I view it from a different approach. The city centers are the most expensive because that's where all the good paying jobs are. Thus if you want to solve the housing crisis, you simply need to shift the density of where these jobs are located. Personally I like the circular city model where you have several CBDs on the outer rings of the city.

The current model basically has all the good jobs in the CBD. The further you go from the CBD - the less jobs and the lower paying they are. So of course you get a housing crisis the closer you get to the CBD.