WASHINGTON – The unemployment rate is suddenly sinking at the fastest pace in a half-century, falling to 9 percent from 9.8 percent in just two months — the most encouraging sign for the job market since the recession ended.

More than half a million people found work in January. A government survey found weak hiring by big companies. But more people appear to be working for themselves or finding jobs at small businesses.

The steepest two-month decline in unemployment since the Eisenhower administration is the latest sign that the economic recovery is picking up speed.

The service sector and manufacturing are growing again at pre-recession rates. The Dow Jones industrial average closed above 12,000 this week for the first time since mid-2008. And retail sales have reached a five-year high.

“It is not all rosy. But we seem to be headed in the right direction,” said economist Chris Rupkey at Bank of Tokyo-Mitsubishi. “The recovery is on track — warts and all.”

Yields on government bonds rose after the unemployment report came out, a sign that bond traders think the job market is improving and will lift the economy after a year and a half of only modest growth.

An unemployment rate of 9 percent remains very high by historical standards. But the swift decline in the rate could also lift confidence at a time when businesses and individuals are already spending more money, fueling more hiring and still-more spending.

Unemployment has not been this low since April 2009.

“It’s the thinking, ‘I survived so far and I’ll make it through,’ ” said economist Nigel Gault of IHS Global Insight. “ ‘I can buy the things I postponed buying over the past several years because I’m not worried about my job.’ “

It could also encourage people who had given up to look for jobs again, which might push the unemployment rate up temporarily.

Still, economists think the unemployment rate will fall below 9 percent by year’s end — a far brighter outlook than they had a few months ago.

Those with jobs are making a little more, too. Average hourly earnings rose 8 cents to $22.86 in January. Over a 40-hour workweek, the increase works out to $3.20, a couple of cups of coffee.

Wages have risen nearly 2 percent over the past year, faster than the rate of inflation, which means people have more spending power.

The Labor Department survey of company payrolls showed a net gain of 36,000 jobs in January. That’s scarcely one-fourth the number needed to keep pace with population growth.

The number of people who described themselves as self-employed rose by 165,000 to 9.7 million in January, the report said. That was the most since May.

In addition, some economists said, the unusually snowy winter might have suppressed hiring at businesses in January. Some construction companies shut down, for instance, and transportation companies cut jobs for couriers and messengers. Neil Dutta, an economist at Bank of America Merrill Lynch, said the figure would probably rebound in February.

“The thumbprints of the weather were all over this report,” he said.