Prices don't rise spontaneously (or not exactly). There's increased demand, which usually means more people living and working in the area. Therefore, in fact the cost of policing and everything else does increase.
Yes, they do rise spontaneously. There are not more people living and working in the area: the prices of the existing houses have risen, just because of speculation and flipping. It's not like there's more people getting packed into these houses (for the most part).
You only get more people living and working in the area if you build more houses. If you do that, then of course you're going to have more property tax, even if the existing house values don't change at all. New houses = new property taxes. That's how you pay for the extra policing, not by soaking the existing residents.
