No, it generally doesn't. It depends on the specific numbers of course (how much of a salary cut, how much cheaper is the housing costs), but a lot of people make that blanket statement because of a mathematical fallacy: The fallacy is if I make half as much, but my expenses are half as much, then I'm just as well off. That statement is mathematically false.
Here's a made up example: San Francisco salary is $140K (or $84K after 40% taxes deducted) and rent or mortgage is $3000/month ($36K/year). Boise salary is half as much at $70K (or $42K after 40% taxes deducted) and rent or mortgage are also half as much at $1500/month ($18K/year). Now calculate disposable income after taxes and housing costs: In San Francisco it's $48K (i.e., $84K - $36K) and in Boise it would be $24K (i.e., $42K - $18K). Clearly, in this example, you'd still be $24K better off in San Francisco.
I know that I made many assumptions above. I'm just illustrating the fallacy that people often make. Halving your income and halving your expenses is not neutral -- it is worse.