Consider a simple tax policy where you plug your income into a function, and it spits your tax rate out. The idea is to do away with the discrete tax bracket system and instead have a continuous system. Consider the generic function

We have and . This satisfies the boundary conditions we want on a continuous progressive tax policy where represents the tax rate ( being %) if your income is . To gain some control over this generic function, we will wish to fix a point on it (i.e. assign a value for ). Suppose the average income is , and that we want . Then we have that . Substituting this gives

Differentiating once shows the slope starts and “ends” at . Differentiating again shows that the maximal rate of tax rate increase is at (that is, tax rates increase the fastest around this income).

We may now attempt to construct another function where is the number of people in a system who have income . We can generalize to define for some , where it will denote the number of people whose income falls in . where is a constant called the **population**. We could attempt to start with a generic function

where is called the **unemployment number** (). In particular we could attempt to determine by requiring

But evaluating the integral poses a problem; and hence so does expressing in terms of the mean (or even median). Thus it is difficult to start with the mean or median. Let denote the mode income and be the number of people with income . Consider the function

Here we have , , and since With tax and distribution functions and we can define the **revenue function** as the Riemann-Stieltjes integral

Similarly we have the **gross product**