The cost of living rose only moderately in September for the third month in a row, continuing relief from the heavy dose of inflation consumers faced during the first half of the year.

The Labor Department reported that its consumer price index increased 0.3 per cent last month, the same as in August and slightly slower than the 0.4 per cent increase in July.

For the last three months the costs of living has been rising at an annual rate of 4.2 per cent. For the three months ended in June it was rising at an annual pace of 8.1 per cent and was near 10 per cent in March and April.

White House press secretary Jody Powell said President Carter was "gratified and encouraged by the continued moderation in consumer price inflation," and said he was "particularly pleased that the slowdown in consumer price increases over the past several months has been broad-based."

Food prices, which surged early in the year because of last winter's record cold, rose only 0.1 per cent in September.

During the three months ended in June, grocery store prices were increasing at an annual rate of 12.8 per cent. In the three months ended in September the rate had slowed to 0.6 per cent.

Powell said that while the "new figures do not mean our problems with inflation are over, they do clearly indicate a welcome respite for consumpters."

Administration economists and most others says that the basic rate of inflation is closer to 6 per cent than the current 4.2 per cent rate. Food prices, which have contributed heavily to holding down the climb in th cost of living, are expected to pick up alightly in the coming months.

While consumer prices were continuing to moderate, interest rates were rising. Only a day after the White House publicly criticized the Federal Reserve Board for its money-tightening, interest-raising policies, several major banks boosted their prime interest rates to 7.75 per cent from 7.5 per cent. (Details on page D10.)

Annalysts said that the recent increases in personal income coupled with the reduced inflation should put consumers in good shape for the heavy Christmas buying season. Slow-downs in consumer buying slowed economic growth markedly in the third quarter.

A spokesman for the President's Council of Economic Advisers said that administration economists are "reasonably optimistic" about retail sales in the final months of the year and noted that sales appear to have picked up sharply in the first few weeks of October.

Food prices are "helping to build some confidence in the consumer sector," according to Richard Peterson, chief economist for continental Illinois National Bank in Chicago.

But Peterson noted that the lull in production activity that has affected the country for the last several months could result in lower hours and reduced paychecks if it continues.

The Labor Department said that its consumer price index stood at 184 per cent of its 1967 average in September, which means that a selection of goods and services which cost $10 in 1967 cost $18.40 last month.

The consumer price index measures chances in prices of about 400 goods and services that are judged to be representative of purchases by the typical urban family of four Changes are adjusted to account for seasonal ups and downs.

Except for a sharp rise in the price of fresh vegetables, most food prices declined, reflecting six months of falling farm prices. While consumers are benefiting from the decline in farm prices, farmers are upset over the low return they are getting on their big harvests.

Congress has approved huge increases in the budget to help prop up farm prices.

Prices of commodities other than food, such as clothing and fuels, rose 0.2 per cent, compared with 0.3 per cent in August. Fuel oil and coal prices incresed 0.4 per cent, about half as fast as they had been rising, but gasoline prices rose 0.6 per cent, after a slight rise in August and declines in June and July.

Gas and electricity rates rose 0.4 per cent, well below recent rises, while the 0.8 per cent jump in medical care services was faster than the 0.6 per cent climb in August.