In the real world, most people can't just relocate their life on a whim, and the majority of citizens are more-or-less locked into a specific area for long periods of time, for a whole bunch of reasons. (Employer, Spouse, Children, Extended Family, Networks, etc)
It's not impossible to relocate or anything. But most people can't do so, and the few who can usually can't do so easily. It's not like window shopping for cars or clothes, these markets are effectively soft-Monopolies.
If landlords in Austin raised rents, most people would just have to eat it, unless it got to dystopian levels that forced them to relocate.
There are a thousand local landlords. If they all decided to double rents, maybe only 5% of the local population would move out, but then 5% of the landlords would have empty properties. They're individually better off to get someone in there at a lower rent than have an empty unit, so they advertise a lower rent and take one of the tenants from a landlord across the street. Now that landlord has the empty unit and they do the same thing.
Which means Kansas City is competition for Austin even if 95% of the people there would never actually move. (Assuming cities where the job market, amenities and so on are similar.)
Also, when rents are higher, there is incentive to build new housing to capture the profits.
Rents only get high when they're not just eager to arbitrarily raise rents but somehow colluding to constrain supply, which is hard for thousands of landlords to do except through zoning. But that's completely independent from how much money the local people have. There are cities that have the same median income and higher rents because the local government imposes housing supply constraints.
If everyone in the US was given $1 million per month, do you think that rent in Austin would go up in absolute terms?
If yes, by what mechanism can you imagine this effect is caused?
If no, I don’t know what to say.
The implied way to do it is, of course, by printing the money. Which would indeed cause inflation in not just rents but everything. But how does that apply to the real case when the money doesn't come from printing it?
The entire reason people are setting up shop in Austin is that it's so much less expensive than San Francisco. If landlords in Austin tried to raise rents then people would go to Kansas City or Buffalo or Louisville.
Some of the landlords would figure out that it's more profitable to have tenants at lower rents than higher rents and no tenants.