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This would be the case in an efficient market, where the risk-adjusted economic profit of renting out homes is close to 0. This is not the case because San Francisco artificially restricts supply of housing. This causes rents to be determined solely by (small) supply and (large) demand. Increasing taxes would not decrease supply of housing or increase demand for it, so it would just serve to decrease the excess economic profits landlords are making.

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You can't really say that without knowledge of how high those additional taxes would be. If the costs rise to the point that a reasonable profit for the level of risk involved can't be made then prices would certainly have to rise irrespective of demand. I suspect given the nature of ever rising taxation in most municipalities that introducing this tax even in a modest form would increase perceived risk[0] of property ownership enough to increase average rents.

[0]If the cost of an unoccupied property or a nonpaying tenant goes up, rents will likely go up in response.

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