# Mathematical idea behind tax bracket

This may be a lame question. The tax bracket is a way to calculate the tax based on the taxable income. For example

Imagine that there are three tax brackets: 10%, 20%, and 30%. The 10% rate applies to income from $\$$1 to \$$10,000; the 20% rate applies to income from$\$$10,001 to \$$20,000; and the 30% rate applies to all income above $\$$20,000. Under this system, someone earning \$$10,000 would be taxed at a rate of 10%, paying a total of$\$$1,000. Someone earning \$$5,000 would pay $\$$500, and so on. Meanwhile, someone earning \$$35,000 would face a more complicated calculation. The rate on the first$\$$10,000 would be 10%; the rate from \$$10,001 to $\$$20,000 would be 20%; and the rate above that would be 30%. Thus, he would pay \$$1,000 for the first$\$$10,000 of income; \$$2,000 for the second $\$$10,000 of income; and \$$4,500 for the last$\$$15,000 of income; in total, he would pay \$$7,500, or about 21.4%.

In my understanding, the idea of tax bracket for computing tax is as follows:

1. first fill the taxable income into the tax brackets in the order of tax rates from low to high,

2. then multiply the amount in each tax bracket, by the corresponding tax rate,

3. finally sum up the the products over the tax brackets to obtain the tax.

I was wondering if such an idea is also used (elsewhere) in mathematics, including statistics, mathematical modelling? If yes, in what situations is this idea used usually?

Thanks!

After some further thought, I realized the following thing. What we called tax rate at a certain amount of income is actually the tax per unit increase from the income, also called marginal tax rate, so the tax rate is a function of the income, i.e. a function from $[0, \infty) \to [0,1]$. The tax for income $X$ is calculated as the integral of the tax rate over $[0, X]$.

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For what it's worth, the calculation is not usually phrased this way. Your hypothetical 35,000 earner would be told that their tax is 3,000, plus 30 percent of all income above 20,000. A few of the examples in the Wikipedia article you link to (Australia, Singapore, South Africa, US) show this. This makes the computation simpler (because certain sums have been precomputed) and gives the same result. –  Michael Lugo Jun 10 '12 at 18:54
@MichaelLugo: Thanks! The way you mentioned is very popular, because it makes calculation more mechanical and automaitic. But I think it makes the underlying idea less clear. Also see my addition. –  Tim Jun 10 '12 at 19:16

The tax paid is a piecewise linear function of the income.

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Thanks! That is true. However the tax is calculated based on tax brackets and income. I think I just figured out something. Please see my addition. –  Tim Jun 10 '12 at 19:18