As the debt ceiling loomed last year, President Obama believed that Republicans had him over a barrel. They had won the midterm election. More important, calling the GOP’s bluff seemed too big a bet: Defaulting on the debt risked plunging the global financial system into chaos.

The president’s mind-set as the “fiscal cliff” approaches is far feistier. He won reelection in a campaign that centered on higher taxes for the wealthy. Even more, in the White House view, the hazard of cliff-jumping is way less than the danger of ceiling-hitting.

So rather than extend the Bush tax cuts for higher-income taxpayers, the president is willing to risk another recession — a move he would blame, and with good reason, on Republican intransigence.

Consequently, the president has adopted a doubly tough posture as cliff negotiations commence. First, he is looking for an amount of new tax revenue far greater than House Speaker John Boehner was unable to deliver last year: $1.6 trillion over 10 years, including a $1 trillion installment to get past the cliff. That is an opening bid — an updated version of earlier proposals — but a daringly high one.

Republicans have signaled a grudging willingness to raise new revenue. What matters, Boehner said last week, is whether this comes from raising tax rates or from “a growing economy, energized by a simpler, cleaner, fairer tax code, with fewer loopholes, and lower rates for all.”

Which brings us to the second, and as it may develop, far trickier issue: tax rates. You might think this would be the easier deal, given how much we’ve heard about the allure of broadening the base and lowering rates.

But the rate standoff could be difficult to bridge. Republicans insist on starting with the current code — the Bush tax cuts made permanent — and cutting from that already low level.

The White House position is that the Bush tax cuts for wealthier households should expire, with tax reform to proceed from there. As the administration sees it, the math of lower rates from loophole-closing does not work in the real world of politics.

“I think that there are loopholes that can be closed,” the president said at Wednesday’s news conference. “But when it comes to the top 2 percent, what I’m not going to do is to extend further a tax cut for folks who don’t need it, which would cost close to a trillion dollars. And it’s very difficult to see how you make up that trillion dollars . . . just by closing loopholes and deductions. You know, the math tends not to work.”

Treasury Secretary Tim Geithner was even more emphatic about the arithmetically possible. “There is a lot of magical thinking about how much money you can raise from tax expenditures,” he said Tuesday at a conference sponsored by the Wall Street Journal. “I don’t see how you do this without higher rates.”

I’d be delighted to see the top tax rates revert to the 36 percent and 39.6 percent level they were under Bill Clinton. The Republican argument that this would stifle job creation by hurting small businesses is bogus — just a sliver of small businesses earn enough to be affected — and is refuted by the Clinton economy.

At the same time, to quote Gen. David Petraeus on Iraq, “Tell me how this ends.” If Obama insists that the Bush tax cuts expire and Republicans refuse to raise rates, where is the potential compromise to avert the plunge?

In part, the question comes down to math. Is the administration correct in its analysis that various reform proposals — cap the value of itemized deductions for the wealthy at a certain dollar amount or a percentage of income, for example — won’t generate enough revenue to cover the cost of extending the tax cuts?

If Republicans believe they have a workable tax-reform solution, they should come forward with it. Because in my conversations, the administration sounds pretty convinced that one doesn’t exist.

Obama was appropriately careful not to draw red lines about specific rates or to slam doors on compromise solutions. Yet those who took away from the news conference the lesson of the president’s conciliatory language may be overly optimistic.

The administration’s reading of the political and economic climate feels less like the debt-ceiling debacle and more like last year’s argument over extending the payroll-tax holiday. It is betting that, threatened with being tagged with raising middle-class taxes and risking a recession, Republicans will capitulate. That is one high-stakes wager.