What we try to do is to calculate what amount of money allow somebody to cover the basic necessities.
Whom is to say what the basic necessities are? They can't be the same for everyone, everywhere, right? Should minimum wage be the same for New York City, San Francisco, Fargo, and Kansas City?
Why even base it on basic necessities at all? Why not productivity / worker output?
At this point $30 sounds more reasonable, doesn't it? But why stop there? Don't you want more people in the middle class? Wouldn't a $50 help lift people out of poverty? I mean, $30 is just enough to live on, right? What about all the poor, single mothers who have to provide for their whole family. Don't they deserve more? Isn't $50 really the right amount?
You can make an argument that any amount is the correct amount. I'm not sure why you think cost of living is the *correct* assessment for minimum wage.
If you want to help people, futzing around with the cost of labor isn't the way to do it. Just do a minimum income / reverse income tax and be done with it.