Hmm, you're right. I didn't mean to suggest that basic income should be based
entirely on cost-of-living, though. My thinking was something more like:
NewUI = OldUI * (CostOfLiving[NewJobMarketLocation] / CostOfLiving[PrevEmploymentLocation])
Thus, if you got paid the going rate for a factory-worker in Michigan, and you moved to SF, then you'd now be getting the UI that a factory-worker
in SF would be receiving if they had been getting paid the going rate for factory-workers
in SF. So you could afford to "switch" to the SF jobs market, or any other jobs market, and compete for jobs with the unemployed factory-workers currently living in the SF area: you'd be able to afford to live roughly where they live, feed the family you moved with to the same standard they were used to back home while you're job-hunting, show up in clothes of the same quality they show up in, maintain your car as well as they do to get to interviews at roughly the same time they do, etc.
But that assumes that there are factory workers in SF. There aren't—running a factory in SF is an unprofitable proposition. Whether or not you could come to the area and job-hunt, supply of your job (or any job at your pay-grade) might just not be there to meet your demand.
And that's not even accounting for the second-order effects: decreasing friction of movement between jobs markets would cause everyone else who ever wanted to live in SF to also come there to job-hunt. So, even if there were some factory-worker openings right at the beginning of this policy, they'd quickly be filled, and the markets in cities like SF would come to resemble Hollywood: a place where you have to put up with unreasonable amounts of shit to get anywhere, because there's so much more talent who has come from all over to vye for attention, than there are productions needing talent, that the productions take absolutely no care in how they treat their talent.
In other words, at equilibrium, you'd just be back to where you started as far as "number of people employed" goes, at most levels†; and you'd also not see any change in the population density, demand for gentrification, or any other such thing in these cities, because Unemployment only lasts a fixed amount of time—you can't afford to stay if you don't succeed in finding a job. The one thing that would change is that companies the nation over would be seeing far more "talent liquidity", and thus thinking of employees as even more fungible/replaceable than they do now. (But, the flip-side of that statement: an individual new hire at an SF company would now have much more even probability of having originated anywhere in the country.)
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† Maybe you'd get more startups, from several people coming to SF (or other expensive cities) and deciding to spend their UI as initial runway to form a company together. But that'd be the exception, not the rule, I think. (Might still have nice knock-on effects for employment if those companies need to grow by hiring later on, though.)