Pay growth and prices picked up, keeping the Fed on track for rate increases.

A wage growth measure that the Federal Reserve watches closely climbed swiftly in the three months through June and prices increased sharply last month, fresh economic reports showed on Friday, developments that are likely to keep the central bank on track for future rate increases even as the economy shows some signs of cooling.

Prices climbed by 6.8 percent in the year through June, the fastest for the Personal Consumption Expenditures index since 1982. Inflation also jumped by 4.8 percent over the past year after removing food and fuel — which economists do to get a sense of underlying trends — a slightly larger increase than the 4.7 percent increase that economists in a Bloomberg survey had expected.

At the same time, a separate report showed that wages climbed briskly, albeit not enough to keep up with inflation. The Employment Cost Index climbed by 5.1 percent in the second quarter compared to the same period last year, and the index’s measure of wages and salaries also picked up strongly.

That combination is likely to reinforce the Fed’s determination to cool down the economy and wrestle inflation back under control. Central bank officials on Wednesday made their second supersized rate increase in a row — three-quarters of a percentage point — as they try to slow down the economy by making money more expensive to borrow.

“Wage increases and labor costs are still showing strong upward pressures, and that’s likely to keep the Fed raising interest rates over the next few meetings,” said Alan Detmeister, an economist at UBS who was formerly a central bank researcher. The employment cost number, he said, “was hot.”

While most Americans are not seeing their pay climb as quickly as prices, wage growth is proceeding rapidly enough that it could make it difficult for price increases to moderate back toward the Fed’s 2 percent annual inflation target. Companies are unlikely to stop raising prices when their labor bills are increasing rapidly, because doing so would eat into profits and possibly wipe out them out entirely.

Inflation has been high for more than a year, and central bankers are focused on trying to restrain demand and drive it lower before it becomes ingrained in the American economy. Once consumers and businesses begin to expect and accept rapid price increases, it may be harder to quash them.

While headline inflation probably cooled in July, because gas prices have dropped sharply this month, it is not yet clear how durable the change will prove. Likewise, while the economy is showing some signs of slowing, it remains difficult to tell just how pronounced the cool-down is: Consumers continued to spend in June, data released Friday showed, even after adjusting for inflation.

Jerome H. Powell, the Fed chair, said during his news conference this week that officials could raise interest rates by three-quarters of a point again at their next meeting in September, though he did not commit firmly to such a move. The Fed has nearly two months, and a lot of economic data to parse, between now and its next rate decision.

“I really do think that it’s important that we address this now and get it done,” Mr. Powell said at his news conference this week. He later added that “we are assigned uniquely and unconditionally the obligation of providing price stability to the American people. And we’re going to use our tools to do that.”

The Employment Cost Index picked up by 1.3 percent from the prior quarter, Friday’s report showed, more than economists in a Bloomberg survey expected. The wages and salaries measure picked up by 1.4 percent, an acceleration from the prior quarter, and is up 5.3 percent compared to a year ago before adjusting for inflation.

The details of the inflation data, meanwhile, showed worrying signs that price pressures remain strong. While the report’s core inflation measure had been slowing down on a monthly basis, heartening news for the Fed, it accelerated to 0.6 percent in June from 0.3 percent the previous month. The June monthly inflation number was the fastest reading in more than a year.

The fresh wage and price data come as the economy shows some signs of cooling. The economy sank for the second quarter in a row after accounting for inflation, data released Thursday showed, which is a common though unofficial definition of a recession. That data will be revised, and many economists cautioned against reading too much into it at a time when job growth remains robust.

“It doesn’t make sense that the economy would be in recession with this kind of thing happening,” Mr. Powell said earlier this week, referencing the millions of jobs employers that have added this year. “So, I don’t think the U.S. economy’s in recession right now.”

In fact, the fresh data suggested that at least some parts of the economy are holding up better than the gross domestic product numbers released Thursday suggested: Personal spending increased in June, even after accounting for inflation.

As the economy offers conflicting signals, investors are carefully eyeing the Fed, trying to guess how much it might raise interest rates. Officials are likely to closely watch inflation and wage readings for July and August — several of which will be released before their Sept. 20-21 meeting — for hints of how the price picture is shaping up as they weigh their decision.

If the economy does slow drastically, it could lead to more moderate price increases going forward. Already, large retailers including Target have said they have too much inventory on hand as consumers pull back, while Walmart has noted that consumers are buying fewer goods as they pay more for food and find their budgets strained.

But when it comes to consumption, the signals are not all lining up in one direction. For instance, used vehicles, which have been a big driver of inflation over the past year, have come back into supply and are showing price declines over time again — which could help to pull inflation lower in the months ahead.

“In our bifurcated economy, used-vehicle buyers are more likely to be more negatively impacted by higher prices for energy, food and rent,” Jonathan Smoke, chief economist at Cox Automotive, wrote in a research note this week.

New cars remain scarce, however, and are selling at higher price points, which could continue to keep some upward pressure on inflation.

“We continue to observe new-vehicle price inflation, limited discounting and record low incentives,” Mr. Smoke wrote.