You are taking as an assumption the very thing we are currently questioning - does lack of supply actually account for the increase in housing prices? Lack of

You are taking as an assumption the very thing we are currently questioning - does lack of supply actually account for the increase in housing prices? Lack of supply would cause prices to rise - but so would cartel behavior on the part of those who control that supply. Housing of the type necessary for relatively low-income individuals, where the location must be close to jobs (see above), is typically controlled by a set of entities who exhibit cartel behavior in how they price the rent.