Americans' personal income increased 0.5 percent in June, but consumer spending continued to rise at a slightly faster pace than income, the Commerce Department reported yesterday.

Personal income rose $16.8 billion in June and personal outlays rose $15.3 billion, Commerce said. Take-home pay -- personal income less taxes -- declined 2.2 percent last month because federal income tax refunds, generally disbursed in February and March, were delayed until April and May, increasing the totals for those months.

Excluding the catch-up in refunds and other one-time factors, after-tax income increased $10.7 billion in June and $11.2 billion in May, Commerce said.

In a separate report, Commerce said that housing starts, which dropped 13.5 percent in May, rose 1.9 percent in June.

Economists said they were disappointed in the housing report, but they expect economic activity to pick up later this year because of the decline in mortgage interest rates in the past few months.

"Mortgage commitment rates are down to their lowest levels in five years, home prices are rising moderately and deposit inflows to savings institutions are improving," said Commerce Secretary Malcolm Baldrige. "The outlook for housing is good for the rest of 1985.

"Construction activity is contributing to current output and employment, and also will help achieve faster overall growth in the second half," Baldrige said.

The personal income report gives economists a clue about whether Americans' finances are strong enough to support continued buying and further economic activity. Housing starts tell economists how strong future activity in construction will be and its indirect effects on other industries such as durable appliances and furniture.

Because of the decline in interest rates, economist Alan Murray of Citicorp Information Services said he was surprised that starts for single-family homes declined in June to an annual rate of 1.027 million units from 1.041 million in May.

"We expected strength to be in single-family houses" rather than in starts of apartment buildings, Murray said. However, he said that several factors, such as lower mortgage rates, "point to some further increase in single-family starts."

Starts for buildings with five or more units increased 10 percent from May to an annual rate of 579,000 units, Commerce said.

One reason for the slowdown in starts of single-family homes as well as the sluggishness of personal income is the failure of the employment situation to improve substantially. For five consecutive months, the unemployment rate has remained at 7.3 percent as the number of workers entering the labor force has just about matched the number of new jobs created, economists said.

With fewer jobs created, incomes have not been able to rise strongly, economists said.

Building permits issued last month dropped 4 percent to an annual rate of 1.71 million.

In a separate report, Commerce said that personal outlays -- personal consumption expenditures, interest paid by consumers to businesses and personal transfer payments to foreigners -- rose $15.3 billion in June, following a $16.5 billion increase in May.

Personal consumption spending rose $13.9 billion in June, after a $15.1 billion increase in May.

Durable goods purchases declined $2.6 billion, compared with a $1 billion increase the previous month. The June decline and May increase were attributed to automobile sales levels.

Purchases of nondurable goods improved, increasing $3.1 billion in June, following a decline of $700 million in May. Purchases of services increased $13.4 billion, compared with $14.9 billion in May.

Personal saving was $105.5 billion, compared with $180.6 billion in May. Personal saving as a percentage of after-tax income was 5.3 percent in May, up from 5.2 percent in April. A June figure was not available.