Glad you asked this, it's something we're constantly pondering. Right now the answer is yes, though we don't do it ruthlessly and our overall philosophy is to show that we truly care about everyone involved. We ask the team member if they feel it's appropriate, rather than forcing it. We've actually not lowered someone's salary in this way yet, though it seems sensible that it would happen.
The fact is that the living costs (largest chunk being accommodation) varies so drastically, that it feels hard to justify having the same salary and basing salary purely on skill, with the assumption that location truly is a choice and people have chosen to live somewhere cheaper. We have people who end up paying $2,500/mo rent and others where it is more like $800/mo or even less.
In addition, we always want to pay people more if they choose to move somewhere more expensive, it seems unfair if someone gets used to a salary in one place and suddenly they move somewhere much more expensive, and we don't help them out.
> with the assumption that location truly is a choice and people have chosen to live somewhere cheaper
That's a terrible assumption to make. People often live where they do due to obligation to others (family usually). I would never work for a company that would consider cutting salary because they moved to a lower cost area. I also know several people that have quit a job rather than relocate to a lower cost part of the country with an employer as the relocation also included a cut in pay.
To be clear, we are agreeing with each other. I think that is an incorrect assumption, too. We consider each case very carefully, and often don't cut the salary (it has happened more that we didn't cut salary).
That said, it is very easy here in the comments to argue how bad it is to cut the salary when someone moves to a cheaper location. However, the flip-side of the same coin is what about when someone moves to a more expensive location? Should we then say "this is the fixed salary and you choose where to live, and if you move somewhere more expensive that is your choice"? I don't think that is fair. So the other side of the coin of "salary goes down if you move somewhere less expensive" is that we increase salary if you move somewhere more expensive. We've done this for people and it is highly appreciated.
Our overall philosophy is that we want to help people to be wherever in the world they can be happiest and most productive. We encourage traveling and we have set ourselves up as a distributed team based around this value.
> However, the flip-side of the same coin is what about when someone moves to a more expensive location? Should we then say "this is the fixed salary and you choose where to live, and if you move somewhere more expensive that is your choice"?
I'd say that's absolutely fair, it's kind of the default expectation in my mind at least - I provide a certain value to the company as a remote worker, for which they pay me accordingly. (edit - as someone who just found a job via a HN jobs thread, I don't want to hijack this with a long response. Please see http://pastebin.com/4SrQJRj6 for my full response to this)