a non-decreasing MTR.A bubble in the MTR function eliminates its non-decreasing character but not the tax progressivity, as long as . Interestingly, such a bubble breaks the principle that the tax burden under income splitting cannot be higher compared to separate taxation [Endres, Spengel, 2015, p. 84]. Figure 11 shows that the joint taxation in case of a bubble in the MTR function could lead to a higher tax burden.This phenomenon can be demonstrated by means of the Polish tax scale.Table 4 presents the tax advantage from the income splitting for a chosen case of a marriage couple.Due to the introduced MTR bubbles, the tax benefit of PLN 74 under the 2016 legal status turns into the additional tax burden of PLN 482 for 2017 and thus the joint taxation is disadvantageous.

Figure 1 .
Figure 1.Average and marginal tax rate -baseline tax scale Source: own diagram based on the tax scale .

Figure 2 .
Figure 2. Average and marginal tax rates -baseline and extended tax scale Source: own diagram based on the tax scales .

Figure 3 .
Figure 3. Average and marginal tax rates -baseline and adjusted for ATR tax scale Source: own diagram based on the tax scales

Figure 4 .
Figure 4. Average and marginal tax rates -baseline and adjusted for MTR tax scale Source: own diagram based on the tax scales T(x) and T(x)*.

Figure 5 .
Figure 5. Average tax rate (ATR), marginal tax rate (MTR) and tax amount (T) in Germany for 2016 and 2017

Figure 6 .
Figure 6.Average tax rate (ATR), marginal tax rate (MTR) and tax amount (T) in Poland for 2016 and 2017 Source: own diagram based on Table3.

Figure 7 .
Figure 7. Average tax rate (ATR), marginal tax rate (MTR) and tax amount (T) on low income in Poland for 2016 and 2017 Source: own diagram based on Table3.

Figure 9 .
Figure 9. Average tax rate (ATR), marginal tax rate (MTR) and tax amount (T) in the USA for the 2016 tax year Source: own diagram based on information provided by [PWC, 2016, p. 47].

Figure 10 .
Figure 10.Tax benefit (TB) from the joint taxation under a non-decreasing MTR Source: own diagram.

Figure 11 .
Figure 11.Tax benefit (TB) from the joint taxation in case of MTR bubble Source: own diagram.

Table 2 .
Table 2 presents the Polish tax scale for 2016 and 2017.Polish personal income tax scale Source: translated table of Art.27 updof (Polish Personal Income Tax Act) in the version of 2016 and 2017.

Table 3 .
Developed Polish personal income tax scale

Table 4 .
Tax benefit (TB) from the joint taxation in Poland Product of the remaining part of the tax bracket with a lower MTR (85,528 -85,000) and the MTR difference:bThe tax benefit explained under a reduced by the product of the income within the bubble and the MTR difference:Source: own calculations based on information provided in Table2 and 3.