The latest sign that inflation is cooling makes it more likely the Federal Reserve will be able to gain enough confidence to cut interest rates this fall.

The odds of a cut in September jumped Thursday after the release of favorable new numbers from the Consumer Price Index (CPI), with traders now pricing in an 83% probability of an easing at the Fed's meeting on Sept. 17-18.

"I think it puts September firmly on pace for a cut," Peter Tchir, Academy Securities macro strategy head, told Yahoo Finance.

Some Fed watchers even think that a cut at the Fed's July 30-31 meeting is now a possibility if some other pieces fall into place.

"The Fed could very well lower rates sooner than September if the labor market softens at a faster clip," said Quincy Krosby, chief global strategist for LPL Financial.

The Consumer Price Index on a "core" basis — which excludes volatile food and energy prices the Fed can’t control — rose 3.3% year over year in the month of June. That was a tenth of a percent below expectations and below the level seen in May.

Month-over-month core CPI was also encouraging, rising 0.1% after increasing 0.2% in May.

The "muted" month-over-month increase “strengthens the case for a September rate cut," said Paul Ashworth, chief economist for Capital Economics.

A lot, however, still depends on the next reading of the Fed's preferred inflation gauge — the "core" Personal Consumption Expenditures index (PCE) — as well as further cooling of the jobs market, Ashworth added.

San Francisco Fed president Mary Daly didn’t commit to any specific timing on rate cuts in a conversation with reporters Thursday following the CPI release except to say that the time is "closer than six months ago.”

"With the information we have received today, which includes data on employment, inflation, GDP growth and the outlook for the economy, I see it as likely that some policy adjustments will be warranted,” Daly said.

Inflation will continue to come down gradually, she added, but "when exactly that happens and when it would be appropriate to make an adjustment of policy is still unclear." She said she still needs to see more information.

Another central bank policymaker, St. Louis Fed president Alberto Musalem, said Thursday that the new CPI number "points to encouraging further progress towards lower inflation" but that "I will be looking for more evidence that inflation can be expected to converge to 2% going forward."

Richard de Chazal, macro analyst for William Blair, said the fact that June marks the third consecutive month of more moderate inflation growth helps "confirm that inflation is firmly back on a downward trajectory."

But he still does not expect a smooth path down to the Fed's inflation goal of 2% because the annual rate of change comparisons start to become tougher through the second half of this year.

"To help justify the start of rate cuts, the Fed will need to shift the focus to a decelerating labor market, rather than continuing to rely entirely on the softening of inflation to do all the heavy lifting on rates," de Chazal said.

"Today’s report, and the subtle shift to a more balanced focus on slowing employment growth by the Fed, helps firmly put a September rate cut in the sights."

Federal Reserve Chair Jay Powell made it clear earlier this week that he is in fact paying closer attention to a cooling jobs market.

"The most recent labor market data do send a pretty clear signal that labor market conditions have cooled considerably compared to where they were two years ago," he said during testimony before US lawmakers. "This is no longer an overheated economy."

For a long time, the Fed has been focused on the inflation side of its mandate, holding rates at a 23-year high in an effort to cool the economy.

But Powell says with the job market getting back to where it needs to be, the Fed is looking closely at both sides of its mission — stable prices and maximum employment.

Daly, the San Francisco Fed president, is also turning more of her focus to the job market as she sees the risks to inflation and the job market coming into more equal balance.

"I'm looking at the labor market which is not only coming into better balance but...suggests that we're to the point now where additional labor market slowing is more likely to result in a rise in unemployment," Daly said Thursday.

The focus on the job market comes after last week's release of an unemployment report for June that showed signs of a gently cooling labor market, with the unemployment rate ticking up a tenth of a percent for the second month in a row to 4.1%.

While the unemployment rate of 4.1% is still historically low, it’s up from 3.4% early last year.

Powell also made it clear this week that the central bank is in fact inching closer to feeling comfortable about rate cuts, telling lawmakers that he was encouraged by evidence of cooler inflation and that more "good data" would help get the Fed to where it wants to be.

The inflation numbers "have shown some modest further progress" after some hotter readings in the first quarter, "and more good data would strengthen our confidence that inflation is moving sustainably toward 2%,” he said in his testimony.

Powell, however, stopped short of saying whether cuts could begin in September. He also cautioned that he is “not prepared” yet to say that the central bank is sufficiently confident inflation is coming back to the goal of 2%.

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