Well, the people advocating for minimum wage aren't doing so out of economic reasons, but political. They're doing it to bribe the bottom sectors of the working public for their votes. One law of economics is that if you set a price cap on a particular commodity, you are going to end up with a shortage of that commodity (unless the cap is set so high as to be irrelevant), and if you set a minimum price, you are going to end up with a surplus. Wages are the price paid for the commodity of labor. Setting a minimum price for labor will result in a surplus of labor, which is better known as "unemployment."

They arrived at the $15/hour figure not by calculating what would be needed for a living, but by offering more than what was already available. This is the best kind of promise for a politician or activist. It requires no spending or distribution, since the money ostensibly comes out of the employers' wallets (but is, in reality, paid by the customers, in prices raised to cover increased payroll, and by the existent workforce, in extra labor to compensate for decreased coworkers), and by the time the bad consequences come around, no one is willing to blame the people who pushed it, because they are all accomplices, having taken the higher paychecks. It's purely a win-win situation.