The debate about whereas or not the minimum wage should be raised may be summarized as follow: on one side, opponents argue that a higher mandatory minimum wage would prevent employers to hire unexperienced / underqualified workers. If you think about it, it makes perfect sense: just imagine what would happen if, somewhere in the United States, the government was to impose a—say—100 dollars minimum wage per hour. It is pretty obvious that it would put people out of work since many companies—especially small companies—would simply not be able to pay such rates. Of course, it is also means that a 1 dollar minimum wage would probably have no effect whatsoever on the unemployment rate since, as far the United States are concerned, even the less experienced / qualified workers are worth much more than that. So it really is a matter of numbers: a $15 minimum wage is likely to put a number people out of work but the question is whether or not that number is significant.
On the other side, the proponents of such a raise argue that, because low-income workers have a higher marginal propensity to consume (which is true), a 15 dollars minimum wage would boost aggregate demand and therefore compensate (or even more than compensate) the cost effect. Admittedly, it also makes sense: modulo the share of imported goods and services, a higher solvable demand is likely to increase the sales of American corporations which, in turn, might boost demand for workers, lower unemployment and therefore push wages up. Again, it’s a matter of numbers: putting aside moralists and the economically illiterates, the whole debate—as I understand it— is about which of these two effects will have the stronger influence on the economy.
Now here is a weird fact: I don’t what the figures are in the United States but in France , it happen that 28% of minimum wage workers are living with an above median income . Yes, really and no, it’s only marginally related to the redistributive system: these workers are simply married with a high-wage earner and, as a result, they actually have a relatively high level of income. Thinking about it, I know many households in that situation starting by my own, a few years ago, when my wife was getting the minimum the wage  while I was probably in the top decile of French wage earners.
So while it’s true that low-income households have a higher marginal propensity to consume, it is not always true—far from it— that minimum wage workers belong to low-income households. Another example that might even not appear in statistics depending on how they are built, is the case of youngsters in their first job that still live with their parents: of course, mum and dad may be poor but they may also be wealthy and even very wealthy.
Taking this into account, a raise in the minimum wage would only have the desired effect—boost the aggregate demand—for a fraction of minimum wage workers. For those who belong to average or high-income households, these additional dollars might end up as savings; especially if the raise causes a rise in unemployment in the first place.
 Source: Insee, 2011 (in French).
 These are after tax and redistribution incomes, adjusted for the number of people belonging to households.
 Her wages were raised many times since then (and much faster than the minimum wage in case you wonder).