Carly Fiorina speaks during a forum on Capitol Hill on March 16 in Washington, D.C. (Mark Wilson/Getty Images)

Women made 82 cents for every dollar men made in 2013. While that may just be an overall number, it's debated in very personal terms. Some explanations for the disparity are uncontroversial. More men work in lucrative professions, such as engineering and computer science. Other researchers have noted that women spend more time at home and less time working, losing opportunities to gain experience. Then there are more controversial explanations, such as outright discrimination.

Carly Fiorina, former chief executive officer of Hewlett-Packard who announced that she is running for president Monday, has her own theory.

Fiorina believes that unions are the culprit — and specifically, the seniority system. Although her reasoning isn't entirely clear, her point seems to be that women who work on contracts negotiated by unions are more likely to be paid based on how long they've worked for a firm, not how much they deserve. Suppose a woman leaves her job to be with her kids when they're little. If she returns to the workforce, she'll have accumulated less seniority, and she'll be paid less. She'll never be able to catch up with her male colleagues who kept working at the same firm, even if she deserves more.

"The single greatest impediment to equal pay for equal work is the seniority system, which pays not on merit and not on performance, but on time and grade," Fiorina has written. "And who is it who supports the seniority system? Unions, government bureaucracies, the very constituencies that Hillary Clinton and the Democratic Party represent and which support them."

There's some logic to Fiorina's argument. Andrew Biggs of the conservative American Enterprise Institute called it a "thought-provoking" explanation of the problem: A woman who takes time off from work to spend with her children may incur a permanent penalty if the formula for her salary in the future will be based solely on experience. If a woman "wants to get ahead and catch up after having kids or whatever, a unionized job is not the place to be," Biggs said.

Indeed, in that sense, Fiorina's explanation is not so different from many others — that women who take time off pay a penalty in earnings.

But by blaming this phenomenon on unions, Fiorina really goes astray. First, Biggs noted that only a fraction of the workforce is unionized. Only 13.5 percent of women are represented by a union, according to federal data, so it is hard to see how organized labor could be an important part of the reason for the gap overall.

In fact, even if mothers in unions aren't being paid what they deserve, many of them may still be better off than they would be outside of a union. The federal data show that women actually make more money if they're in unions, both in absolute terms and relative to men.

Full-time female employees who were represented by unions were paid 91 cents for every dollar their male colleagues earned in 2013, compared to 81 cents for those women who were not. The typical female worker in this category made $893 a week, compared to $676 for a woman working without labor representation.

These figures reflect raw data and are not adjusted for differences in experience or education.

"It's good when we see our Republican friends talking about the pay gap as a real problem," said Lisa Maatz of the American Association of University Women, adding that Fiorina's statements betrayed a "fundamental misunderstanding" of why the gap exists.

Thinking more carefully about Fiorina's argument, it isn't surprising that women's and men's paychecks are more alike in unions.

Many unions have demanded that equal pay for equal work be stipulated in their contracts. Of course, women often work in different capacities from men, but generally speaking, unions tend to raise wages for the workers in the shop who are paid the least and to lower them for those make the most. To the extent that women are working in occupations at a firm that pay less, they'll be better off in a union.

Also, even if collective bargaining puts mothers at a disadvantage, they're even worse off bargaining on their own. Research done by Linda Babcock, among other economists, has found that women hesitate to ask for a raise, knowing they are less likely to succeed if they do, and that their bosses may view them less favorably afterward.

"Unions are the great equalizer in terms of pay," said Babcock, a professor at Carnegie Mellon University.

On a more basic level, Fiorina's objection to the seniority system — on the grounds that it rewards employees solely for experience, not their merit — has problems. Some women may deserve more than their experience would suggest, but others deserve even less. A system that bases pay on experience won't necessarily bias pay for women in one direction or the other. Indeed, on average, experience is a good guide to how productive workers are, which is why economists rely on it to try to understand differences in pay.

Indeed, outside of unions, pay appears to be even more strongly dependent on experience, explained Claudia Goldin, an economist at Harvard University and an expert on women and the labor force.

The largest differences in pay by gender are in white-collar professions in which employees rarely belong to a union. Female lawyers make 79 cents for every dollar that men make in the same occupation. Female physicians and surgeons are paid 72 cents. Female managers make 76 cents.

For chief executive officers — Fiorina's old job description — the gap is somewhat smaller, but still larger than the average gap across all professions. Female executives take in 80 cents for every dollar that their male counterparts make. Again, this federal data is not adjusted for differences in education and experience.

There are other data that suggest that unions help bridge the pay gap. By one estimate, nearly all workplaces with a union presence offer parental leave, compared to about four in five without a union. Even if a mother makes less when she gets back on the job than she would if she hadn't taken time off, at least she can count on having a job. That's another reason that many women might look for work in a unionized industry, particularly if they're thinking of starting a family.

"I don't get it," Goldin said when asked about Fiorina's argument. "The places where unions have been the strongest are the places that have very narrow gaps."