This is an empirical claim. It looks like new monthly permits are down from ~4k a month in 2021-2022 to ~3k a month. They're still up significantly from before 2020. The building boom has slowed but it's still elevated. Not particularly close to zeroing out.
Not to mention, Austin could just make it cheaper to build to offset lower sell prices. Gruber et al estimate permits are 50% of an LA home's price. Fix permitting and rent goes down 50% by your assumptions that developers build until margins are 0.
Assume austin is only half as bad as LA, a 25% rent decrease would be incredible
> Gruber et al estimate permits are 50% of an LA home's price
That is not what Gruber shows. The 50% figure is that there's a permit adds a 50% premium on the value of raw land. In other words, the permit cost is "only" 33% of the cost of the permitted raw land. That's significant and a big problem, but very very far from "50% of on LA home's price."
But yeah structurally the solution is indeed to reduce the cost of production. Which if your 50% figure was correct, would be huge. But it's not correct.
The data is here: https://fred.stlouisfed.org/series/AUST448BPPRIVSA.