High density zoning is only a short term solution, and it seems with current policy enactment speed, technology (and especially building regulations), demand will always outpace supply for the foreseeable future.
They had a housing shortage for the past few years and removed a lot of restrictions. Recently they actually overbuilt despite a shortage of skilled construction workers.
You can build hella fast when the demand is there and the government wants buildings fast. We're talking skyscrapers in 12-14 months.