# Marginal tax rate

Marginal tax rate function (general example)

The marginal rate ( marginal tax rate) is the rate at which the respective next unit of the tax base is charged. It indicates which part of an additional taxable euro (or other currency unit) has to be paid as tax.

The marginal tax rate is of practical importance in connection with progressive income tax rates as used worldwide (e.g. in Germany, Austria and Switzerland). The marginal tax rate is dependent on the amount of taxable income (assessment basis). It is zero up to the amount of the basic tax allowance, after which it increases from the initial tax rate to the top tax rate . Looking at all parts together, the result is an average tax rate that is always lower than the marginal tax rate.

The marginal tax rate is (only) relevant to the question of the extent to which it is worthwhile to increase the taxable income (zvE) or to reduce it (through business expenses , income-related expenses or other allowances ). Since the marginal tax rate only relates to the additional income, but not to the total income, an increase in income does not necessarily lead to a lower residual income. The latter would only be the case if the marginal tax rate exceeded 100 percent.

## General

### Introductory example

The taxable annual income (hereinafter: zvE) of a taxpayer is € 60,000. According to the general tariff example in the picture above right, € 11,250 income tax has to be paid. That is 18.75% of the zvE (= average tax rate). The marginal tax rate is calculated by looking at income growth .

If the zvE increases by € 3,000 to a total of € 63,000, a total of € 12,322.50 income tax must be paid. On this increase in income of € 3,000 there is an income tax of € 1,072.50, or 35.75%. That would be the “differential tax rate”. If you now consider very small changes in income taking into account the tax amount function , you get the marginal tax rate. This is so called because it applies exactly at the limit of the next additional euro earned. In the example, the marginal tax rate is initially 35%, but 36.5% after the salary increase.

It can be seen that the marginal tax rates differ from the differential tax rate, which, however, can be neglected in the case of low income growth.

### definition

Mathematically, the marginal tax rate is the differential quotient (the derivative) of the tax amount function . If the tax amount function is differentiable for a certain value of the tax base , its derivation is the marginal tax rate . ${\ displaystyle S (B)}$ ${\ displaystyle B}$${\ displaystyle s_ {g}}$

With = marginal tax rate and = assessment base (taxable income) the following relationship applies: ${\ displaystyle s_ {g}}$${\ displaystyle B}$

${\ displaystyle s_ {g} (B) = S '(B) = {\ frac {{\ mbox {d}} S (B)} {{\ mbox {d}} B}}}$

This is also the slope of the tax amount function for the assessed value of the tax base.

The initial and top tax rates are also marginal tax rates. The starting rate refers to the income part immediately above the basic allowance. The top tax rate applies to the part of the income above the top collectively agreed basic income value and remains the same from then on.

### Rounding rule

In addition to the pure tax amount function, there is often a rounding rule according to which the ZVE and the tax amount are rounded to specific amounts. This leads to a deviation from the basic tariff schedule. In the case of very coarse tax tables, this leads to distorting effects that can lead to the problem of the sequence being reversed. With modern formula tariffs with rounding to a full euro, this effect can practically be neglected.

## Germany

Development of the marginal tax rate (dashed lines) in Germany in 2018

The development of the marginal tax rate in Germany depending on the taxable income can be seen from the graphic opposite for the year 2018 (dashed lines). With a taxable income of € 48,000, the marginal tax rate is 38.94%. This is the marginal tax rate for individual assessments. For a jointly assessed couple with the same joint income, the marginal tax rate is only 28.34%.

The rounding rule that applied until 2003, according to which taxable income was rounded down to a full 36 euros, resulted in small income levels that noticeably distorted the linearity. The tax amount has risen much more continuously since 2004 because it has been rounded down to a full euro since then.

The income tax rate in Germany is basically a linear-progressive rate , which, however, is divided into two sections. This creates the two creases that can be seen in the picture. The additional tariff level at around 250,000 euros with the top tax rate of 45% (not visible in the picture) also deviates from the linear course. This trend is often criticized and a shift in the benchmark at the end of the linear progression zone (42% at around 55,000 euros) is called for. How this shift should look, for example with or without an increase in the marginal tax rate, is controversial.

For the historical development of the marginal tax rate in Germany since 1934 see the tariff history of income tax in Germany .

## Austria

Development of the marginal tax rate (thin lines in the lower diagram) in Austria

The income tax in Austria is for a stage marginal rate tariff determined. The course of the marginal tax rate depending on the taxable income (zvE) is shown in the adjacent graphic for the year 2016 (lower diagram, thin red line).

The marginal tax rates are:

• 00% for income components from 00.000 to 11,000 euros annually
• 25% for income components from 11,001 to 18,000 euros annually
• 35% for income components from 18,001 to 31,000 euros annually
• 42% for income components from 31,001 to 60,000 euros annually
• 48% for income components from 60,001 to 90,000 euros annually
• 50% for income components from 90,001 to 1,000,000 euros annually
• 55% for income parts over 1,000,000 euros annually

This tariff was introduced after the 2015/2016 tax reform.

## Switzerland

Trend in marginal tax rates (dashed line) in Switzerland in 2012

In Switzerland, income tax is levied at the federal, cantonal and communal levels at the same time. The diagram opposite shows the development of the marginal tax rates using the example of Zurich from 2012 onwards. The marginal tax rates are shown for each level and for the total as a dashed line.