Should wages be the same in Nelson as Auckland?

Stuff reports:

Foodstuffs pays its South Island supermarket staff $2 less an hour solely because they are in the South Island, the FIRST union alleges. 

An ongoing dispute over wages for Foodstuffs’ staff will head to the Employment Relations Authority (ERA) next week as contract negotiations remain gridlocked.

New World Nelson, Pak ‘n Save Richmond and Pak ‘n Save Invercargill’s local owners rejected the FIRST union’s attempt to bargain for pay rises in collective agreements, FIRST’s Nelson organiser Rachel Boyack said. …

Boyack said employees at Foodstuffs’ South Island stores were paid about $2 less an hour than North Island staff without explanation.

“[Foodstuffs] have said that they’re paid less because it’s the South Island.

“The jaw drops on our side of the table. I have never heard an employer say that South Island staff should be paid less.”

I don’t see why wages should be the same in every city. Wage rates are a factor of supply and demand, and this varies by area. It costs more to live in Auckland so wages tend to be higher in Auckland.