We've already tried this experiment. It's called financial aid for college. Tuition prices just keep tracking federal aid increases.
http://college.usatoday.com/2015/08/20/report-federal-aid-ri...
If they gave the money as credits to developers to build more housing, then they might have a chance at accomplishing their goal.
Colleges and other shared-expense systems (see also: healthcare, military spending, public spending generally) present and pose other challenges, but they are not the same as the situation as applies to housing, specifically.
You don't give builders credits to build houses. You tax them for the potential revenues of a given quantity of land. Since land is nonfungible, the consequence is to both develop undeveloped lots, and to increase the density of utilisation of developed losts.
One can argue that taxation in all its forms, including inflation, is the singular cause of all poverty in the USA-- it is the largest most massive wealth transfer fro the masses to the rich ever. And I don't mean "rich don't pay enough taxes" I mean taxes are stealing from the poor and giving to the rich.
That's a key reason they're so markedly different from other goods.
Incidentally, your username and curiosity prompted me to check to see if Hazlitt's infamous little book makes any mention at all of either rents or monopoly. It doesn't.
I find that quite interesting.