In the Bay Area, this would probably just increase rents by $1000/month because the housing market is very inelastic.
This reminds me of Berlin where rent control is being proposed. For reference the average rent prices there are 7€/m² and rent control will limit it to 5€ per m² for buildings that are older than 50 years (not an exact number).
How do you expect to ever profitably build something for 5€/m²? It's impossible and therefore won't happen. New buildings will probably start at 10-12€/m² and go down once they are 20 years old.
The state did institute rent control.