User talk:Fyre4ce/Marginal tax rate

Variable Marginal Rates with Section 199A Deduction
Define variables:

$$ \begin{align} T(N, Q) & = \text{tax liability as a function of income} \\ N & = \text{non-Qualified Business Income} \\ Q & = \text{Qualified Business Income} \\ TB & = \text{tax bracket (eg. 24 percent)} \\ D & = \text{below-the-line deduction (eg. standard deduction)} \\ UL & = \text{upper limit of deduction phase-out (eg. 415,000 for MFJ 2018)} \\ R & = \text{phase-out range (eg. 100,000 for MFJ 2018)} \\ \end{align} $$

This analysis will assume a single tax bracket, although because the equation will be differentiated, the results will apply to any tax bracket.

The total tax liability can be written as follows. If total taxable income is below the beginning of the deduction phase-out ($157,500 for single or $315,000 for MFJ, for 2018), total tax is:

$$T(N, Q) = (N+Q-D) \cdot TB - 20% \cdot Q \cdot TB$$

In the phase-out range ($157,500-$207,500 for single or $315,000-$415,000 for MFJ), total tax is:

$$T(N, Q) = (N+Q-D) \cdot TB - 20% \cdot Q \cdot \left ( \frac{UL - (Q + N - D)}{R} \right ) \cdot TB$$

Above the phase-out range ($207,500 for single or $415,000 for MFJ), total tax is:

$$T(N, Q) = (N+Q-D) \cdot TB $$

Above the phase-out, the Section 199A deduction has no effect. Below the phase-out, the marginal tax rates with respect to QBI and non-QBI are found by taking the partial derivative of T with respect to N and Q:

$$\frac{\partial T(N, Q)}{\partial N} = TB$$

$$\frac{\partial T(N, Q)}{\partial Q} = (1 - 20%) \cdot TB = 80% \cdot TB$$

In the phase-out range, the marginal tax rates with respect to QBI and non-QBI are found by taking the partial derivative of T with respect to N and Q:

$$\frac{\partial T(N, Q)}{\partial N} = TB - 20% \cdot Q \cdot \left ( \frac{-1}{R} \right ) \cdot TB$$

$$\frac{\partial T(N, Q)}{\partial N} = TB \cdot \left ( 1 + \frac{20% \cdot Q}{R} \right )$$

$$\frac{\partial T(N, Q)}{\partial Q} = TB - \frac{\partial}{\partial Q} \left ( \frac{20% \cdot TB}{R} \cdot (UL \cdot Q - Q^2 - N \cdot Q + D \cdot Q)\right )$$

$$\frac{\partial T(N, Q)}{\partial Q} = TB - \frac{20% \cdot TB}{R} \cdot (UL - 2Q - N + D)$$

$$\frac{\partial T(N, Q)}{\partial Q} = TB \cdot \left ( 1 + \frac{20%}{R} \cdot (2Q + N - UL - D) \right )$$

The second partial derivatives are:

$$\frac{\partial^2 T(N, Q)}{\partial N^2} = 0$$

$$\frac{\partial^2 T(N, Q)}{\partial Q^2} = \frac{2 \cdot TB \cdot 20%}{R}$$

$$\frac{\partial^2 T(N, Q)}{\partial Q \partial N} = \frac{TB \cdot 20%}{R}$$

Example
A MFJ couple has $100,000 of non-QBI income and also earns QBI. They take the standard deduction of $24,000. Below $315,000 of taxable income they are in the 24% bracket. Their marginal tax rates for non-QBI and QBI income are:

$$\frac{\partial T(N, Q)}{\partial N} = 24%$$

$$\frac{\partial T(N, Q)}{\partial Q} = 80% \cdot 24% = 19.2%$$

The phase-out begins when their taxable income, after the standard deduction, equals $315,000. This corresponds to a QBI income of $239,000 ($315,000 + $24,000 - $100,000). Note that although their taxable income is at the 24%/32% threshold, the Section 199A deduction pulls them well down into the 24% bracket. At this income, their marginal tax rates are:

