If you consider anecdotal evidence then I have received, with no negotiation, similar (statistically identical) offers in low cost of living areas and high cost of living areas.
A different company was in a different country than Omaha and the naive cost of living was higher, but the cost of living, after adjusting for taxes, was actually very similar.
I hope that gives enough information. I really enjoy living in high cost of living areas so that's where I ended up, but I did have the choice to live somewhere cheaper.