One common argument against new housing is that permitting it causes land to become more valuable, thus leading to higher rather than lower rents. It seems to me that this argument is unpersuasive for a few reasons.
First, if it was true, places with permissive zoning would have higher rents rather than lower rents, as the possibility of building would cause land values to explode. Obviously this is not the case.
Second, the argument leads to absurd results. If downzoning reduces land values, obviously the best way to ensure low rents is to prohibit as much housing as possible. Perhaps we could prohibit all housing not on five-acre lots. But suburbs with large-lot zoning tend to be pretty expensive, suggesting that such policies are more likely to increase property prices than to lower them.
Third, the argument suggests that land costs are the primary determinant of rents. But in fact, land values are much more volatile. The Lincoln Institute has created a database of land value data, and shows huge swings in land prices. For example, in the New York City metro area, the land price (apparently per house) swung from $99,000 in 1996 to just over $433,000 in 2006, down to under $225,000 in 2012, and up to about $250,000 today. It goes without saying that rents and housing prices follow very different patterns.