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)