Geez, are you a professional economist or something?

They should have given the one group a housing voucher and the other cash, but then you'd have to control for how much of the cash was spent on housing.

Alternatively, they could have done some sort of intent to treat design and just teach one group how to use real estate sites, fill out housing paperwork, etc. Then, if that group used more housing attribute the difference to the intervention and proceed from there.