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“When you do a budget for your house or for your company, you don’t just do a total budget,” he said. “You do a budget that has all the details. And you make tough choices about where you’re going to spend more and where you’re going to spend less.”

The solution, Mr. Walker believes, is for Mr. Obama and Congress to raise the borrowing limit through the end of this year while working on an elusive “grand bargain” on deficit reduction, to include long-term spending cuts on entitlements and other programs, as well as tax reform and tax increases. But many Republican lawmakers say the recent “fiscal cliff” deal, which raised taxes on families earning more than $450,000, was the only tax increase Mr. Obama will get.

Addressing the debt ceiling, historically, has been a bipartisan affair. Republicans point out that Democrats in 1979 created the so-called “Gephardt rule,” named for former House Majority Leader Richard A. Gephardt of Missouri, which attached increases in the debt ceiling to annual budget bills. Although it fell out of favor periodically and was discontinued in 2011, Congress did use this budget-related tool to agree on borrowing limits four times.

A report by the Congressional Research Service last month said the debt limit “provides Congress with the strings to control the federal purse.” Congress enacted a debt limit in 1917 and has raised the limit more than 70 times since 1962. Until recently, those actions were usually routine.

But with the cumulative federal debt now surpassing the value of the nation’s annual total economic output, the question of raising the borrowing limit has grown more contentious. With the stakes so high and the opposing positions so entrenched, the proposed solutions have ranged from the president invoking the 14th Amendment, which says the validity of the nation’s debt “shall not be questioned” (the White House has ruled out this option), to having the Treasury mint a $1 trillion coin and deposit it in the Federal Reserve (an option that the White House hasn’t specifically ruled out.)

Mr. Walker called the coin option “one of the dumbest ideas in the history of mankind.”

“Why would we waste the valuable metal?” he asked. “Why not just use Monopoly money?”