I think the formula for minimum wage should be as follows
Minimum wage adds up to enough per month that the average number of hours worked times the minimum wage are large enough for the national average for rent to make up at most 40% of a minimum wage worker’s budget.
If a county (or equivalent subdivision) has a higher average rent than that average, the minimum wage there is the same formula but for that higher average rent instead of the national average.
If there’s a discrepancy between where someone lives and where they work, their minimum wage is the higher between the two.
The only modification I think it’d need is maybe finding a scientific way to replace monthly rent with “average monthly expenditure” so that folks getting billed for medicine aren’t being screwed over, but frankly a country that can get the above proposition done is probably one that made UHC happen too anyways.
Also just to keep those landlords from thinking they can get wise, rent can only be raised on lease renewal, and only by as much as the reported inflation rate at the time of lease renewal. Also, constructive eviction to try and get around this is a felony and you lose all rights to own housing property other than your own home ever again.