While this may have played out in San Francisco, this is the exception not the rule. Rent caps introduced in New Jersey and Massachusetts did not lead to a housing shortage. Properties were constructed in the same volume as in cities without rent controls. In New Jersey, the volume of properties on the rental market actually increased, as landlords saw financial benefits in splitting larger properties into multiple units.
Studies indicate that the housing and labour markets do not follow the established supply and demand economic models of other industries. Rent controls do not lead to fewer houses on the rental market, nor do they disincentivize the construction of new buildings.