I am working on the same project described here. But I will provide all the information needed here to understand the math.
Here is a function that I created. I believe this is an example of a partial fraction. $$R_{oM} = \frac{R_t R_n^2}{R_n^2 + c_M p r_m R_t}$$
$R_t$ & $R_n$ represent two different measures of total land value in a given area. $p$ represents population. $R_{oM}$ is meant to be calculated for each city in a given region, with their own values of each. $c_M$ is a a value that we set to whatever we want. It will be the same for each city. $r_m$ will be the same value in all our calculations.
Now here is a function that can be calculated for a region, using the $R_{oM}$ output of each city in the region as an input. In this case, the values of $R_t$, $R_n$, & $p$ are the aggregate for the entire region. $$R_{oR}=\frac{(\sum{R_{oM}})R_n^2}{R_n^2 + (c_R-c_M)p r_m (\sum{R_{oM}})}$$
$R_{oR}$ was successfully designed to never exceed $\sum{R_{oM}}$, as long as we set $c_R$ to a higher value than $c_M$. $R_{oM}$ never exceeds $R_t$ if $c_M$ is positive (which will always be the case). Setting both c-values equal results in both having the same output. The following relation is true if all cities in the region have the same $R_n$ & $R_t$ over population. $$R_{oR} = \frac{R_t R_n^2}{R_n^2 + c_R p r_m R_t} $$
I wanted to replace $R_{oM}$ with a function closer to a straight line. Here is the one I created: $$R_{oM2}= \frac{p R_t^2 + 2 R_t R_n^2}{2 R_n^2 + (1+3c_M r_m)p R_t}$$
However, this function has a different behaviour when using it as an input to a similar regional function. It doesn't have the same relationship that $R_{oR}$ has with $R_{oM}$.
$$R_{oR2} = \frac{p(\sum{R_{oM2}})^2 + 2 (\sum{R_{oM2}}) R_n^2}{2 R_n^2 + (1+3c_R r_m)p (\sum{R_{oM2}})} \neq \frac{p R_t^2 + 2 R_t R_n^2}{2 R_n^2 + (1+3c_R r_m)p R_t}$$
Another problem I have found, experimentally, is that $R_{oR2}$ is more sensitive to changes in borders between cities than $R_{oR}$. They can both be manipulated by drawing borders in such a way that cities have more variation in $R_t/R_n$. This is counter-intuitive, as $R_{oM2}$ is closer to a straight line than $R_{oM}$.
What is the reason for this difference in behaviour. In particular, why is $R_{oR2}$ more sensitive to the distribution of $R_t$ & $R_n$ values between cities than $R_{oR}$?
And lastly, is there any way I can modify $R_{oR2}$ so that it behaves as I want it to?