Market forces will cause more houses to be built since the price increases are due to excess demand.
Not necessarily. Not a black and white issue, lots of factors involved. Mainly depends on the planning commissions or city planners, and what they allow. City planners always interfere so there is no true market force by itself at work.
For example, with only market forces at work an equilibrium would be reached with all buildable space occupied but liveable. City planners always interfere, some with good results that restrict building to create more quality of life for existing residents; often with bad results due to overbuilding and unfair rent control practices and congestion. There are quite a few towns that preserved open space and prevented congestion while allowing market forces to operate. I don't care which way it goes, I just want a minimal local government that won't interfere too much with the lives of people. If they do, I'll leave.