You are missing the point. Houses for poor people are almost never built, instead poor people live in 50 year old places that were built for the rich or middle class.

What happens is that 10 people move into those new places and 10 people move into their old one and the vacancy works it's way down the line. Eventually some guy down the bottom gets slightly cheaper rent or a better place.

Imagine that instead 100,000 high-end apartments were built in the Bay Area. None of them will have poor people move into them but instead places like (to quote elsewhere in the thread) "downtown PA (university Ave) its right around $2800-3000 for a 1 bedroom, for places with 20 year old appliances and no upgrades." might now be $1000/month cheaper and affordable by somebody on a lot lower income.

I think you are exactly right, and in a healthy market that's what would happen. In practice, I believe we are so far from equilibrium that it would take such a long time for the trickle down effects to push housing costs in a range which is affordable for people that aren't well off.

Telling people that within 10-15 years if you keep building several hundreds of thousands of houses eventually rents will be affordable is not a winning slogan :)

Well, a mansion should be able to house more people than a shack, so it would increase supply more, and pressure prices down more.

New housing would start off being expensive because all housing in SF is expensive. Not the other way around.

Maybe the root of this fallacy is that it assumes prices are constant. If you build something today and charge $5k/month for it, that's how it will somehow remain forever. But in reality prices are set by supply and demand, for both new and old housing.