I don't know whether my answer describes real situation — it is an idealized model.
Firstly, you get income. Based on this, let us define:
- base Ma'aser income, , based on which Ma'aser is paid. It means that you should give (where ) to your local Rebe.
- base state tax income, , analogously — tax is payed based on this value. In general, what you should pay is a function , it does not have to be necessarily linear.
Therefore, you need to calculate solution to equation
The analytical solution doesn't have to necessarily exist. You can use any iterative method of root calculation to get the numerical result for , for instance Newton's method. When you finally have it, you pay Ma'aser and of tax.
This equation is analytically solvable in special cases. For instance, let us assume that
It is a linear progression --- if you pay 17% of tax, . In such case
Solution to this equation for is
The answer (how much should you pay) in case of linear progression is thus
of Ma'aser, where
EDIT: In answer to OP question in comments --- checking the convergence that is --- let us generalize variables to have explicit time dependence. Thus,
- is base income used in calculation of Ma'aser in —th month. The superscript is not a power — it is just an index of sequence,
- is base income used in calculation of state tax in —th month.
So that in one particular (—th) month one owes Ma'aser and pays of tax (in linear model, I assume constant as well). We can write equations analogous to the previous ones, but now what matters in —th month is how much one paid in previous one:
Reindexing ( and ) the first equation and putting second into first yields
This iterative equation can be solved the following way: we split into constant and variable part:
The choice is of course not unique. We want to choose
such that equation for has simple form. Combining the two previous equations yields
The equation for is really simple when
i.e., the constant terms are canceling each other. Then,
and since , when . It means that in long time , which is
This is exactly the answer for calculated in previous section. Therefore, regardless of initial conditions, if one only follows the rules that tax payed in —th month determines amount of Ma'aser in —th month (and vice versa), he/she eventually reaches the proper (`equilibrium') amounts of Ma'aser and tax.