# Looking for functional forms for urban distance

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.

• If $P$ is a constant across all people (and therefore with respect to $r$), then you really just have $P = P_0 + P \int_{r=0}^\infty 2 \pi r f(r) \, dr = P_0 + 2 \pi P \int_{r=0}^\infty r f(r) \, dr$. You just need that definite integral to be bounded, so you can use $1/(r^k+1)$ for some $k$. But it might be useful to prove something for any valid $f(r)$. – Brian Tung Oct 23 '20 at 16:44
• Sorry about edit noise, a bit of brain fade, I guess. :-) – Brian Tung Oct 23 '20 at 16:46
• Thanks. I got antiderivatives for k=1 and k=2 but they blew up at infinity. Couldn't find anything for k=3 (not sure I wanted to). So I just used a linear function that goes to zero at some radius $r_L$ and the result was an uninspiring $P = P_0/(1-bunchastuff)$. – Daniel Oct 23 '20 at 22:05

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.