The former. We measurably just have less housing stock per capita than other developed Western nations. Maybe the latter happens too, but I'm skeptical, because rent money is as (non-)green as the rest, and it's not required to explain the situation.

Why we haven't built more houses is less clear. In the absence of hard data or guesses from more educated people, I'll speculate a bit:

One tradesman I know is billing high enough to feel personally guilty about it, and is still swamped with work. From what I hear this is typical. The US and Canada officially have the same rate of construction work employment. We don't have undocumented immigrants working construction really, while it sounds very common in the US. Put together, my guess is there's a sizable labour shortage, but it's masked by data collection issues in the main other jurisdiction that builds largely with wood framing.