The obvious reason to prevent bubbles is that crises are bad for the economy. Seven years after the peak of the 2008 crisis, the Fed still has not been able to drive unemployment or inflation back to normal levels. The unemployment rate has fallen to 5.1 percent — a level usually associated in the past with a robust economy — but that figure overstates the current health of the labor market.

Image Eric Rosengren, head of the Boston Fed, has argued that financial stability should be part of the objectives of monetary policy. Credit... Pat Greenhouse/The Boston Globe, via Getty Images

Prices, meanwhile, rose just 0.3 percent in the 12-month period ending in August, far below the 2 percent level the Fed would like to achieve to support healthy spending and investment.

The basic problem for regulators is that crises are hard to predict. A 2012 study by the International Monetary Fund concluded that only about one-third of credit booms ended in crashes. Many instead produced permanent economic benefits. And even in retrospect, the researchers found it hard to identify clear warning signs.

So popping bubbles probably means curtailing some beneficial booms, too.

“We’ll have to make a choice about how much growth we are willing to give up in good times to limit the likelihood of a future financial crisis,” said Loretta J. Mester, president of the Federal Reserve Bank of Cleveland.

Other officials and experts also share Mr. Dudley’s doubts about the Fed’s power to pop bubbles. American regulators have fewer tools than some of their European counterparts, and those tools are distributed among a number of agencies that have little history of cooperating either quickly or effectively.

“The current U.S. institutional setup is likely to fail in a crisis and will do less to prevent a crisis than it should,” said Adam S. Posen, president of the Peterson Institute for International Economics. “And we are likely to suffer from this.”

Frederic S. Mishkin, a former Fed governor, noted that financial firms tend to resist increased regulation, often with considerable success. “They’re going to hire a lot of lawyers to figure out how to get around these regulations and undermine them,” Mr. Mishkin, a Columbia University economist, said.