WASHINGTON — Senate leaders on Sunday failed to produce a fiscal deal with just hours to go before large tax increases and spending cuts were to begin taking effect on New Year’s Day, despite a round of volatile negotiations over the weekend and an attempt by Vice President Joseph R. Biden Jr. to intervene.

In seesaw negotiations, the two sides got closer on the central issue of how to define the wealthy taxpayers who would be required to pay more once the Bush-era tax cuts expire.

But that progress was overshadowed by gamesmanship. After Republicans demanded that any deal must include a new way of calculating inflation that would mean smaller increases in payments to beneficiaries of programs like Social Security, Democrats halted the negotiations for much of the day.

The Republican leader in the Senate, Mitch McConnell, made an emergency call to Mr. Biden in hopes of restarting negotiations, and the White House sent the president’s chief legislative negotiator to the Capitol to meet with Senate Democrats. Soon after, Republicans withdrew their demand and discussions resumed, but little progress was made.