Solving progressive tax calculation for pre-tax income

Question

Progressive Tax Rate Explanation

Progressive taxation works by taxing income within a certain bracket at different rates. For example:

Bracket #	% tax rate within bracket	Min amount (exclusive)	Max amount (inclusive)
1	10%	0	50,000
2	20%	50,000	60,000
3	25%	60,000	n/a (>60,000)

In the example above, the taxes on 100,000 in income would be 17,000 (5,000 from bracket #1, 2,000 from bracket #2, and 10,000 from bracket #3).

I'm currently computing this as the dot product of two vectors (with a "zero" bracket of 0%/0 max): $$ ([\textrm{bracket 1 rate}, \textrm{bracket 2 rate}, \textrm{bracket 3 rate}]-[\textrm{bracket 0 rate}, \textrm{bracket 1 rate}, \textrm{bracket 2 rate}]) \cdot [max(0, \textrm{income}-\textrm{bracket 0 max}), max(0, \textrm{income}-\textrm{bracket 1 max}), max(0, \textrm{income}-\textrm{bracket 2 max})] $$

So, in the example above we do: $$ \begin{align} ([0.1, 0.2, 0.25]-[0, 0.1, 0.2]) \cdot [max(0, 100000-0), max(0, 100000-50000), max(0, 100000-60000)] \\ ([0.1, 0.1, 0.05]) \cdot [100000, 50000, 40000] \\ 100000*0.1+50000*0.1+40000*0.05 \\ 17000 \end{align} $$

Question

Using the progressive taxation formula above, we can get "take home pay" (income less taxes) by subtracting taxes from the income: $$ \begin{align} \textrm{take home pay} & = \textrm{income} - \left([\textrm{bracket 1 rate}, \textrm{bracket 2 rate}, \textrm{bracket 3 rate}]-[\textrm{bracket 0 rate}, \textrm{bracket 1 rate}, \textrm{bracket 2 rate}]) \cdot [max(0, \textrm{income}-\textrm{bracket 0 max}), max(0, \textrm{income}-\textrm{bracket 1 max}), max(0, \textrm{income}-\textrm{bracket 2 max})]\right) \end{align} $$

This means that in the example above, the post-tax "take home pay" is 83,000 (100,000 - 17,000).

I'm trying to solve this equation for income, so that I can plug in a given take home pay, and get the amount of pre-tax income I need to get that take home pay.

Any clue how to solve for income, or even approximate income necessary for a given take home pay?

Answer 1

Let $I$ be the total income and $P$ be take-home pay. Then the taxes paid is $T = I - P$, where $$T = \begin{cases} 0.1 I, & I \in [0, 50000] \\ 5000 + 0.2(I - 50000), & I \in (50000, 60000] \\ 7000 + 0.25(I - 60000), & I \in (60000, \infty). \end{cases}$$ Hence $$P = \begin{cases} 0.9 I, & I \in [0, 50000] \\ 0.8 I + 5000, & I \in (50000, 60000] \\ 0.75 I + 8000, & I \in (60000, \infty). \end{cases}$$ Now all that remains is to invert this piecewise function. We do this by solving each piece for $I$ and expressing the interval for each piece in terms of $P$. So for instance, $$P = 0.9I, \quad I \in [0, 50000]$$ implies that $$I = \frac{10}{9}P, \quad P \in [0, 45000].$$ The other cases are handled similarly; we obtain $$I = \begin{cases} \frac{10}{9}P, & P \in [0, 45000] \\ \frac{5}{4}(P - 5000), & P \in (45000, 53000] \\ \frac{4}{3}(P - 8000), & P \in (53000, \infty). \end{cases}$$ This furnishes the total income $I$ needed to earn a take-home pay of $P$.

Answer 2

You can easily see that the function you want should be case-defined. Let's split into three cases. We let $P$ to be the take-home-payment and $I$ be the income.

If $I\le\$50000$, then certainly it only needs to pay tax $1$, so you have $P=0.9I$.

If $I\in (50000,60000]$, then $P=(I-50000)\times0.8+50000\times0.9=0.8I+5000$.

For $I>60000$, $P=(I-60000)\times0.75+(50000\times0.9+10000\times0.8)=0.75I+8000$.

In sum, the function you wants should be $$P(I)=\begin{cases} 0.9I&\text{for } I\le 50000\\0.8I+5000&\text{for } 50000<I\le60000\\0.75I+8000&\text{for } I>60000\end{cases}$$

Edit. Sorry for misunderstanding the question. But the idea is roughly the same. Since the above is an injective function, you can just take the inverse back.

For example, in $0<I\le 50000,$ $P=0.9I$, so $I=\dfrac{10}{9}P$ for $0<P\le0.9\times50000=45000$.

With similar manner, we can show that $$I(P)=\begin{cases} \dfrac{10P}{9}&\text{for } P\le 45000\\\dfrac{5P}{4}-6250&\text{for } 45000<P\le53000\\\dfrac{4P}{3}-\dfrac{32000}{3}&\text{for } P>53000\end{cases}$$

Answer 3

Start by assuming that the pre-tax income is in the lowest bracket, in which case you need take-home/0.9 Check whether the pre-tax is under 50000. If yes, you are done. If no, the first 50000 of pre-tax generates 45000 of take-home. Subtract 45000 from the required take-home and assume you are in the second bracket, so the required pre-tax is (take-home - 45000)/0.8 + 50000 Check whether the pre-tax is under 60000. If so you are done. Otherwise go into the third bracket the same way.