As a campaign to raise the minimum wage as high as $15 has achieved victories in such places as Seattle, Los Angeles and New York, it has bumped up against a harsh reality: Plenty of scofflaw businesses don’t pay the legal minimum now and probably won’t pay the new, higher wages either.

Some economists, labor activists and regulators predict that without stronger enforcement, the number of workers getting cheated out of a legal wage is bound to increase in places where wages rise.

Estimates on the size of the problem vary, but the Bureau of Labor Statistics said that in 2014, about 1.7 million U.S. workers — two thirds of whom were women — were illegally paid less than the federal minimum of $7.25 per hour.

Other studies put the number higher. A report by the Labor Department in December estimated that in New York and California alone, there are 560,000 violations of the law every week, representing $33 million in lost income.

The Labor Department investigates violations and is doing a brisk business in enforcement cases. During the past fiscal year, it said, it recovered $270 million in back wages for 270,000 workers.

But the agency’s roughly 1,000 investigators, who police 7.3 million businesses employing 135 million workers, don’t enforce state and local wage laws, for the most part. That means cities and states that increase their minimum wage above the federal rate of $7.25 are on their own.

That’s causing some concern that, without a robust enforcement mechanism, many workers could wind up being left behind.

Twenty nine states have a minimum wage higher than the federal rate, but anti-poverty activists have been campaigning hard for municipal lawmakers to bypass both Congress and their state legislatures and set wages much higher.

— Associated Press