If only some things are automated, then yes those places will have an advantage when it comes to labor costs, but they will use it to pad their profit margin.
Factory X employs humans ands sells widgets for $100, but their costs are $90, so profit is $10.
Factory Y is automated and they have costs of $10. Do they sell their widget at $30, thus earning a handsome profit of $20? No, they charge $90 and earn an obscene profit of $80. The worker gets screwed, the shareholders get richer.
This whole thing only works if workers have ownership in the corporation, but that's gasp socialism and we can't have that.