By Kate Davidson

Most Federal Reserve officials agree the economy is at or getting very close to what economists consider full employment, the rate below which inflation starts to rise.

What they can’t agree on is what should happen next.

As the unemployment rate hovers below 5% and job gains moderate, one of the key debates at this week’s Fed policy meeting will be how much farther the Fed can let it decline without risking runaway inflation.

On one side are officials, including governors Lael Brainard and Daniel Tarullo, who say there’s still room for improvement in the labor market. Allowing unemployment to fall further below its current 4.9% level would give more Americans—especially minorities—a chance to come back to the labor force and share in the gains of the expansion, they say.

The counterargument from officials such as San Francisco Fed President John Williams is that letting the jobless rate get too low could cause prices to surge, forcing the Fed to ratchet up short-term interest rates faster than they’d like. That could trigger a downturn that would hurt minorities most.

“I understand the desire to try to help everybody in the economy,” Mr. Williams said Sept. 6 in Reno, Nev. “But I think that just running an overly hot economy for too long does risk…creating the conditions that then could lead to a recession that undoes all of that.”

When faced with inflation jumps in the past, the Fed doesn’t have a great track record of combating rising prices without causing a sharp upturn in the unemployment rate.

In the 1960s and early ’70s, the Fed let the jobless rate fall below estimates of full employment. Several bouts of runaway inflation and the severe recessions that followed have made officials wary of testing that boundary today, despite little evidence of inflationary pressures.

Fed Chairwoman Janet Yellen has expressed sympathy for both camps, and added in a footnote to a March speech that the Fed could be wrong in its estimate of the full-employment level. Officials’ median estimate then was 4.8%, and she said the true level was likely lower. And if so, “a lower level of unemployment might be needed to fully eliminate slack in the labor market, drive faster wage growth, and return inflation to our 2% objective.”

With officials divided over whether to raise rates, they are inclined to leave them unchanged at their meeting Tuesday and Wednesday. They have held their benchmark short-term rate since December in a range between 0.25% and 0.5%.

The debate among Fed officials is occurring against a backdrop of growing pressure from liberal lawmakers and labor activists, who urge the central bank not to raise rates right as African-Americans, Hispanics and other low-income workers are starting to see economic progress.

They say some measures of labor market slack suggest there are still gaps to be closed. The share of prime-age workers participating in the labor force—those 25 to 54 years old—is still below where it was before the crisis, at 81.3% in August, compared with 82.9% in August 2007.

The median U.S. household income rose 5.2% last year to $56,516, the biggest annual gain recorded since the survey began in 1967, the Census Bureau said last week. Still, it remained 1.6% below the 2007 level and 2.4% below the all-time recorded high reached in 1999.

At the same time, the Fed’s preferred inflation gauge continues to run below its 2% objective, and inflation expectations have softened.

“In the presence of uncertainty and the absence of accelerating inflationary pressures, it would be unwise for policy to foreclose on the possibility of making further gains in the labor market,” Ms. Brainard said in a speech in Chicago Sept. 12.

Part of the challenge for both sides is that nobody knows where that sweet spot is—that is, how low the jobless rate can go and for how long before inflation starts to rise too much.

The Congressional Budget Office estimates that full employment—also known as the nonaccelerating inflation rate of unemployment, or Nairu—is 4.8%, the same as the Fed’s estimates in March and June and a hair below the current jobless rate. But CBO’s estimates also have varied over time.

That’s enough for some officials to worry that hotter inflation could be right around the corner, based on the long-held belief that falling unemployment eventually generates higher wages, which in turn will push up prices and generate stronger inflation. But that so-called Phillips curve relationship hasn’t been working the way many Fed officials have come to expect.

“No one can prove them wrong, except for the fact that we keep coming around the corner and there’s no inflation,” said Jared Bernstein, a senior fellow at the Center on Budget and Policy Priorities and a former Obama administration adviser.