"There have been 47,500 units added in the metro region since 2008, and yet prices and rents have climbed while vacancy rates are extremely low."

This literally means nothing without reference to the relative population growth in that same time period.

"How many units would have to be built to really bring down, or even stabilize prices?"

There is actually someone who has done a study, and it's quite interesting.  Take a look: https://medium.com/@andersem/a-guy-just-transcribed-30-years-of-for-rent-ads-heres-what-it-taught-us-about-sf-housing-prices-bd61fd0e4ef9#.w7eb463pe