Huge amounts of the demand just moves $x up the price curve, but that doesn’t suddenly conjure supply into existence, so you have the same demand willing to pay higher prices across the same supply: that’s just higher prices. In HCOL areas the vast majority of rent prices stem from land rent, and that supply is not just less elastic than demand, it’s totally fixed.

Most insidiously: the demand that will move most aggressively up the price curve is lower income people seeking better lower/middle income housing. So those prices will move the fastest.