(As an equally plausible hypothesis I'd guess that outside of extreme regions like Antarctica, higher wages are a prime reason for real estate values, and thus contribute to higher costs of living.)
Labor costs and the cost of living both affect the other. Arguing which came first is pointless since neither exists in a vacuum.
Furthermore even people in marginal jobs will make more if they live in a high cost area. For instance a gardener can charge more per hour in Los Angeles than in Ohio, even though the productivity of the two are identical.
Wages are fundamentally set by supply and demand. Productivity affects demand, and hence affects wages. But a high cost of living changes the supply curve, and also affects wages.
> For instance a gardener can charge more per hour in Los Angeles than in Ohio, even though the productivity of the two are identical.
Yes, that's because gardening isn't traded.
BTW, to be having this conversation makes me feel like I'm trapped in Chapter 33 of 'A Connecticut Yankee in King Arthur's Court' (Mark Twain, 1889):
http://www.archive.org/stream/connecticutyanke00twai#page/32...
Though, Twain first lays it out on p. 304 of the same edition:
http://www.archive.org/stream/connecticutyanke00twai#page/30...
A man who hasn't had much experience, and doesn't think, is apt to measure a nation's prosperity or lack of prosperity by the mere size of the prevailing wages; if the wages be high, the nation is prosperous; if low, it isn't. Which is an error. It isn't what sum you get, it's how much you can buy with it, that's the important thing; and it's that that tells whether your wages are high in fact or only high in name.