Regulator expenses in building houses may only be 20-40% of the cost, but this isn't true of the price! In healthy markets, houses are priced close to the cost of construction. In SF (in the 90s), house prices were over 140% the cost of construction [1]. I'd expect the difference to be even larger today because regulations are more ridiculous.
I agree, if we eliminated all regulations, the cost of building a house would decrease perhaps 40% at the most. I'd expect some of this savings to be passed on to tenants. If this were the only consideration, housing would still be unaffordable. But, without regulation, we'd also see at least 10x increase in new construction. Housing would be built until it were no longer profitable. IE, the price of apartments would approach the cost of building them (from astronomical to merely expensive).
The Bay simultaneously welcomes any and all new jobs while it refuses to build housing concomitant to the demand. It will never be cheap to live here, but maybe we could drop the distinction as the most expensive city in the country.
1. http://www.researchgate.net/publication/228180251_Zoning%27s...