The government's index of leading indicators--guide to future trends in the economy--rose by 0.6 percent last month, the first increase since July and a sign that the recession may be nearing bottom, the Commerce Department reported yesterday.

The December rise in the leading indicators was greeted as "welcome news" by Commerce Secretary Malcolm Baldrige. Five of the 10 indicators contributed to the improvement, with a rise in building permits the largest factor. The indicators rose in only three months of 1981, and had fallen in successive months since a 0.1 percent increase in July.

But despite the climb in December, Baldrige cautioned that a further decline in output and a rise in unemployment in the early months of this year are still likely. Commerce economist Robert Ortner warned that later revisions to the December leading indicators could pull the figure down. But the rise reported yesterday "is a further suggestion that the worst of the declines in the economy have occurred already," he added.

Early estimates by the Congressional Budget Office, however, indicate the economy is likely to show a slight drop in the real Gross National Product for 1982 over last year, sources said yesterday. The administration has predicted a small increase in real GNP between 1981 and 1982, with the growth concentrated in the second half of this year.

Sources said the CBO forecasts inflation of about 7 1/2 percent this year, and an unemployment average of 8.9 percent. The forecast shows the jobless rate climbing above 9 percent early in the year before dropping back. The budget experts predict federal deficits of about $160 billion in 1983 and $195 billion in 1984 with no new spending cuts or tax increases, sources said.

Treasury Secretary Donald Regan said this week that the administration expects the deficit to decline by $10 billion a year, after coming in just below $100 billion in 1982. Part of the discrepancy is that the administration assumes the enactment of new domestic spending cuts to be outlined in its budget on Feb. 8.

Although many analysts expect the economy to bottom out this spring, there is considerable disagreement among them over how strong the recovery will be. High interest rates and tight money could limit the upturn, some say. Administration economists predict further interest rate declines and growth at an annual rate of about 5 percent in the second half of the year.

The Labor Department, in the meantime, reported that the recession led to a steep decline in productivity in the final quarter of 1981 as business cut back production more quickly than it cut back employment. Productivity plunged by 7.2 percent at a seasonally adjusted annual rate, the fastest drop since records began in 1947, the department said.

In another economic development, the Commerce Department reported an improvement in the merchandise trade balance in December, as imports dropped by 13.3 percent and exports slipped only slightly. Oil imports plunged during last month from $6.44 billion to $5.16 billion. The December trade gap totalled $1.59 billion, a drop of $2.8 billion from the November trade deficit. For 1981 as a whole, however, the United States posted a $39.68 billion merchandise trade gap, up from $36.36 billion in 1980.

The Commerce Department reported that part of the increase in the indicators came from a rise in the money supply. The Federal Reserve is now trying to slow money growth. Other factors contributing to the increase were a rise in stock prices, a slowdown in the layoff rate and an increase in new orders for consumer goods and materials.

The five indicators pushing the overall index down included the average work week and contracts and orders for new plant and equipment. President Reagan has predicted that business will step up its investment because of the tax incentives legislated last year. So far, however, the recession has dampened investment. Baldrige commented yesterday that "prudent managers will continue their efforts to hold down costs while, at the same time, making plans to meet the strengthening demands which lie ahead."

One piece of bright news from the productivity report was that hourly compensation rose by only 5.7 percent at an annual rate in the last three months of 1981, the smallest increase for several years. If continued, such a slowdown in wage costs will help to slow inflation.

Despite the productivity decline recorded in the fourth quarter, 1981 as a whole showed a 1 percent increase in productivity over 1980.