A desirable city with very limited new construction will lead to increasing prices. You can use rent control to give cheaper housing to a smallish number of peo

A desirable city with very limited new construction will lead to increasing prices. You can use rent control to give cheaper housing to a smallish number of people, who kind of win in this lottery and get cheaper housing the others. But there will be those others, lots of them, who were not among the "lottery winners" and who will have to resort ti living with a much longer commute. If demand is high, building new homes for people should be allowed.