As the economy slowly and haltingly recovers from the recession, a major question about the job market looms, namely, whether those who gave up looking for work in the past few years are ever coming back.

A related question, with more immediate implications for the Federal Reserve in particular, is what will happen to the highly elevated number of long-term unemployed workers.

With any luck, Friday’s jobs report will help shed light on the answers to those questions.

The Congressional Budget Office estimated in February that the U.S. at the end of 2013 was 6 million jobs short of full health. That is, between the elevated unemployment rate and the the depressed labor force participation rate, there are 6 million Americans who would like a job but can't find one.

As the unemployment rate falls toward the normal 5-6-percentage-point level, however, Chairwoman Janet Yellen and other Fed officials will have to try to determine by how much the job market remains short of full health. That will be a major decision to be made over the course of the next few months as they consider the proper stance of monetary policy.

As for the long-term unemployed, recent research suggests that long-term unemployment might be less relevant in affecting inflation and wage growth than short-term unemployment -- another issue that Yellen and company will have to address.

Researchers at the Federal Reserve Bank of New York reported in February that models of wage growth focusing on short-term unemployment better explained the economy than ones that use the headline unemployment rate.

And a new report from a group of economists that includes former Obama economic adviser Alan Krueger found a similar dynamic.

Not only does the long-term unemployment rate exert little pressure on wages and inflation, the researchers found, but people out of work for long periods of time also seem disconnected from the labor market in a number of ways. The study’s authors report that even in states experiencing faster recoveries, the long-term unemployed have trouble finding work.

That line of research has obvious implications for understanding today's labor market. There were 3.8 million Americans out of work for 27 weeks or longer in February, according to the Bureau of Labor Statistics -- 37.0 percent of the total number of unemployed, a historical high ratio. In fact, the short-term unemployment rate is already back to pre-crisis levels.

At her first press conference earlier in March, Yellen was asked if the new research on long-term unemployment suggests that the labor market is tighter than it normally would be with the current headline rate of unemployment. She said that the Fed would continue looking at the new evidence, but that “it would be tremendously premature to adopt any notion that says that that is an accurate read on either how inflation is determined or what constitutes slack in the labor market.”

Over the weekend, Goldman Sachs economists also pushed back against the idea that the high long-term jobless rate meant that the weakness of the labor market was overstated. Instead, they wrote that the other studies' results were likely driven by the shift in the Fed's response to inflation beginning in the 1980s and that the “behavior of U.S. wage and price inflation since the late 1990s does not support the notion that the short-term unemployment rate is better than broader measures of labor market slack for predicting wage and price inflation.”

In other words, it’s an unresolved issue, and one that could prove important in the months ahead as the unemployment rate continues falling.

The BLS will report on the jobs market for March on Friday. Deutsche Bank’s projections are for 275,000 new jobs and a 6.5-percent unemployment rate, down from 6.7 percent in February.

Although February's 175,000 new payrolls number was weak relative to the average month's gains during the recovery, it was stronger than many had expected. Job growth has been weak in recent months, but it's not yet clear to what extent the slowdown was influenced by weather -- a strong March report would suggest underlying strength.

As for other events happening this week, the Senate is expected to debate both extending the long-term unemployment insurance program that expired in December and the Obama administration's proposal to lift the minimum wage from the current $7.25 an hour rate to $10.10, indexed for inflation.

On Tuesday, the Senate Budget Committee is scheduled to hold a hearing on opportunity and mobility featuring Nobel Prize-winning Columbia economist Joseph Stiglitz and Harvard economist Raj Chetty, one of the authors of an influential new study showing that the rate of mobility in the U.S. hasn’t changed significantly in decades.

This post was updated for clarity at 9:51 a.m.