[hysan]

The question feels too open ended in the sense that there is no baseline premise on how the adjustment is being made. Cost of living varies greatly, can change at different rates, and if you're talking about the US, there are some aspects of your pay that aren't directly reflected in your salary - health benefits.

Let's take that latter point as an example. Health care plans differ per state and you are usually better off having your health care plan be from the state you are living in. At all the companies I've worked at, the plan is usually from the state where they have their HQ and subsequently, most employees. If you live in a branch office, that means your plan isn't as useful to you (this actually factored into comparing similar offers to me recently).

Now an employee moving knows the risk and how this will affect their coverage. For a company to adjust their salary based on where they live, how will they take this into account? Are they going to do a per-state/per-city level analysis of say, how their provider network coverage differs? Then pay you more if your coverage is worse off?

I'm not a risk taker, but I'd be willing to bet that most debates won't take something like this into account. However, this and many other factors matter a lot in determining what is considered "fair" compensation for a given area. If the premise of what a fair adjustment is cannot be established, then I believe that this isn't even a debate because the hypothetical reduces to, "what excuse can we use to cut someone's pay?"