Talking about "cost of living" in a place like Seattle, as if it were an ordinary economic issue, masks the real problem, which is that liberal US cities have made it very difficult to construct new housing. If a million families want to live in a city, and the city only allows half a million houses to be built, then a lot of people are going to be forced out. It's not a question of economics, it's a simple question…
There's no need to toss "liberal" in there. Dense cities tend to have rent-control, and Dense cities tend to have people with policies generally referred to as "liberal" (to the extent that "liberal" means anything. But NIMBYism and status-quoism is not just a liberal issue, and "not changing the neighborhood" is right in line with the definition of "conservative", which means a preference for how things already are…
Regardless of dictionary definitions, if we agree that building restrictions are a bad policy, it is a valid and IMO interesting empirical question whether most people supporting it self-identify as liberal, conservative or something else. (I honestly don't think I could guess the answer.)