This economics point is often misunderstood, because at the high-end of the earnings spectrum, a large portion of income can go to savings, and cost-of-living differences make no impact on that.

That is, if you're making a $200k salary in NYC, let's say you spend 50k a year on housing and 50k a year on other expenses, but the rest you can put towards savings.

Then you move somewhere that is 50% cheaper, but you make 50% less. So now you can get the same quality apartment for $25k and your other expenses are only $25k, but you're only saving $50k a year instead of $100.

Obviously this is a gross oversimplification, but the vast majority of time if you move to a place with a much lower cost of living, your reduction in salary, if looking at local jobs, is unlikely to make up for it. Never mind the fact that for many industries there are only a few places where you can live if you really want a career: if you work in high finance or fashion in the US, you probably want to live in NYC. Film or TV, you'll be in LA, etc.