The 2012 presidential election will present voters with contrasting visions of government, brought into sharper focus by Mitt Romney’s selection of House Budget Committee Chairman Paul Ryan as his running mate. At the heart of each vision, however, is a dangerous myth that undercuts the potential for serious policy discussion.

The fundamental Republican myth is that the country’s fiscal problems can be tackled without new revenue. Or, in the more sophisticated version of this argument, that they can be addressed by rejiggering the tax code in a way that would promote economic growth, and therefore produce additional revenue, without asking any households to pay a larger share of their income in taxes.

In the Romney-Ryan version of the myth, marginal rates can be cut even further, eliminating popular deductions — deliberately unspecified — and counting on unduly optimistic projections to fill the gap. As the Tax Policy Center has demonstrated, that approach would “provide large tax cuts to high-income households, and increase the tax burdens on middle- and/or lower-income taxpayers.”

The Romney-Ryan approach fails to acknowledge the level of revenue required to sustain even the smaller government the candidates envision. The population is aging, and even with entitlement reform, health-care costs will continue to grow. The world remains a dangerous place, which is why Romney says defense spending should consume 4 percent of gross domestic product.

Yet the Congressional Budget Office has found Ryan’s plan would whittle spending on everything but Social Security, health programs and interest on the debt to less than that. The country cannot afford to continue all the Bush tax cuts, at a cost of some $5 trillion over the next decade, let alone pile on trillions more from Romney’s cuts.

The sensible approach of the Bowles-Simpson deficit commission, which Ryan voted against and Obama allowed to languish, was to slightly lower tax rates and dramatically broaden the base — and use the ensuing revenue, along with spending cuts, to pay down the debt.

The flawed Romney-Ryan approach is to believe that the debt problem can be solved entirely on the spending side.

That is a mathematical and moral impossibility.

The fundamental Democratic myth is that the country’s fiscal problems can be solved by focusing on, and asking for sacrifice from, only a tiny, vilified slice of the population: the wealthy.

The central promise of President Barack Obama’s 2008 campaign, reaffirmed in 2012, was that he would not raise taxes on the middle-class, defined as households earning less than $250,000 annually. At the same time, Democrats want voters to believe that Social Security and Medicare can be fixed, to the extent they need fixing, barely touching ever-rising benefits.

Certainly, in an era of growing income inequality, the wealthiest should be tapped first. Certainly, too, when it comes to entitlement programs, the wealthiest can be asked to pay more (as has happened, already, with Medicare premiums), and the benefits of the least well-off should be shielded.

Obama’s blended approach of spending cuts and tax increases is far more sensible than the no-new-taxes insistence of Romney and his fellow Republicans. Obama, however, has understated the magnitude of the task.

Debt threatens the nation’s prosperity and global standing. The deficits are growing slowly enough that a solution remains feasible, but the problem cannot be solved only by asking for more from the top 1 percent or 2 percent — still less by cutting them even more slack.

So long as both campaigns rely on such mythology, neither candidate can be elected with a mandate to handle the hard choices to come.

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