Opposing political positions have complicated the issue of poverty wages far beyond what is necessary. What we have, in law, is the result of a century of unproductive struggle, that branches out into other parts of the economy.

We are still living the legacy of the Great Depression, then the New Deal, and then also the attempts to unravel the New Deal.

To address your comment more directly, companies can't fire all poor people. They are needed to run the businesses (until their automaton replacements are built). But one of the reasons poor people are poor is that they cannot afford to not work for long enough to make potential employers hurt enough to offer them higher wages.

One point of a minimum wage is to put a stop on the race to the bottom for wages. Raising the minimum wage will certainly put some people out of work, many of them permanently. All those who cannot generate enough labor value to pay for the cost of their employment will lose their job, if they had one, and be unable to find other work.

But companies that require wage laborers will have to pay them enough to live on, without the fear of being undercut by someone more desperate.

But then you still have the problem of all those people who are unemployable economically, because they're just not productive enough to work for an employer, and lack the capital, credit, or capability to support themselves with self-employment.

So you have to pair minimum wage with something, so that those people don't resort to crime, the career of last resort. Whatever that is would certainly be sustainable, if and only if the employers that are setting the prevailing wages were somehow made to pay the costs of the externalities they force upon the society in which they operate, mostly brought on by ruthlessly cutting their labor costs as closely as possible to the bone, and diverting a greater proportion of their revenues to owners and managers. For the most part, for non-luxury goods and services, labor cost funnels into customer disposable income. You can't sell mass-market consumer-grade goods and services unless someone pays their workers enough to afford them.

And that's what the minimum wage does. It forces all employers into a cartel, such that everyone must pay their workers enough to survive on, plus a little extra disposable income to spend on stuff that no one needs, but requires economies of scale to exist. If the rich owner of a business likes blockbuster movies with big production budgets, they can pay the hundreds of millions of dollars all by themselves, or they and all their rich buddies can pay their workers enough that they can all afford a $10 movie ticket once in a while. If the rich owner of a business likes fine dining, they can pay for a personal chef and the upkeep for kitchen and pantry, or they and all their buddies can pay their workers enough that they can all afford a $50 meal once in a while. When Wal-Mart pays poverty wages, they are reneging on the cartel agreement. Their lowest-paid employees can't even buy a Big Mac without budgeting for it in advance, much less go to the movies or eat a steak dinner. Those employees cannot support other types of business when all their pay goes to rent, utilities, public transportation, and food.