Anyone interested in reading more on this topic, I rec. this book. It's about people who are near-homeless, not fully homeless, but it's one of the best things I've read on the topic. The author lived with the subjects for several years & does a superlative job of telling their story in that sort of explanatory way (not judgmental or absolving) that HN readers seem to appreciate.
https://www.amazon.com/Evicted-Poverty-Profit-American-City/...
(excerpt of the book here: http://www.newyorker.com/magazine/2016/02/08/forced-out)
In theory that's true, but in practice many U.S. municipalities have restricted the development of any new housing to the point that Section 8 vouchers are impractical due to costs and simple apartment availability (http://www.vox.com/cards/affordable-housing-explained/supply...). Without doing something about NIMBYs and local zoning processes, Section 8 vouchers will not be effective.
Matthew Desmond's book Evicted is pretty good on this point (http://www.amazon.com/Evicted-Poverty-Profit-American-City/d...). I've written or worked on Section 8 proposals, as well HUD 811, 202, HOPE VI, and related programs (see http://seliger.com/2008/07/27/reformers-come-and-go-but-hud-... if you're curious); the people who run them, especially in high-cost cities like LA, SF, NYC, and Seattle are well aware of the problems that local zoning imposes on affordable housing.
As someone who lives in one of those cities, it seemed like a clear correlation could have been made. It was chance to unite a rural and an urban problem across race and I thought it was a missed opportunity. I still enjoyed the book, but think it is even more powerful when read in tandem with a book like Evicted [1] that views the problem from another perspective.
[1] https://www.amazon.com/Evicted-Poverty-Profit-American-City/...