New evidence of a fast-developing recession was released yesterday by the government, and a top Carter aide said unemployment could hit 8 percent early in 1981.

Alfred Kahn, the president's chief inflation adviser, told a mayors' conference here that unemployment, 6.2 percent in March, could reach 7.5 percent by the end of this year and 8 percent a short time later.

"The country now faces the dilemma we have so long feared, the twin ugly evils of accelerating inflation and the long-predicted recession," the outspoken Kahn declared.

As Kahn spoke, the Commerce Department reported that its index of leading indicators, which usually fore-shadows changes in the economy, plunged 2.6 percent in March, the third largest one-month decline since the series began in 1947.

The Labor Department added its own gloomy news on unemployment. Initial claims for unemployment benefits, seasonally adjusted, were filed in the week ended April 19 by 605,000 people -- the highest level in the 13 years such information has been collected.

Separately, the Labor Department said that, because of inflation and higher taxes, as of last fall it took an average $12,585 for a family of four to have even a "lower" standard of living in an urban area. For the same family to have an "intermediate" standard of living required $20,517, and a "higher" standard took $30,317.

In each case, the Washington metropolitan area figure was the fifth highest of the 24 areas surveyed across the nation.

The budgets are calculated for a hypothetical family with a 38-year-old husband employed full time, a non-working wife, a boy of 13 and a girl 8.

Such a family living in the Washington area would have to spend $13,631 to have the lower standard of living. Costs are higher only in Anchorage, Honolulu, San Francisco and Seattle.

An intermediate standard of living in Washington took $22,206, surpassed only in Anchorage, Honolulu, Boston and New York. The higher budget took $32,636 here, and was higher in the same-four cities.

The lower budget rose 9 percent between 1978 and 1979, the department said. The intermediate and higher budgets rose more, 10.2 percent and 10.6 percent respectively, because of large increases in homeowner costs and Social Security taxes.

The Commerce Department released other economic statistics indicating a recession is underway.

New orders for factory goods fell 0.9 percent in March to $154.1 billion as a result of fewer purchases of autos, steel and other metals. The drop was the largest since last July.

On the other hand, orders for non-defense capital goods -- items such as machine tools and other equipment in which business invests -- rose 6.3 percent to $23.5 billion because of purchases of aircraft and non-electrical machinery. Some economic forecasters believe the recession will not be servere unless business investment falls sharply from present levels.

Also yesterday, the F.W. Dodge Division of McGraw-Hill reported that the value of new construction contracts declined market's reaction to anti-inflationary restraint is no longer limited to housing," the division's chief economist, George Christie, said.

Kahn's remarks surprised other administration economists, who said the official forecast calls for a 7.2 percent unemployment rate in the fourth quarter of this year, a further increased to about 7.5 percent in the first half of 1981, and a drop to 7.3 percent in the final three months of next year.

A minority of private economists, however, are predicting unemployment rates as high as 8 percent during this recession. Most do not believe it will go that high.

The plunge in the index of leading indicators was the result of sharp drops in stock prices, building permits and inflation-adjusted measures of the money supply and new orders for consumer goods from manufacturers.

Also contributing to the decline were a reduction in the length of the average workweek in manufacturing, an increase in the layoff rate in manufacturing, and a drop in some raw materials prices.

In a separate report, the Labor Department said that from February to March fewer people quit their jobs, fewer were hired, and more were laid off.