Where does the 40*3 term come from? I'm assuming the 5/(40*2) is 5 years of subsidy out of two 40-year careers. I can't figure where the 3 term is coming from unless it's 2 children from 3 total people or something else that I'm missing.

Yes it’s three careers. I say just before it if you have two kids close together. Two years is a normal gap. So seven years of subsidy out of three forty year careers. The mother and two children each have forty years of “productivity”.

(I think) That only works if neither child is eligible for this leave during their career.

If you imagine Adam and Betty have two kids Charlie and Debbie who have two kids Earl and Felicia who have Gary and Helen… each generation receives N years of leave income from (2M - N) years of work income. You can’t take Betty, Charlie, and Debbie’s work income (or total productivity) against just Betty’s leave income.

I think you take Adam and Betty’s combined leave against their combined M years and M years minus N years of leave, C&D’s N against their 2M-N, etc. (you can time-shift it however you like if you want to think of some generation paying for a different generation, but the long-run math doesn’t change for replacement rate births)

In your example, this seems like 7 years of 73 years (9.6%) is going to this program, not 7/120 (5.8%)

It’s quite hard to follow what you mean there so I’ll assume the extra detail is correct. If you wanted to model more detail think about the growth of the effect if those children have children. That really starts to drive down that percentage.

My point was to show roughly that worrying about the cost of productivity loss when having children is the only way to sustain human productivity doesn’t add up.