By John D. Stoll

California Gov. Jerry Brown has until the end of September to sign legislation that would require public companies based in the state to put a certain number of female directors on their boards. Before picking up the pen, he should give weight to how ineffective these quotas can be.

Such quotas have been snaking their way through Europe’s boardrooms for more than a decade. Researchers say they can help boost morale among aspiring female business leaders and potentially boost profits.

But mandates don’t effectively address two frustrating issues women face in corporate America: a stubborn pay gap and a thick glass ceiling.

For instance, in Norway—the egalitarian culture credited as the pioneer of such quotas—women CEOs are nearly nonexistent even though roughly 40% of directors are female. The Nordic nation’s pay gap is slimmer than the U.S.’s, but data from the Organization for Economic Cooperation and Development indicate Norway lags behind about 25% of the countries it tracks.

The debate over quotas is not inconsequential. California’s giant technology companies have enormous sway in our daily lives and in the stock market, but they are often criticized for being slow on advancing the careers of women and lacking foresight when it comes to how their products will affect society. Diversity in the boardroom could help Silicon Valley clean up its act.

‘I would like to see board governance get better in and of itself, regardless of gender.’

—Cindy Schipani, professor of business law at the University of Michigan’s Ross School of Business

One of the theories held by state Sen. Hannah-Beth Jackson, the bill’s co-sponsor, is that women on boards are more apt to look out for the women in their companies. “Women tend to have a different social circle than the men who have been running things at the higher echelons,” she said in an interview. “By expanding the circles, we learn about the tremendous talent we would never know about.”

The research, however, shows that good intentions don’t result in meaningful change.

Marianne Bertrand, a professor of economics at the University of Chicago’s Booth School of Business, co-wrote an analysis of Norway’s quotas in 2014 about a decade after they took effect. A key conclusion: “We find no robust evidence to support the view that the mandated greater share of women on the board improved outcomes for women employed in [these companies].”

Ms. Bertrand told me her team updated the research recently and “the results did not change much.”

What has changed is that there are more countries that have adopted quotas, providing further evidence that quotas are little more than a Band-Aid. In France, one of the bigger economies to adopt Norway-like reforms, 50% of directors at the largest companies are women, but 14% of senior executive officers are women.

What’s also troubling is France’s sizable pay gap, where the OECD says women make 90% of the median pay of men. French President Emmanuel Macron recently said that narrowing the compensation divide between men and women is a “grande cause nationale.”

Betsy Atkins, a board member at France’s Schneider Electric SE , Sweden’s Volvo Cars and Las Vegas-based Wynn Resorts Ltd. , said that when it comes to quotas in Europe “it certainly has not panned out.”

Ms. Atkins says she is encouraged by recent progress in the U.S. that have come without the overhang of government mandates. Female board representation has grown, particularly amid the #MeToo movement, even if it remains disproportionately low in California. And officials last year reported the first narrowing of the gender pay gap in the U.S. since the recession, with the female-to-male earnings measure improving to 80.5% in 2016 from 79.6% in 2015.

Fueling the momentum is the pressure big institutional investors are putting on companies.

BlackRock Inc. earlier this year, for instance, posted on its website that it wants to see at least two women directors on every board. The firm has written to 300 companies in the Russell 1000 that have fewer than two female directors and asked them to disclose their approach to boardroom and employee diversity.

“We believe that a lack of diversity on the board undermines its ability to make effective strategic decisions,” BlackRock’s head of investment stewardship, Michelle Edkins, wrote in those letters.

Cindy Schipani, a professor of business law at the University of Michigan’s Ross School of Business, said she applauds quotas that add women to the mix, but said “tokenism worries me.” In order for higher female representation to make a difference, “you have to be cognizant of how you organize the boardroom.”

Board selection has historically been a “networking game,” Ms. Schipani said. “I would like to see board governance get better in and of itself, regardless of gender.”

Ms. Atkins, the board member, said improved board governance requires more from directors than may be in the job description. At Volvo, for instance, she routinely meets with female employees at the car company and acts as a mentor even though Volvo’s chief executive isn’t specifically asking for the help.

“I happen to believe that is part of my job as a director,” Ms. Atkins said. If more female board members take this type of initiative, “leadership teams and CEOs will be thirsty and receptive.”

Scott Wagner, CEO of GoDaddy Inc., said he welcomes board scrutiny when it comes to meeting diversity targets. While the bulk of directors and executive officers at GoDaddy are men, an increasing number of senior leaders are women at the tech company and he often discusses this with the board.

Mr. Wagner, running a company once known for producing racy commercials, also makes public data about pay equality and gender representation. As of 2017, GoDaddy said it increased women at the company to 26% from 24% the year prior, and said it is paying men and women in like roles at “near parity across the company.”

“This is legitimately hard work,” he said, referring to continuing efforts to overhaul practices that govern pay and hiring. “The board can play a role in success, but they are not the answer in any way shape or form.” He said laying responsibility at the board’s feet is an empty gesture unless management is willing to be transparent.

“We started with just the facts,” Mr. Wagner said, referring to the company’s reliance on hard data as the basis for diversity initiatives. “The truth will set you free in all aspects of life.”

Write to John D. Stoll at john.stoll@wsj.com