Tuesday's GOP presidential debate fell on the same day that fast food workers in more than 270 cities went on strike for higher pay. So Fox Business Network moderator Neil Cavuto dutifully asked the candidates about the nationwide push for a $15 federal minimum wage. Their responses were standard: No dice. Not gonna happen. Terrible idea.

If you follow the GOP candidates' logic, their arguments make coherent internal sense — but for a really horrifying reason.

To explain this, let's go back to that tried-and-true analogy of the economy as a giant pie. Every worker's income represents some tiny slice of that pie. Add all those worker slices up into one giant slice, and you have the overall share of national income going to labor. The goal is to keep the overall size of the pie growing every year. Maybe we can get it to grow faster, maybe not. But as long as the pie grows, and as long as everyone's slice stays the same size relative to all the other slices, then everyone's income and living standards grow in real terms.

But what if the worker slices shrink as a share of the pie?

That's happening right now. The slices that go to low-wage and mid-wage workers are shrinking as a share of the pie, while the slices going to high-end workers and people who own wealth are growing. And if the shrinking slices shrink fast enough, it will cancel out or even reverse the effect of the rest of the pie's growth. In other words, if inequality increases fast enough, it will actually reduce the real incomes of low- and mid-wage workers.

This dynamic puts conservatives in a tight spot. Because you want everyone to have jobs and to stay attached to the labor force. But if a group of workers are stuck with a shrinking portion of the pie, then the only way to keep the same number of slices in that portion of the pie is for the slices to shrink too.

This is really the heart of conservatives' objection to the minimum wage. A government-mandated wage floor sets a legal limit on how much those slices can shrink. And if the overall portion of the pie going to low-end workers shrinks, but the individual slices in that portion can't shrink past a certain point, then eventually only one thing can happen: The number of slices has to go down. Some workers have to lose their jobs.

Thus, if you accept rising inequality, but you also want to maximize how many people can have a job, the minimum wage looks like a very bad idea.

Ben Carson in particular implied that the social importance of keeping people in the workforce trumps the importance of raising wages. Recalling his own youthful employment, Carson said, "I would not have gotten those jobs if someone had to pay me a large amount of money. But what I did gain from those jobs is a tremendous amount of experience, and how to operate in the world and how to relate to different people, and how to become a responsible individual."

Donald Trump, being Donald Trump, was even more blunt: "Taxes too high, wages too high, we're not going to be able to compete against the world. I hate to say it, but we have to leave it the way it is."

Now, Carson's social-importance-of-work argument isn't necessarily wrong, though it's a highly romanticized portrait of low-end employment in America. But even if you agree with it wholeheartedly, it doesn't change the fact that the GOP's two lead candidates both straight up said American workers — especially low-wage workers — are just going to have to put up with being poorer if they want to stay in the workforce.

See what I mean about making internal sense, but in a horrifying way?

Now, the minimum wage, in limiting how much the slices can shrink, also forces businesses to give more of their revenue back to their poorest workers as higher incomes. That means the very communities in which these workers live get more aggregate demand, on net, and thus more job growth. And if you look at the data, the two effects seem to balance each other out: The minimum wage neither noticeably increases nor decreases the jobs supply.

But that means, on its own, the best the minimum wage can deliver is a kind of holding pattern. Maybe, if you're lucky, the demand effect slightly outpaces the job-loss effect, and you move back to full employment very slowly over time. So, while the Democratic candidates appear to support hiking the minimum wage, they still need to offer accompanying policies to drive up aggregate demand and increase the supply of jobs.

Marco Rubio caught on to this point Tuesday, and tried to tell a story about how we could increase the jobs supply, and thus increase worker bargaining power, and thus increases wages, all without touching the minimum wage. Unfortunately, it was also a well-trod litany of economic fallacies.

Still, the Democrats don't really have a competing macroeconomic story to accompany their support for a higher minimum wage. All their offerings are basically Rubio-light: federal spending that's not quite as low; a tax code that doesn't favor massive accumulations of wealth at the top quite as much, and so on. The only exception is Bernie Sanders, and even he can be touch and go.

So: The Republicans oppose the minimum wage, and the larger story they tell for why they oppose it makes internal sense — even if the tale ends horribly for workers.

The Democrats, meanwhile, are right to support a minimum wage hike. But they're gonna need a surrounding story — and a surrounding set of policies — in which their support makes sense.