[pasabagi]

I think generally speaking, lowering housing costs isn't to allow a homeless person to save up for a deposit and rent an apartment. It's to increase the likelyhood that the sibling they always got along with will have a spare room, or to increase the buffer between losing a job and being on the streets, or to make it so the old friend doesn't mind it that much that somebody is sleeping in their spare room, since they have a spare room.

On the individual level, I expect the reason why people stay is that being homeless makes you very dependent on local knowledge and social network. Knowing what bins contain food, where is good shelter, which police officers are dangerous, and so on - is very important. If you move city, you're not suddenly going to have the kind of money to pay a deposit. You'll just be homeless in a place you don't know. That, and a lot of homeless people have jobs.