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I don't think cost is the main limit on population density in preferable regions. I would think zoning plays a larger role and likely drives cost. I also don't think we necessarily have a real limit on the number of preferable regions other than land area. Most of what makes a location desirable are the amenities that it has access to, we know how to build schools, hospitals, libraries, parks, and other public services. Rather than making people duke it out for access to such things, maybe we should consider making more of those things available to people?

You can build more, but there will still be competition for what's there. Look at Manhattan as an example.

I'm not sure it's even possible to overbuild in an area that's not economically depressed. If London had three times the indoor residential area, a very possible case would be that it's still expensive, but that people previously living in small flats end up with larger ones or terraced houses etc.

It's hard to imagine how the poor are ever going to be able to meaningfully compete with people who have (at a minimum) hundreds of thousands of pounds of capital. The bottom tier of worker will be bid to their limit at almost any size of dwelling I would think.

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