You are a very smart person! I certainly agree with you, raising the minimum wage would raise the cost of everything. So even if the minimum wage was raised to the level of "a living wage" by the time the prices were adjusted to make it possible for employers to pay that higher wage ... the minimum wage would be below the living wage once again!
There could be a solution to this dilemma, but it would take a group of legislators with backbones to vote for the right legislation ... fat chance of that happening in Washington any time soon!
One of the biggest expenses for a family right now in this country is medical care. A "single payer" type health care system, like most of the industrialized world has, would move that expense out of the budget for most families. Basic, necessary medical care available to all families through government supported clinics, hospitals, rehabilitation facilities and long term care facilities would certainly make that minimum wage paycheck go farther.
How could such a plan be implemented? Start with government paid medical training using the same system military doctors, nurses and medics are trained with. These individuals would "work off" their education by working in a government owned and operated medical facility, just as countless doctors and nurses do in today's military.
Not only would this program bring quality medical care to low income (or no-income) families, it would provide low income and economically disadvantaged people a chance at becoming doctors or nurses!
Yes, the program would require a tax increase ... but the long term effects of such a program would be well worth the increase.