Your reply suggests you're thinking inside a very strange (to me) box.
Have you considered… not rationing housing?
As in, you can actually pay people to build more houses, houses are not a fixed resource. Likewise roads, schools, shopping centres etc., they're all things you can just pay to get built. Only thing a little harder amongst the usual talking points is hospitals, but that's because medical qualifications take so long, not because you can't do it.
