Marginal tax rate

Definition

The marginal tax rate of a given taxable entity at a given income level (per unit time) is a dimensionless quantity that measures the fraction of a slight increment of income that goes into additional taxes.

Differential definition

If $I$ denotes the income in a given time unit (say a year) and $T(I)$ is the tax liability as a function of income, then the marginal tax rate is defined as the derivative of $T(I)$ with respect to $I$, i.e., it is given as:

$\operatorname {Marginal} \ \operatorname {Tax} \ \operatorname {Rate} =T'(I)={\frac {dT}{dI}}$

If the tax liability is considered to depend on many factors other than income, we use the partial derivative symbol $\partial T/\partial I$ instead.

Interval definition

This definition measures the marginal tax rate between two different but close levels of income, and is defined as the difference quotient of tax liability with respect to income. If the tax liability at income level $I_{1}$ is $T_{1}$, and the tax liability at income level $I_{2}$ is $T_{2}$, then the marginal tax rate is defined as:

$\operatorname {Marginal} \ \operatorname {Tax} \ \operatorname {Rate} ={\frac {T_{2}-T_{1}}{I_{2}-I_{1}}}={\frac {\Delta T}{\Delta I}}$

Subtleties about tax slabs

• Jumps for simplicity of tax calculation: For simplicity of tax calculation, tax laws often have small slabs where the tax amount is kept constant, rather than the tax liability being a continuous function of income. For instance, in the United States, the tax liability increases in jumps at intervals of $50 for taxable income. Thus, the differential definition does not work in a strict mathematical sense, but the interval definition works well if we choose an interval that is a multiple of $50.

Variation of marginal tax rate

• The marginal tax rate is not a constant number but depends on the income level in general. The nature of variation is described below:

Type of tax	How does the marginal tax rate vary with income?	What can we conclude about average tax rates?
progressive tax	marginal tax rate is increasing, or non-decreasing, with respect to income.	average tax rates are increasing, and for any income level, the average tax rate is less than or equal to the marginal tax rate.
flat tax	marginal tax rate is constant with respect to income (the term flat tax is often also used for a flat tax with a base exemption, in which case the marginal tax rate is zero up to a point and then jumps to a nonzero value, at which it stays constant).	In the absence of a base exemption, the average tax rate is constant and equals the marginal tax rate. In the presence of a base exemption, the average tax rate is increasing and asymptotically approaches the marginal tax rate.
regressive tax	marginal tax rate is decreasing, or non-increasing, with respect to income.	The average tax rate is decreasing, and for any income level, it is greater than or equal to the marginal tax rate.

In most cases, even the case of flat and regressive taxes, the total tax liability is an increasing (or, at worst, non-decreasing) function of income, so the marginal tax rates are still positive (or, at worst, zero). The terms progressive and regressive refer to the sign of the second derivative of tax liability with respect to income, not the sign of the first derivative.

Note on abrupt changes in tax rates: The marginal tax rate itself may vary continuously or in discrete jumps. For instance, if there is a basic income exemption, then the marginal tax rate up to that basic income is zero, and after that, it suddenly jumps to a nonzero value.