I don't have all the information on Copenhagen yet. The stats from 1970-s are not available online, so I commissioned someone to get the data from the archives.
The available data basically shows that prices were stagnant during the 70-80-s and started rising in the 90-s.
> After all, building more housing in the city isn't mutually exclusive with building housing outside the city.
I think it is mutually exclusive, exactly because of the population growth (the lack thereof). Each dense apartment in a city core means one less house in a rural area somewhere.
Japan, that I gave as an example, has literally free houses that anyone can get for nothing but the government real estate transaction fees. Just 3-4 hours away from Tokyo.