Can you, though? Any given neighborhood has limited space to build things. At some point, you have to start building high-rises all over to pack more people in. This also has a limit, so beyond that you'd have to start building over parks etc.
Of course, without government-imposed zoning laws, you will also have residential units competing with commercial and industrial ones, too...
Oh, and who will take care of the extra infrastructure that is required for more people?
So it would seem to me that this approach would result in extremely densely packed neighborhoods with poor infrastructure, and undesirable placement of (cheap) residential units next to, say, an industrial park.
It would also be pretty effective at inducing more rich flight from urban areas into the suburbs, which is likely to hurt the cities even more (by reducing their tax base).
I don't think this is all that simple.