[eclipsetheworld]

Allocating public housing could be done through an auction system. Bidders would submit offers for annual rent. The surplus, after deducting costs, could then be allocated to buying or building additional housing for this program. Alternatively, it could be used directly to subsidize rent for low-income individuals.

In the end, this would solve the allocation problem while maximizing the available public housing. It would take a couple of years or decades to reach an equilibrium state I guess.

I'm probably missing something obvious here. Can somebody point out my mistake?

[bko]

You've literally described a market system.

A developer charges as much as they can for their rental units. If profitable they take that surplus and build new units up until the point that the marginal cost of providing an apartment is equal to the marginal revenue for renting such a unit. This isn't due to benevolence but how you maximize profit. The developer also pays taxes which pays for public services

So yes, I am in favor of this system.

[UncleEntity]

The problem with this so called 'market system' is it allows people to chose their neighbors through bidding only on properties they know the poors can't afford.

If the French wanted citizens to have the freedom of (dis)association they would have written it into their constitution.