Home building continued at a very high rate in June, the Commerce Department reported yesterday, surprising many economists who expected a slowdown in construction.

The Commerce Department also reported that personal incomes of Americans rose $14.4 billion in June, a 0.9 percent increase.

Williams Cox, chief economist of the Commerce Department, said the rise in personal incomes was "not bad, but not thrilling either. It was a respectable increase, but considering the not-respectable rate of inflation, there's not much left for real income increases."

The consumer price index for June will not be released for two weeks, but it is expected to be at a rate close to the rise in personal income - which includes, among other things, wages and salaries, rental return and dividends.

As inflation continues to eat into incomes, economists expect that consumers will begin to hold down their purchases of goods and services. Since consumers have been the driving force of the three-year-old economic recovery, business investment in plant and equipment will have to take up some of the slack to keep the economy growing - although at a slower pace than in recent months.

But economists received a pleasant surprise yesterday when the Commerce Department said construction of homes and apartments started at an annual rate of 2.099 million in June, up from a 2.081 million rate in May.

Michael Sumichrast, chief economist of the National Association of Home Builders, said the June level of housing starts was a "startling good figure." He said the report has caused him to revise upward the number of homes and apartments he expects to be started in all of 1978.

Economists had expected new home building to start declining by June - in part because of rising interest rates and in part because permits to build, an indication of future construction activity, turned down in April and May.

But permits rebounded sharply in June, in part the Commerce Department cautioned, because home builders in California sought to get new homes approved before a July 1 deadline setting stringent energy construction standards. Houses approved before July 1 were exempt from the rules.

Cox, of the Commerce Department, said that most economists still expect construction to drop off. "The question is how soon and by how much," he said.

Economists who expect a weak third quarter of economic growth are heartened by the housing starts figures too. Since spending for home construction generally reaches a peak two to three months after ground is broken, the June housing starts should guarantee strong construction spending employment in the third quarter.

In another development, Chase Econonometric Associates, a well-known forecasting concern, said that the United States, West Germany and Japan probably will not be able to fulfill the pledges made at this week's economic summit in Bonn.

Chase said that Japan will try, but be unable to reduce its trade surplus with the United States, that West Germany's plan to stimulate his growth rate is not likely to be successful and the United States will not reduce its inflation rate.