The problem is some places are more desirable than others. Penthouses, even if they're not big, they are, for most people, more desirable than the bottom floor.

The problem is some places are more desirable than others. Penthouses, even if they're not big, they are, for most people, more desirable than the bottom floor. They have a view. They're away from noise. They're often away from rats, roaches, ants which start at the bottom. Similarly any place on the side of a hill, a cliff, a beach, ... You're unlikely to ban such places (all buildings 1 story?) Even if you managed that they're the house on the quiet street and the loud house next to the highway/trainline/airport You could try to make such places merit based but that only turns into a different kind of bribe and cronyism. Money is at least, somewhat obtainable by anyone. Plenty of people were not rich and became rich, maybe by luck, but at least it's a higher rate of churn than central authority has ever been.