In recent months, Republicans have seemed impervious to outside pressure. They don’t fear the president. They don’t fear the debt ceiling. They don’t fear the market. But on Tuesday afternoon, when the Congressional Budget Office announced that House Speaker John A. Boehner’s (R-Ohio) proposal for raising the debt ceiling would save significantly less money than promised, Republicans’ response was instantaneous. They backed down. Immediately.

Within hours, Boehner vowed to rewrite the bill to deliver the promised savings. The CBO soon concluded that Boehner’s revised plan would save $915 billion — news that quickly resuscitated the proposal.

As in previous high-stakes debates, the CBO has become a key arbiter in the legislative process as lawmakers scramble to react to its analyses. But how has a small agency of budget analysts gained such authority in Washington, helping to determine whether a bill lives or dies?

Congress created the CBO through the 1974 Congressional Budget Act. Until then, Congress had relied on the budget expertise of the executive branch, and the CBO was intended to give legislators an independent source of information.

The CBO director — currently Douglas W. Elmendorf — is appointed by the House speaker and Senate president pro tempore and oversees some 250 economists, policy analysts and other experts. Elmendorf was previously at the Brookings Institution, the White House’s Council of Economic Advisers and the Treasury Department — all institutions his predecessors have passed through. Although some CBO directors have worked in the private sector, most have remained within the realms of policymaking, think tanks and academia.

Although legislators regularly contest the CBO’s findings, they have deferred to its authority more often than not. Lawmakers’ frustrations are rarely attributed to bias on the part of the number-crunchers. Instead, they are more often related to the agency’s methodological caution: The CBO uses existing evidence and historical analogues when judging legislation, and some say that makes the agency too cautious when assessing new approaches.

On several major occasions, the CBO’s numbers have almost single-handedly doomed some of Congress’s biggest priorities, regardless of which party had appointed the agency’s director at the time. In 1994, for example, under a Democrat-appointed director, the CBO decided to include the cost of premiums in evaluating President Bill Clinton’s health-care proposal. As a result, the estimated cost of the bill skyrocketed — a moment many believe was the beginning of the end for that proposal.

Such experiences have strengthened the CBO’s credibility by proving the agency’s political independence. At the same time, the CBO’s power has led some to question whether the agency has an undue amount of influence on Washington policymaking. “You know you’re not God,” Senate Finance Committee Chairman Max Baucus (D-Mont.) told Elmendorf in 2009. But some think such complaints speak more to the frustration of dealing with an honest agency than the CBO itself.

Whether legislation passes “depends on the judgment of members of Congress,” Elmendorf told The Post in 2009. “We’ll provide information that helps them make that judgment. But the decisions are theirs.”