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I can't speak for sex ratios, but generally the reason some places have a lower cost of living is generally because either no one wants to live there (often for good reason) or it's more difficult to earn money there.

If you're thinking of trying the sex ratios thing, I strongly suggest controlling for age, and preferably social class.

Do you have any data to do this? I think the results would be interesting.

Sadly, I was considering including just such an appeal with my original comment. If you find some, let me know?

"I don't know how much longer these arbitrage opportunities will exist"

Forever. Unless there would be a world where every city costs the same and has the exact same gender ratio. And, most people don't move, thus leaving the arbitrage intact.