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As a lay person I apologise if this is the wrong forum please let me know if it isn't.

In flat taxation regime you pay x% of some value (income for arguments sake). This remains at x% as value decreases or increases.

In a progressive taxation regime you pay a varying % dependant upon some variables bases on a function of the 'some value'.

So for example the rate may be y% (where y < x) for where some value is less than the above or possibly be z% (where z > x) for some value is greater than in the able example.

What I'm trying to find is if there is name for a function where a proportion varies according to the size of the whole. If you see what I mean.

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Usually taxation is "piecewise linear": within a bracket it is a linear function, but there are breakpoints where the slope changes (in this case increasing).

For a more general question, it could just be a quadratic. The percentage is then proportional to income, but at high enough income could exceed 100%.

You can have more complicated functions that do different things. Some of them have recognized names.

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In the first case, the "marginal rate" would be "piece-wise constant", also known as a "step function". – Gerry Myerson Aug 29 '12 at 1:12
Pedantry: I thought a "step function" was part of a more restricted class of functions where the width of each constant piece is constant. – Eric Stucky Aug 29 '12 at 1:55
@EricStucky: Often I have seen the name step function for one with only one transition (the or one with only one step up and one down as in In Fourrier analysis the function has to be perodic, so all the steps are the same width. – Ross Millikan Aug 29 '12 at 2:03
@Ross: Thanks! For the thoughts and the links. – Eric Stucky Aug 29 '12 at 3:05

I think that what you're describing is the differential equation

$$ \frac{dL(x)}{dx} = ax $$

Using your example, $L$ would be the tax liability and $x$ would be the income. $\frac{dL(x)}{dx}$ is just code for "how your tax liability depends on your income." The solution in this case is just

$$ L(x) = \frac{ax^2}{2} + c $$

Where $c$ is the amount of money that you charge someone with no money.

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