By Lydia DePillis, CNN Business

(CNN) -- Nearly seven years ago, a week after Black Friday, a few dozen workers walked off their New York City fast food jobs to demonstrate for higher pay. The median wage for fast food workers was $9 an hour, CNN reported at the time. The demand that would soon emerge as the movement's rallying cry — $15 and a union — seemed hopelessly ambitious, like a wild-eyed opening bid.

This week, the House of Representatives will vote on the Raise the Wage Act, which would make $15 an hour a reality for nearly all American workers by 2024, up from the current national minimum wage of $7.25 an hour. And it's expected to pass.

Although the bill still has to make it through the Republican-led Senate and past the President's desk — a near-impossibility, at least in its current form — the House vote sets the stage for a final victory if Democrats win back the Senate and White House in 2020. That would break the longest period America has gone without raising the minimum wage since it was originally instituted, back in 1938.

The path from then to now has been a remarkable journey. The first place to adopt a $15 minimum wage, in 2013, was the tiny town of SeaTac, which is almost entirely taken up by the airport that serves Seattle and Tacoma. Then came Seattle, San Francisco, New York City, all of California and New York State. Conservative states like Arkansas and Missouri started raising their minimum wages through ballot initiatives. Big companies like McDonald's and Walmart started inching their way up as well, and Amazon adopted a $15 minimum last year.

"It created this snowball," says Mary Kay Henry, president of the Service Employees International Union, which helped organize the Fight for $151 movement behind the scenes. "And then employers started raising the wage — all these people we didn't expect. It became natural to think about, why not try to do it at the federal level?"

The other reason raising the wage nationwide became important: Blue cities in red states that set local minimum wages, like Birmingham and St. Louis, were knocked back down by their conservative state legislatures. Other cities in places like Texas and Virginia never even tried, because their state laws already pre-empted localities from acting on their own.

Those are the places where a $15 minimum wage could make the most difference in peoples' lives, because so many workers there currently make less than that. For the same reason, they're also the places that face the highest risk of job loss as a result, because businesses will have to spend more to comply.

Will it cost jobs, and does it matter?

The fight over the effects of raising the minimum wage goes back to the 1990s, when economists Alan Krueger and David Card published trailblazing work casting doubt on the canonical understanding that artificial wage floors kill jobs. The most recent research, which now has a large number of minimum wage increases to draw from, generally finds little to no job loss following minimum wage increases — and that that they may actually create more jobs by bringing people into the labor force.

But most of those increases have been in relatively hot job markets, where — at least in the past few years — employers have no choice but to cough up to stay fully staffed, even if it means raising prices. That could be a different story in places with weaker local economies, where businesses are operating on thinner margins and don't have as much room to charge more.

The debate reached its apogee last week, when the nonpartisan Congressional Budget Office published a study taking into account most of the latest research and projecting the effects of the Raise the Wage Act. (The legislation also indexes the minimum wage to inflation after 2024, and phases out the tip credit, which allows employers to pay as little as $2.13 per hour if employees make up the difference in tips.)

The verdict: By 2025, the legislation would raise the wages of 27 million people and lead to 1.3 million fewer jobs, or 0.8% of total employment. They arrived at that conclusion by averaging numbers that represent what other studies had found to be the employment "elasticity" — the sensitivity of jobs to changes in the minimum wage.

The CBO heavily emphasized the uncertainty around those estimates, but both sides immediately seized upon parts of the report as evidence to support their positions, while some criticized its methodology.

Arindrajit Dube, a professor of economics at the University of Massachusetts-Amherst, said the CBO erred in equally weighting some of the older papers. He is among a group of authors to publish a study forthcoming in the prestigious Quarterly Journal of Economics that finds the elasticity is actually positive, and thinks that the CBO's report overestimates how many jobs will be lost.

"What we had to do is reconcile a lot of the past literature," says Dube, who provided input to the CBO on an earlier draft. "We specifically show what the problems are with some of those papers. What I see is peculiar choices that, in my opinion, are not justified that clearly affect the final outcome."

Others saw the CBO study as an effective synthesis of a complicated literature, and cause for scaling back the bill to something more modest. Daniel Hamermesh, a distinguished scholar in economics at Barnard College who has also published papers on the minimum wage and reviewed CBO's report ahead of time, called it "extremely good and extremely fair." Because of the employment effects of the $15 option, he favors $12, which is projected to have only a mild impact.

"This is closer to religion than anything else," Hamermesh said, of the two sides of the minimum wage debate.

Methodological quibbles aside, progressives argue that it's important to focus on the overall welfare impacts of raising the minimum wage, as well as possible government responses to help anyone who may end up without a job because of it.

David Howell, a professor of economics and public policy at the New School, says that raising wages significantly for those at the bottom of the income distribution is more important than simply preserving low-paid employment.

Half of the projected 1.3 million lost jobs are those held by teenagers, he points out, many of whom could be employed through work-based learning programs that would provide a better on-ramp to a career. Everyone else, given the strong demand in America for workers in fields like nursing and construction, could be retrained for higher-paid work.

"The American problem isn't quantity of jobs," Howell says. "It's quality of jobs."

Both sides dig in

After several months of inter-party lobbying, the Democratic caucus seems to have come around to support the bill. Members of the moderate New Democrat Coalition were reassured by the addition of an amendment that would require a study of the effects of the bill a few years in, leaving room to make adjustments if more jobs have been lost than anticipated.

The bill now has 203 cosponsors, and National Employment Law Project government affairs director Judy Conti says she's "very confident" the votes are there to pass it. Even Representative Terri Sewell, who had been trying to rally support for an alternative bill that would have tailored the minimum wage to local economic conditions, now says she'll vote for HR 15.

But no Republicans have voiced support for it. The minority leadership of the House Education and Labor Committee sent a press release slamming the Democrats' "radical and unprecedented mandate," and that "one job lost is one too many." White House economic advisor Larry Kudlow said recently he doesn't think minimum wages are a good idea at all, raising doubts that Trump would sign it. And although the powerful US Chamber of Commerce has said it's willing to meet in the middle on some higher minimum wage, $15 is out of the question.

"If the House proceeds and passes $15, that's a dead letter, it's going nowhere in the Senate, and any momentum for a compromise that could be enacted into law dies," said Neil Bradley, the Chamber's chief policy officer.

Bradley sent a letter to all House members last week outlining his willingness to support a number in the double-digit range. But he also requested labor law changes to help businesses absorb the increased costs, such as allowing after-hours electronic communications to not be counted for the purposes of overtime, and letting employers audit their own payrolls rather than having the Department of Labor do it.

To the charge that businesses already should be able to stomach higher wages, given the massive corporate income tax cut they received in 2017, Bradley responds: "We don't believe that the government has a claim on everyone's income and gets to return it when it sees fit."

There may be room for compromise on the House side in order to get something through the Senate this session.

"If they have an interest in talking, we'll obviously talk," said Representative Bobby Scott, who, as chairman of the Education and Labor Committee, has pushed the bill forward. He's coy, however, on how far he's willing to go. "I'm not going to say I'll accept this or that."

Even if nothing passes before the 2020 election, the union that pushed $15 onto the national stage sees the debate as a way to galvanize support among voters. The aim is to elect a Democratic Congress that wouldn't have to compromise at all — or persuade Republicans to change their minds.

"We want to use the Raise the Wage Act going to the Senate as a way to inject it into Republican-dominated states and districts," said Henry, of the SEIU. "The fight we want to have is to convince them that the political price they would pay for voting against it is not worth it."