The misunderstanding goes like this: “If my salary jumps to $42,000 I might jump a tax bracket and take home less money.” In short: no you won’t, not ever. The tax rate listed beside your tax bracket is a marginal tax rate, as opposed to how much tax you actually pay, which is the effective tax rate.
In a few words, all the money you make before your tax bracket is taxed at those lower bracket rates. The misunderstanding is an easy one to make, and anti-tax politicians are happy to let people misunderstand. In one poll, respondents were asked to choose between a correct and incorrect explanation of marginal tax rates. 62% of Republicans and 37% of Democrats failed to answer correctly.
I’m not going to write a better explainer than one of the many already out there, but I’ve wondered for a while what the rates actually look like when laid out in one continuous graph (emphasis on the continuous!). So I pulled out the ol’ plot software today and hacked it together, using the 2019 tax brackets:
The math behind it:
For each tax bracket, we consider the percentage $p$ of total income $I$ that is taxable under that bracket:
\[\begin{align*} b_l &= \text{ lower bound of bracket } \\ b_u &= \text{ upper bound of bracket } \\\\ p(I) &= \begin{cases} \frac{b_u-b_l}{I} & \text{if } b_l < I \le b_u \text{,}\\\\ \frac{I-b_l}{I} & \text{if } b_u < I \text{,}\\\\ 0 & \text{otherwise} \end{cases} \end{align*}\]Use that percentage to scale your income from each bracket, factoring in the tax rate $r_n$, and sum them all up.
\[\begin{align*} E(I)=\sum_{n=1}^{7} I \cdot r_n \cdot p(I) \end{align*}\]Example: if you make $50,000, you need to consider three tax brackets:
That calculation goes like so:
\[\begin{align*} E(I) =&\ 10\% \cdot \frac{$9,951 - $0}{$50,000} + \\ &\ 12\% \cdot \frac{$40,526 - $9,951}{$50,000} + \\ &\ 22\% \cdot \frac{$50,000 - $40,526}{$50,000} \\\\ =&\ (10\% \cdot 19.9\%) + (12\% \cdot 61.2\%) + (22\% \cdot 18.9\%) \\\\ =&\ 13.5\% \end{align*}\]One motivation for wanting to make these plots is that I wanted to play around with an idea I’m calling the Sigmoid Tax. A sigmoid is an s-shaped function, in this case based on the formula:
\[\begin{align*} \frac{1}{1+10^{-x}} \end{align*}\]The idea would be to replace that gnarly tax bracket system with something smooth and simple. I added some parameters so I could toy around with sliders and see how the plot compares to the bracket system. Surely I’m not the first person to consider such a thing, but I thought it would be interesting to explore. Enjoy!
\[\begin{align*} S(I) = (r-h)+\frac{h}{1+10^{\frac{-2(I-m)}{w}}} \end{align*}\] ]]>