Yes, there is a housing shortage in many major US cities. This is evident both from the high rent in these cities, and the large share of workers who commute long distances to work in these cities.

It can be hard to immediately see that there is a shortage of something like housing, because it’s rare that people will actually end up without housing because there isn’t enough to go around. But this is because people will rationally choose to live further and further from their jobs and commute longer and longer distances as the shortage drives prices up near their workplace, rather than choosing to go without housing.