Theresa May hadn’t been sworn in as Great Britain’s prime minister yet, but already some feminists were uncorking the champagne. “Is Theresa May Britain’s most feminist Prime Minister ever?” asked The Telegraph’s Radhika Sanghani. May is “passionate about women’s rights,” gushed Catherine Meyer, a former treasurer of the Conservative Party. The Globe and Mail’s Elizabeth Renzetti even went so far as to urge feminists to “get over Theresa May’s politics” because feminism is “about the primacy of choice in people’s lives: In this case, Theresa May has chosen to dedicate her life to a set of conservative political beliefs.”

The celebratory mood—and the claims that May’s ascension represented a victory for women—echoed many feminists’ response to Hillary Clinton’s clinching of the Democratic Party’s presidential nomination. “Pinch yourselves again, ladies,” Lowery wrote. “Come 2017, it won’t just be a woman running America. It could be women running America. And it will be women running the world.” Indeed, if Clinton wins, women will be leading the three largest economies in the West—the U.S., Germany, and the U.K.—and they will be running the Federal Reserve Board and the IMF.

We know what this means for the female leaders themselves: more power, influence, and wealth. What does it mean, though, for the women below them? Sarah Kliff and Matthew Yglesias, both of Vox, argued that Clinton’s election as president would be, in Kliff’s words, a “huge deal” for women, resulting in more female candidates and more pro-women policies. This echoes the argument that Sheryl Sandberg popularized in her best-selling self-help book, Lean In. “More female leadership will lead to fairer treatment for all women,” Sandberg wrote (emphasis hers). But the available evidence doesn’t support these bold assertions.

Some studies show that private firms with more women in senior management experience a narrower gender wage gap and enable more female employees to advance. But other studies have found that female leadership produces few benefits for women; this paper, for example, shows that female managers at a large grocery retailer were not associated with better pay for women. Studies of women in the sciences reveal that women faculty members evaluate female students and subordinates just as harshly, and unfairly, as men do. And researchers who looked into the effects of Norway’s gender quotas for corporate boards found that the only women who benefitted were the recently appointed board members. For women employees, pay, advancement within the firm, and career aspirations showed no improvement.

Research about the impact on women of female political candidates and elected officials is similarly mixed. Political scientist Jennifer Lawless, co-author of the new book Women on the Run: Gender, Media, and Political Campaigns in a Polarized Era, notes in a recent scholarly article that past research has showed that female legislators were more likely to focus on “‘women’s issues’ such as gender equity, child care, employee flex time, abortion, minimum-wage increases, and the extension of the food-stamp program.” But the most recent studies of Congress have found that legislators’ gender has no effect on roll call votes, even when the analysis is restricted to votes on “women’s issues.” Similarly, The Washington Post last year ran a simulation of congressional roll call votes to find out what would happen if Congress, which is 80 percent male, were 80 percent female instead. They found that a gender reversal would change nothing because “the results of congressional action are determined almost entirely by party identification, not by gender.”

That said, it’s not true that the gender of political candidates and elected officials makes no difference at all. There is some evidence that female candidates can act as role models and inspire other women to run. However, the effect is temporary and fades over time.

Research has also consistently shown that women tend to exhibit a more cooperative style of political leadership. Theresa May, however, may be an exception to that rule; she’s been criticized for being uncollegial and a loner. And though she’s by no means a conservative hardliner (she supports same-sex marriage and spoken out against racism in the British police force), she is not exactly a feminist. Yes, she once wore a T-shirt that said “This is what a feminist looks like,” but she’s also advocated restricting abortion rights and pushed for punitive anti-immigrant policies (as Home Secretary, she presided over the mistreatment and sexual assault of immigrant women at the notorious Yarl’s Wood detention facility).

Like her German counterpart, Angela Merkel, and another of the world’s most powerful women, IMF head Christine Lagarde, May is an advocate of fiscal austerity. According to The Financial Times, May has given no indication that she would reverse “the cuts to public services and benefits that have attracted criticism for hitting the poorest the hardest.” Austerity is a major women’s issue because austerity policies significantly undermine gender equality, women’s well-being, and women’s rights. When governments slash budgets, women lose out. They lose jobs because women are disproportionately likely to be employed in the public sector. And when governments dismantle services for children, the elderly, and the disabled, families depend on women’s unpaid care work to pick up the slack.

The records of women like May, Merkel, and Lagarde call into question the entire trickle-down feminism project: the idea that if you put a few elite women in leadership positions, the benefits will flow to the female masses. Ensuring that women enjoy equal opportunity to advance and that women’s voices are well-represented in the corridors of power is vital, of course. But what’s far more crucial is that our society eradicate the structural barriers to women’s equality.

Many feminists have noted America’s “stalled gender revolution”—the fact that women’s labor participation has flatlined, and that over the past decade, the gender pay gap has barely budged. Researchers have examined the reasons for these stubbornly persistent inequalities; however, they have not cited the lack of female leadership as a significant factor. Instead, they have pointed to structural issues, such as growing economic inequality, the lack of public programs like paid family leave and part-time work entitlements, the declining value of the minimum wage, and inflexible workplace policies that disproportionately reward long hours.

If Hillary Clinton is elected president, it will be an extraordinary moment in human history. Women will literally be running the world. Some feminists may be pinching themselves, but others like me are far more ambivalent about this scenario. The pulverizing of so many of the world’s highest, hardest glass ceilings is enormously gratifying. But far more important to the fate of the world’s women is the ideology of these leaders and the political constituencies they represent. On that score, the world’s most powerful women have a mixed record at best.