Cost of living is one of the factors that everyone should consider when they are relocating, but with a caveat: If you are choosing between two cities with different costs of living, you must also assess the salary you would earn in those cities.
If your salary is significantly more in the city with a higher cost of living, you may save more money through that move, compared to earning less in a lower cost of living area.
Individuals who have fixed salaries, because they work from home or run their own business, should definitely consider moving to an area with a lower cost of living.