It's not so simple. Some cities have too much housing. So much so the value of land goes down because of the house built on it.
Sure building more housing, pushing prices down might induce more demand, but there are a dozen other factors (not in the control of the city) that also influence demand - job vacancies, crime levels, quality of schools, etc, etc.
Sure building more housing, pushing prices down might induce more demand, but there are a dozen other factors (not in the control of the city) that also influence demand - job vacancies, crime levels, quality of schools, etc, etc.