On the pay gap, interesting paper published just this month that looks at the gap in Denmark, and determines that the remaining gap is largely accounted for by the 'child penalty', and that this child penalty has grown in importance as a factor of the overall gap from 40% to 80% since 1980 (I interpret this as other factors gradually declining until the child penalty is by far the most important unaddressed element of the gap).
www.nber.org/papers/w24219.pdf
Interesting paper. It shows that the child penalty impacts women roughly equally in 3 ways - labour force participation (women stop working), hours of work (they go part time), and wage rate (their earnings stop growing at the same rate after child birth, they are less likely to achieve a managment position after having kids).
One really interesting bit was that the paper finds statistical evidence that 'child penalties are transmitted through generations'.
"We find that female child penalties are strongly related
to the labor supply history of the maternal grandparents, but not the paternal grandparents, even
after controlling for a rich set of family characteristics. For example, in traditional families where
the mother works very little compared to the father, their daughter incurs a larger child penalty
when she eventually becomes a mother herself. Our findings are consistent with an influence of
nurture in the formation of women’s preferences over family and career"
So, if you were born into a traditional household where the woman prioritizes children over career, you're likely to make the same choices.
The real question, for me, is how much you ascribe this to the free choices of women who want to sacrifice some earnings for family life, versus an undesirable and socially-pressured limiting of women's options based on inflexible notions of a woman's role and unhelpful structures for fairly distributing the 'child penalty'. I don't think this paper sheds much light on that, but it's an interesting look at the numbers, regardless.