Yes, it can; SF built 5,000 units last year, a record, and rents and owner move-in evictions finally declined. At the very least, building more housing can prevent further large price rises.
> BTW this is what's happening here in Portland OR, where a building boom has been going on for years. The area population has increased, housing prices are still out of range for many people.
The question is what would happen if you didn't build the housing. I contend that the people moving in would outbid everyone for the existing housing stock, which would drive rent increases even more than rents are currently going up. It may be that the new housing is just keeping things at a base level.
Flipped on its head the logic here doesn't really make sense; destroying existing housing stock would be disastrous for rents and prices, it makes sense that building more would help prevent increases.