$$\frac{\partial T(N, Q)}{\partial N} = 24% \cdot \left ( 1 + \frac{20% \cdot $239,000}{$100,000} \right ) = 35.472%$$

$$\frac{\partial T(N, Q)}{\partial Q} = 24% \cdot \left ( 1 + \frac{20%}{$100,000} \cdot (2 \cdot $239,000 + $100,000 - $415,000 - $24,000) \right ) = 24% \cdot \left ( 1 + \frac{20%}{$100,000} \cdot $139,000 \right ) = 30.672%$$

At some point in the phase-out range, the couple will cross into the 32% bracket, and then into the 35% bracket. At the top of the phase-out, when QBI income is $339,000 ($415,000 + $24,000 - $100,000), marginal tax rates will be:

$$\frac{\partial T(N, Q)}{\partial N} = 35% \cdot \left ( 1 + \frac{20% \cdot $339,000}{$100,000} \right ) = 58.73%$$

$$\frac{\partial T(N, Q)}{\partial Q} = 35% \cdot \left ( 1 + \frac{20%}{$100,000} \cdot (2 \cdot $339,000 + $100,000 - $415,000 - $24,000) \right ) = 35% \cdot \left ( 1 + \frac{20%}{$100,000} \cdot $339,000 \right ) = 58.73%$$

Above $339,000 QBI, the Section 199A deduction is completely eliminated and the marginal rates become:

$$\frac{\partial T(N, Q)}{\partial N} = \frac{\partial T(N, Q)}{\partial Q} = 35%$$

--Fyre4ce 05:55, 30 May 2019 (UTC)

Maximum possible rates
By inspection of the above formulas, the maximum possible rates occur at the very top of the phase-out range, when:

$$N + Q - D = UL$$

The maximum also occurs when Q is largest, so:

$$N = 0$$

and

$$Q = UL + D$$

Substituting these values into the above equations for the marginal rates give:

$$\frac{\partial T(N, Q)}{\partial N} = TB \cdot \left ( 1 + \frac{20% \cdot (UL + D)}{R} \right )$$

and

$$\frac{\partial T(N, Q)}{\partial Q} = TB \cdot \left ( 1 + \frac{20%}{R} \cdot (2 \cdot (UL+D) + 0 - UL - D) \right ) = TB \cdot \left ( 1 + \frac{20%}{R} \cdot (UL + D) \right ) $$

Note that the two formulas are the same.

For single filers for 2019, UL = $210,700 and R = $50,000, and we will assume the standard deduction D = $12,200. $210,700 taxable income is barely into the 35% bracket, which begins at $204,100 taxable income, so TB = 35%.

$$\left (\frac{\partial T(N, Q)}{\partial N} \right )_{max} = \left (\frac{\partial T(N, Q)}{\partial Q} \right )_{max} = 35% \cdot \left ( 1 + \frac{20% \cdot ($210,700 + $12,200)}{$50,000} \right ) = 66.206%$$

The QBI required to achieve this rate is:

$$Q = $210,700 + $12,200 = $222,900$$

For married joint filers for 2019, UL = $421,400 and R = $100,000, and we will assume the standard deduction D = $24,400. $421,400 taxable income is barely into the 35% bracket, which begins at $408,200 taxable income, so TB = 35%.

$$\left (\frac{\partial T(N, Q)}{\partial N} \right )_{max} = \left (\frac{\partial T(N, Q)}{\partial Q} \right )_{max} = 35% \cdot \left ( 1 + \frac{20% \cdot ($421,400 + $24,400)}{$100,000} \right ) = 66.206%$$

The QBI required to achieve this rate is:

$$Q = $421,400 + $24,400 = $445,800$$

Note that the single and married joint maximum rates are the same. If itemized deductions are larger than the standard deduction, the maximum rate will be slightly higher. --Fyre4ce 23:20, 12 June 2019 (UTC)