East Bay is a good example - that is indeed how the upper bound on rents is set, typically. You can move to a smaller apartment, or commute from further out (my original comment).

It doesn't work for everyone - some will be worse off if they either a) move (lower wages - small town retail vs in the city) or b) commute (costs will be too high - fuel, working shorter hours), or c) they already live in the cheapest accommodation

Such people obviously exist - they live 'paycheck to paycheck' and make up a large minority of the population. They do not have money they could spend on more rent! In fact, they rely on welfare for necessities like food and medical care and yes even rent https://home.treasury.gov/policy-issues/coronavirus/assistan...

Now everyone gets their $100/week. The bottom 1%, with few accommodation options, accept paying higher rents. This means the people slightly closer to the city can't move further out, without paying the extra $100/week. Following this chain, eventually people in East Bay mostly accept the $100/week increase, and hence so do the people in SF. This isn't exact - the new equilibrium might not be quite X + 100, but history shows it will be very close.

What if the fairy took $100/week? Now landlords must decrease rents in the outer areas as people can't pay, and hence the rent decrease also propagates to the inner suburbs, else people would take advantage and move further out. These inner suburbs residents get the discount even though they had more than $100 week spare in the first place.

This argument is straight out of classical economics