So if a flat tax hit all households with, say, a 15 percent rate that would mean taxes on the bottom fifth would increase more than sevenfold, while the top 1 percent would see their taxes cut almost in half.

To get around that, some flat tax proposals would boost the threshold a household could earn tax-free. GOP candidate Sen. Ted Cruz has proposed leaving the first $36,000 untaxed and fellow Republican hopeful Sen. Rand Paul wants to leave the first $50,000 untaxed. But that would still represent a huge tax cut for those on higher rungs of the income ladder.

"The flat-taxers like to run around with Lake Wobegon economics where we are all going to pay less and still have the same amount of money, but the world doesn't work that way," according to Dean Baker, co-founder of the Center for Economic and Policy Research, in a U.S. News & World Report blog post.



That's one of the main criticisms of the flat tax proposals: They wouldn't bring in nearly enough money to pay the government's bills, swelling the national debt with huge budget deficits that most of the GOP candidates have pledged to eliminate.



Read MoreGOP debate comes up short on tax, budget solutions

So far, four of the GOP candidates have proposed flattening the tax system to a single rate, all of which would eliminate trillions of dollars in tax revenue needed to balance the federal budget. Carson, who originally proposed a flat 10 percent on personal and business income, has recently upped that to 15 percent.

But, as CNBC's Becky Quick noted in the Oct 28 debate, the proposal would leave a roughly $1 trillion hole in the federal budget, an analysis confirmed by Politifact, a website devoted to fact-checking candidate claims.

The math underpinning the other candidates' flat tax proposals are equally problematic, according to the Tax Foundation analysis.



Cruz, for example, has called for a 10 percent flat tax on personal income and a 16 percent tax on businesses.

"The revenue loss under this plan, using reasonable rather than insane estimating assumptions, would be absolutely staggering," according to a blog post by New York University law professor Dan Shaviro, who has written about tax policy.



Other researchers agree. An analysis by the Tax Foundation, which Cruz has cited, found that his flat tax proposal would create a $3.6 trillion shortfall over the next decade.

Paul wants a flat 14.5 percent tax on personal and business income, a plan that would lose $1.8 trillion over 10 years, the Tax Foundation found. Even at a 20 percent flat tax rate, former Pennsylvania U.S. Sen. Rick Santorum's tax plan would come up short by $1 trillion, according to the research group.

The candidates argue that the deep losses in revenue would be offset because tax cuts would spur growth and create more jobs which, in turn, would generate more taxes. But that assumption doesn't take into account the impact of larger federal budget deficits or potential spending cuts on the overall economy. (Widening deficits would almost certainly increase pressure to cut government spending, which would create a drag on economic growth.)

