The days are ticking off to a January 1 fiscal implosion, and Republicans are trying to MacGyver a fiscal fix. Three days after the election, President Obama claimed the voters had given him a mandate to raise taxes on the rich. With Pew reporting that 58 percent of Americans feel the rich pay too little in taxes, it’s hard for Republicans to contradict the president.

Unless Congress intervenes to prevent the Bush tax cuts from expiring, high earners will face significantly higher income taxes in 2013. For the top two brackets, the rate will rise to 36 percent (from 33 percent) and 39.6 percent (from 35 percent). And there’s a good chance these taxpayers will also face deduction limits and higher taxes on capital gains and investment income.

But as Congress considers which segments of society should be funding public profligacy, definition matters. During the campaign, Obama embraced Warren Buffett’s idea of a “millionaire’s tax.” But millionairedom is oddly nebulous. The Obama administration has lowered the wealth bar; by the president’s reckoning, becoming a millionaire is as easy as earning $200,000 annually for a single person, or $250,000 for a married couple. But, as many have noted, in cities like L.A. and New York, families earning $250,000 are not really rich. So Republicans are pondering a counter-offer: What if we defined millionaires as those who make $1 million or more a year, and limit the tax increases to them?

Let the math (and the groaning, and the slogging) commence. A few caveats: Calculations are inevitably complicated by the many tax-policy variables that could be factored in. And, because projections necessarily include assumptions about future taxable income, the numbers become even foggier.

Earlier this year, Congress’s Joint Committee on Taxation reported that if the Bush tax cuts were allowed to expire for “the rich,” as defined by the president, the federal government would collect an additional $829 billion in taxes over the next ten years. However, if the tax cuts were extended for all but those who make $1 million or more a year, that additional tax revenue would drop to $463 billion. That’s $366 billion (or 44 percent) less over a decade.

That’s a projection; here’s how the scenario plays out with historical data. Drawing from the most recent numbers from the Internal Revenue Service — rather than projected future assumptions, such as those the JTC study used — Dan Indiviglio of Reuters Breakingviews assessed the impact of raising tax rates only on those who make $1 million a year or more. He found that if the marginal tax rate for those people were raised from 35 percent to 39.6 percent for all income above $374,000, the federal government would raise only one-third (not 44 percent) less in additional taxes while sparing around 2.1 million people from hikes — “which is great for political and even economic impact,” he tells National Review Online.

While studies disagree on the amount of tax revenue that would be raised by changing the taxpayer threshold for rate increases, they reveal something more important: America’s deficit problem can’t be solved simply by taxing the rich. Regardless of which definition of “millionaire” is applied, the tax revenue just doesn’t cut it. Furthermore, while Obama has stated that his purpose in calling for higher taxes is to reduce the deficit, it’s reasonable, given his history, to expect that much of any new revenue will go to spending, not deficit reduction.

But even if taxes on the rich did go directly to deficit reduction, it wouldn’t be enough. Earlier this month, the Congressional Budget Office put the 2012 deficit at $1.1 trillion. It noted that this was the fourth straight year of trillion-plus deficits. This year, the deficit is 7 percent of the gross domestic product. CBO projections don’t show much change in the annual deficit over the next decade. So even if the additional income taxes amounted to the $829 billion figure, the extra revenue would not eliminate the annual deficit, and it would do nothing to pay down the accumulated national debt of $16 trillion.

Furthermore, the hikes would carry significant economic ramifications. Obama’s definition of “millionaires” certainly includes the super-rich, but it also manages to rope in a lot of small-business owners. (Unlike their corporate counterparts, many small-business owners use Schedule C of the individual 1040 form to report their business income and expenses, and a lot of them make over $200,000 a year.) An Ernst & Young study published in July reported:

The increase in the top tax rates would reduce long-run output by 1.3% when the resulting revenue is used to finance additional government spending. Employment is found to fall by 0.5%. In today’s economy, these results would translate into a reduction of gross domestic product (GDP) of $200 billion and employment by 710,000. Investment and the capital stock (net worth) would fall in the long run by 2.4% and 1.4% respectively. Real (non-inflationary) after-tax wages would fall by 1.8%, indicative of the decline in living standards relative to what would have occurred otherwise.

Those are the stakes. The question will be whether the marginal amount of additional tax revenue is worth the economic consequences.

— Jillian Kay Melchior is a Thomas L. Rhodes Fellow for the Franklin Center for Government and Public Integrity.