This is styled as an economics problem, but it's very mathy so I put it here.
Wages are higher in cities. The reason is that people are more productive when they are in close proximity to other productive people. So I modeled this as
$$P = P_0 + \int_A P f(r) dA $$
where $r$ is the distance to other people and $f(r)$ is a decreasing function. To make it simple I assume everyone is the same, i.e., the same $P_0$ and $f(\cdot)$, and the city is infinite so we can ignore boundary conditions.
I tried to solve this (in polar coordinates), but the problem is $f(r)$. I tried $\frac{1}{r}$ and $\frac{1}{r^2}$, but they blow up when $r\to0$. So any help would be appreciated.
You could move this to the economics stack exchange, but a traditional functional form would be something like $Ae^{-pr}$ to get around that problem, where $A$ and $p$ are some weights that could be 0. Also makes the integration particularly easy.