I'm no expert by far. But I wonder how the US deals with this?
We are actually 50 small nations banded together with a common language and currency. Obviously we have a unified monetary policy for all 50 states. Economically speaking, EU shouldn't be too different.
You have some nations with high productivity and some nations with lower productivity. We have some states with high productivity and some states with lower productivity.
Things in California and New York are outrageously expensive compared to Mississippi and Alabama. Of course, wages are much higher in California and New York than in Mississippi and Alabama.
I wonder if there's something EU can learn from the US.