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“We will only solve this problem when we finally get the spending under control,” Mr. Toomey said. “And the debt ceiling, and after that, the continuing resolution expiration, those are the vehicles that give us the opportunity to insist on making progress on the real problem.”

But Rep. Sander M. Levin of Michigan, the ranking Democrat on the Ways and Means Committee, echoed the president’s view that the fiscal cliff agreement has created a guideline for future talks on deficit reduction.

“This package is vital for future deficit-reduction efforts, setting the stage for a balanced approach from here on out by delaying sequestration through 1-1 revenue-to-spending cuts,” Mr. Levin said.

Both sides dispute just how much the fiscal cliff bill accomplishes. While the president is hailing it as a deficit-reduction measure, the agreement would boost the deficit by $4 trillion over the next 10 years, when compared with a scenario in which Congress did nothing.

Democrats said the legislation will lower deficits by about $600 billion over 10 years compared with current policy, because doing nothing was never a realistic option. The nonpartisan Congressional Budget Office said the bill will increase spending over the next decade by about $330 billion.

The agreement includes an extension of lower tax rates for families earning up to $450,000, and a patch to the alternative minimum tax that adds roughly $3.6 trillion to the deficit over the next decade. Other business and energy tax extenders would add a further $76 billion. The extension of unemployment benefits for about 2 million people will cost roughly $30 billion, and the so-called “doc fix” on Medicare reimbursements will cost an additional $25 billion.