I have been working on creating a system for distributing revenue from a land value tax to multiple administrative levels of government. While related to economics, the problems I currently have are mathematical.
In my system, a state imposes a land value tax (similar to property tax) on all private property. The state is subdivided into administrative divisions called Regional Districts, which are themselves composed of municipalities or cities (which I will use interchangeably). Each city will give a portion of it's revenue to the regional district according to a formula, and then the regional district will give a portion of that revenue to the state according to a similar formula. I want to make sure the regional district never owes more money than it receives.
Here is the formula I currently have for how much revenue a city owes to the regional district.
$$R_{oM} = \frac{R_t R_n^2}{R_n^2+c_M p R_{mS} R_t}$$
Here is the meaning of variables:
$R_t =$ The total land value tax revenue from property in the city. This value is affected by zoning, as most zoning restrictions reduce land value.
$R_n =$ A measure of total land value that's independent of zoning. This will usually be higher than $R_t$, as it doesn't consider the burden of zoning. Not restricted to private property, but includes other land in the city's control such as parks and roads.
$p =$ Population of the city.
$R_{mS} =$ The land value tax revenue per-person in the entire state.
$c_M =$ A value that can be adjusted based on how much funding we want cities to have. All cities in a regional district, and perhaps the whole state, should be given the same $c_M$ value.
Now here is the formula for how much money a regional district owes to the state government:
$$R_{oR} = \frac{R_t R_n^2}{R_n^2 + (c_M+c_R)p R_{mS} R_t}$$
In this case, $R_t$, $R_n$, & $p$ includes all land and population in the entire regional district. $R_{mS}$ should be the same. $c_R$ is set to the same value for all regional districts, depending on how much funding we want this level of government to have.
One problem I've noticed with this function is that it's possible for the regional district to owe more money than it receives from it's constituent municipalities. This happens when the ratio of $R_n$ to $R_t$ varies too much between different municipalities.
Replacing $R_t$ with $\sum{R_{oM}}$ in this function (as can be seen in the Desmos link) guarantees that this never happens. However, this formula is less fair because the location of borders between municipalities has too much of an effect on how much money a regional district owes to the state. The unmodified $R_{oR}$ function is completely unaffected by internal borders.
If I use the unmodified $R_{oR}$ function, can I guarantee that it will never exceed $\sum{R_{oM}}$ by replacing $R_m$ in $R_{oM}$ with the output of a new function?
I had already considered replacing it with $\sqrt{R_{mS} R_{mR}}$ to increase funding for poorer regional districts, as the unmodified $R_{oM}$ & $R_{oR}$ would result in their member municipalities keeping most of it. But this doesn't guarantee that $R_{oR}$ never exceeds $\sum{R_{oM}}$.
I can probably figure something out that involves finding the ratio of $\sum{R_{oR}}$ to $\frac{R_t R_n^2}{R_n^2 + c_M p R_{mS} R_t}$. However, I'm hoping for a mathematically simpler solution that doesn't require every entity to be calculated twice.
What function can I use to substitute $R_{mS}$ in $R_{oM}$, to guarantee that $R_{oR}$ never exceeds $\sum{R_{oM}}$, without reducing funding for the poorest regional districts?
While this is the preferred approach, feel free to modify $R_{oR}$ if you want.
I have found a solution that doesn't replace $R_m$ with a different value, but changes $R_{oM}$. It's quite mathematically complicated though. Unfortunately, it makes other cities in a regional district pay for the bad zoning in one of their neighbours, making it an inelegant solution.
Here is a new formula for how much a municipality owes:
$$\frac{R_t R_n(R_n + p((R_n/p)-(R_{nR}/p_R)))}{R_n(R_n + p((R_n/p)-(R_{nR}/p_R))) + c_M p^2 R_{mP} ({R_mR} + ((R_t/p)-R_mR))}$$
$R_nR =$ The regional district's $R_n$ value. $p_R =$ The regional district's population.
Using this function for municipalities results in the regional district always receiving at least as much as it owes. After seeing how punishing this is on neighbouring cities, this isn't the approach I would like to take. A new formula for the regional districts would be better.