What emerged was a picture of young men reporting a slightly less feeling of well-being when they contributed twice as much to the household piggy bank as did their wives. Married young men who contributed equally reported feeling a great sense of well-being.

Related: 7 reasons women face greater retirement challenges than men

The difference wasn’t huge — the equal contributors’ average score was 3.33 on a 4-point scale, compared to 3.27 for the double contributors. But, Munsch told CNN Money, that slight difference wasn’t the whole story. The score dropped to 3.17 for men who were bringing in 100 percent of the dough.

Meanwhile, women in the study who were completely dependent upon their hubbies for household money reported the lowest overall well-being score: 3.08. And those who were the least economically dependent had an average score of 3.17 — equal to the “low” score of the men who were the main breadwinners.

Munsch said she thinks the trend is a reflection of the desire among millennials to have more “egalitarian relationships.” Men and women in that age group aren’t hidebound by the “traditional” roles their parents learned as children. And, in fact, it suggests that more men want to stay at home than in any past generation.

Related: Life choices aren't to blame for gender pay gap

“For women, because there isn't the expectation that you’re going to make as much money as possible for your family no matter the cost, I think they're much more likely to ask themselves, ‘Oh, is this what I really want to do?’” Munsch says. “And so when women do go into these high-level occupations, there’s more of a match between their preferences. So their psychological well-being is going to improve, whereas for men, it's not. … This is what [millennials] prefer when you ask ‘what kind of relationship do you want?’ They envision both partners contributing financially and both partners taking care of domestic responsibilities.”