You already have shanty towns with all the regulation.
Making housing cheaper also means making higher quality housing cheaper.
Making housing cheaper also means making higher quality housing cheaper.
https://en.wikipedia.org/wiki/List_of_tent_cities_in_the_Uni...
After you've done all this, you'll find that your city is more expensive than the surrounding cities without, not less.
[0]: https://www.theatlantic.com/business/archive/2015/08/more-am...
Or, much more likely, builder profits higher.
Competition will sort that out.