The myth of matriarchy is one of patriarchy's oldest inventions. Those stories of primitive warrior queens, buxom mother goddesses and tribes of Amazons are no evidence at all that women did once rule the world. As most anthropologists have recognised for decades, these are cautionary tales invented by men to justify their own dominance. The important point about matriarchy in most of these myths is that the women rulers made a terrible mess of things or they imposed regimes of such terror (who would like to be ruled by the Amazons, after all?) that there was no choice but to overthrow them. And that, according to the logic of the stories, is why we now have patriarchy.

Most of the literary fantasies about women in charge, ancient or modern, make the same point. In Aristophanes' 5th-century BC comedy, Assemblywomen, for example, women have taken charge of Athens, and bring in a whole series of hare-brained pseudo-egalitarian measures – including the requirement that men had to sleep with ugly old women before pretty young ones. It would have been enough to reconcile the average Athenian man to any kind of male government, no matter how incompetent.

Hanna Rosin's The End of Men is yet another version of the old story, but with a twist: she claims that we are now actually living under a matriarchy, or very soon will be. This, I imagine, will come as something of a surprise to most women on the planet. But Rosin's book teems with examples of the decline of modern man and the rise of woman in terms of money, education, employment and power. Men, she argues, have been the major victims of the recession, failing to adapt to the decline in manufacturing industries and to the challenges of post-modernity. If you believe her, vast numbers of the male American working class have been transformed from proud breadwinners to unemployable couch potatoes. Their womenfolk, on the other hand, have seized the opportunities offered by the economic changes of the last few years; they have retrained, requalified and taken the driving seat.

In 2009, she points out, American women outnumbered men in the workforce for the first time. They now outnumber men on degree courses by a ratio of three to two, and are even beginning to "crowd out men" on science and engineering courses. More and more families depend on the woman as the main breadwinner (almost 64% in Washington DC). And of the 15 most expanding job categories in the US, 12 are now dominated by women. "Indeed," she writes, "the US economy is in some ways becoming a kind of travelling sisterhood." And it is not just in North America. Examples from all over the globe – the first openly lesbian head of state in Iceland, the 80% of college-educated Brazilian women who aspire to top jobs, the female majority in the Rwandan parliament – make the new matriarchy seem more or less a fait accompli.

I don't know exactly what to make of all these facts and figures, but I strongly suspect that they have been rather carefully selected and presented. Rosin herself admits that the dominance of female breadwinners in Washington DC "is largely because the city has so many poor single mothers". It is also the case that the majority of those expanding job categories she refers to are in the caring (ie relatively low-paid) professions, and it is these that underpin the "travelling sisterhood" that supposedly defines the US economy; or, as she puts it, "professional women leave home and enter the work force, creating domestic jobs for other women to fill" (not "sisterhood" in the traditional feminist sense then). Besides, she has very little to say about the fact stressed in most studies of gender in the US workforce: namely that women's pay still lags far behind that of men, across the board, with women's salaries averaging 20% less than those of their male counterparts. If this is a budding matriarchy it is a surprisingly poorly paid one.

The headline thesis of The End of Men is, of course, eyecatching (and, according to the book's optimistic publicity material, the title has already "entered the lexicon" as dramatically as The Second Sex or The Feminine Mystique). But, on a closer reading, it is not entirely clear how far Rosin believes it, or at least how far she thinks that women are the straightforward beneficiaries of the power-shifts she detects.

At one point, she writes of visiting a community college in Kansas City and getting into an elevator with a young woman still dressed in her medical assistant's scrubs – who promptly "fell asleep between the first and fourth floors, so tired was she from studying, working, and taking care of her kids by herself". This sounds like the usual downside of women's careers for 100 years or so: "getting on" requires superhuman energy and stamina.

That is certainly the message of several of the case studies in which she explores a new style of male-female domestic partnership that goes hand in hand with greater female power. She calls these "see-saw marriages", in which men and women divide the responsibilities differently at different periods of the relationship – with sometimes the man, sometimes the woman taking on more of the domestic, or more of the money-raising duties. It looks fine on paper, but again, it turns out to be the same old story of two jobs not one (wage earner and domestic goddess) for the women, even the relatively affluent.

Take the unfortunate Sarah, a successful lawyer married to a "mediocre house dude" (his words) called Steve, who stays at home to mind the baby. Sarah's daily routine includes making the family's breakfast before she leaves for work and washing the diapers when she gets back in the evening. Steve defines his domestic tasks as "smearage containment", which means putting the dirty diapers in the sink, but not actually washing them. Sarah, needless to say, is her own worst enemy in refusing to use disposables. So much for the new matriarchy.

It is perhaps to the credit of Rosin (and of her honesty) that if you did not know the title of the book you might very likely imagine that whole sections of it had been written to support precisely the opposite argument. One chapter, entitled "The Top", is a fairly standard discussion of how women still haven't made it there, or not at any rate to the very pinnacle. She does stress a few exceptions. "The job of Secretary of State has been virtually reserved for a woman," she insists (though I'm not sure that three out of the last eight really suggests female dominance). Or: "The number of female heads of state, although still small, has suddenly doubled in the last several years" (is that from four to eight, or five to ten?).

But for the most part she puzzles, as many have before, about women's absence from the boardroom, and their apparent unwillingness to demand the pay rises that their male counterparts achieve. "Women," she explains, "carry psychological baggage into the workplace: a lingering ambivalence about their ambition, a queasiness about self-promotion, a duty to family that they can't or won't offload on to their husbands, etc etc."

So how does all that fit with the arguments about the new matriarchy? All Rosin can offer is the old gradualist answer. "The world does not flip upside down overnight. Men have been in charge for about forty thousand years, and women have started edging them for about forty. So of course there are still obstacles at the top."

It turns out then that we haven't actually put a new matriarchy in place. This version is as mythical as any other – and Rosin, to give her her due, probably knows it.

• Mary Beard's Pompeii: the Life of a Roman Town is published by Profile