Both unemployment and the inflation rate dropped in June, the Labor Department repoerted yesterday, but neither bit of good news seemed likely to affect President Carter's apparent search for a broad new energy and economic policy.

Unempolyment dipped from 5.8 percent in May to 5.6 percent last month, the lowest level since August 1974, the department said.

The price of finished goods, as measured by the Producer Price Index, formerly the Wholesale Price Index, rose 0.5 percent, the smallest monthyl rise since last August.

Administration economists welcomed the improvements but cautioned that neither was a sign of fundamental charge for the better in the economic outlook.

Even President Carter has begun to warn that a recession is "more likely than it was before" because of recent oil price hikes and economists generally expect inflation to remain high for months to come.

The improvement in inflation was due almost entirely to a 1.2 percent decline in consumer food prices. The cost of other finished goods -- paced as usual by energy prices -- rose 1.1 percent in June, and annuual rate of 13.7 percent in June, and home heating oil was up 8.4 percent. None of the latest round of crude oil price increases set by the Organization of Petroleum Exporting Countries (opec) was reflected in last month's hikes for gasoline and heating oil.

The improvement in the unemployment picture was due entirely to a decline in teen-age joblessness, the Labor Department said. Teen-ages' unemployment rate fell from 16.8 percent to 15.3 percent, while the number out of work dropped by 130,000 persons, to 1,455,000, on a seasonally adjusted basis.

But Commerce Department economist William Cox said the apparent drop could have been a one-month aberration due to the problem of adjusting numbers for the usually seasonal influx of teen-age workers into the job market when school close.

Total employment rose by 440.000 according to the department's survey of households. This large gain and the slight increase registered in May followed a large drop in April caused mostly by the nationwide Teamsters strike.

Even with the gains of the last two months, total employment, at 96.8 million in June, was almost the same as it was in March. "The economy seems to have slowed to something close to a halt," observed Commerce's Cox.

Furthermore, the length of the average work week for production and nonsupervisory workers dropped from 35.7 hours in May to 35.6 hours last month. This was a 0.3 hour decline compared to the March level. This implies that total production in June probably was less than in March, analysts said.

The latest numbers are expected to have little if any impact on the midyear economic forcast and revised budget estimates the administration will present to Congress next week.

That forecast will predict a period of "zero growth" in gross national product but not a recession, according to administration sources. At the same time, Carter economists will acknowledge the distinct possibility that a recession will occur, blaming it on the impact of the OPEC oil price increases, as the president had done.

Vitually all private economists are now predicting a recession that would be worse than that of 1970 but not nearly as severed as 1974-75.

The drop in the prices of consumer foods was the third consective monthly decline. Between March and June, foods fell 2.7 percent but little of that improvement has showed up in prices in grocery stores.

Profit margins of food wholesalers and retailers have increased since March as producer prices have dropped, according to economists at the Agriculture Department. However they expect more of this drop of begin to affect retial consumber prices soon.

Consumer durables prices rose 0.4 percent in June, following a 0.7 percent increase in May. Passenger car prices charged by manufactures rose a bit less last month than in May, but precious jewelry and household flatward prices rose sharply.

In another report yesterday, the Federal Reserve Board said that consumer installment credit increaed by $3.7 billion in May, down a notch from April's $4 billion level.

Credit for automobile purchases rose only $1.2 billion compared to $1.3 billion in April. That was the fifth consecutive month in which the amount of auto credit outstanding has increased by less than in the previous month.

Sales of new cars have dropped sharply as consumers, struggling to keep pace with inflation, begin to reduce postponable purchases. The large decline in retail sales, led by the auto dealers, is one reason economists are so confident the economy is heading into a recession.

During 1978, the amount of consumer credit outstanding rose 19 percent. The growth in May was at a 16 percent annual rate, further evidence of a decline, particularly when adjusted for inflation. CAPTION: Graph 1, Unemployment Rate, Month by Month for 1978 and 1979, The Washington Post; Graph 2, Wholesale Price Index For Finished Goods, The Washington Post