Pay is usually max(COL, local market rate). You don't need a super high COL if the local market is very competitive, but COL can raise the pay because you stil

Pay is usually max(COL, local market rate). You don't need a super high COL if the local market is very competitive, but COL can raise the pay because you still need to convince people to move to your location and most people won't do that if it has an abnormally higher cost of living without a corresponding pay increase.