Actually, living in welfare housing may be cheaper for the state than living on the street. Welfare housing costs a few hundred Pound a month. The healthier living conditions, and thus reduced costs to the NHS, alone may be worth it.

(And in the USA that would be even more pronounced, because people without medical coverage often wait until their conditions become an emergency.)

There was an article a few years ago about some homeless people in L.A. or so cost the state so much, that just paying for their housing is cheaper.