Ezra Klein has a chart that puts the offer Barack Obama gave John Boehner in context:
What Obama offered Boehner was an opportunity to take the Bush tax cuts off the table. So though $800 billion in revenue sounds sizable, it’s only half as much in total revenue as the White House’s April proposal, two-fifths as much as Simpson-Bowles wanted, and one-fifth what we’d get if the Bush tax cuts expire next year.
And that's in addition to substantial cuts to entitlements. The White House may be playing this well politically in the short term, but the idea that this kind of offer is part of some incredibly complex rope-a-dope defies reason. Obama is giving the GOP almost everything they want in order to raise the debt ceiling, and they're saying no. But it's not because he wants them to say no. He wants a deal, bad enough to take a really awful one.
Some of the quotes passed on by White House aides in response to the cuts Republicans are requesting are amusing, but let's not forget what the "balanced" approach to deficit reduction the president is offering looks like (also from Ezra):
This is "balance" and "shared sacrifice" only in the loosest possible definition of each term.