“It’s not a charade. It’s not a gimmick — unless what the president sent us is the same,” said Rep. Mick Mulvaney, a freshman Republican from South Carolina who sponsored Mr. Obama’s proposal for purposes of the debate. “I would encourage the Democrats to embrace this landmark Democrat document and support it. Personally, I will be voting against it.”

But no Democrats accepted the challenge.

They have their own alternative they wrote, which closely tracks the president’s deficit numbers, though it changes the details of his plan. That plan will receive a vote on Thursday, as will Mr. Ryan’s proposal.