There may be a housing shortage in the expensive cities, but I doubt there is one nationwide. I’ve come to believe that the real driving phenomenon here is the “you have to be in one of five cities” idea that took hold in the early 2000s.
But this increase in pay means the assets tied to those cities must also increase in value, as there are now more cash chasing the same (or similar) amount of assets - exacerbated by zoning, which restricts supply.
Children follow the pension fund of their parents.