This is correct, it does apply to all forms of wage increases. I wouldn't say "instantly," but yes that's why cities are perpetually expensive despite constantly increasing productivity. They are highly productive ergo they have high wages ergo they can sustain high rents ergo they have high rents ergo they "are expensive" compared to adjacent markets.

Mentioned above but I'll put it here too for other readers: the reason rent is unique is that in high COL areas, it's driven primarily by the price of land which is has zero supply elasticity. Higher prices induce supply in all other forms of goods and services.