Why is the erech (Vayikra 27:1–8) of a woman less than that of a man? And why does erech depend on age? Sources, please.
Rabbi Samson Raphael Hirsch's approach, if I recall correctly, is that the number 3 represents the family (father+mother+child), and an additional 2 represents the community. Men are obligated to be involved in both, whereas women are only obligated in the former.
- mature female = 10 [= completion] x (3 [=family]) = 30 shekels
mature male = 10 [= completion] x ( 3 [=family] + 2 [=community]) = 50 shekels
toddler girl, proportional fraction of adult duties = 3 shekels
toddler boy, proportional fraction of adult duties = 3 + 2 = 5 shekels
young female, working towards completion (10) in the family realm = 10 shekels
- young male, working towards completion (10) in both family and public realms (x2) = 20 shekels
(Note that the Torah's stage of "working towards completion" is ages 5 to 20, or 15 years. If we split that in half, we get two stages: 5--12.5, and 12.5--20. That fits well with the concept of bar/bat mitzvah -- half that time is maturing without the technical obligation, and half the time with.)
- retired female, back to her prior state of responsibilities = 10 shekels
- retired male, given a reduced role of communal responsibilities = 10 (family) + 1/2 * 10 (community) = 15 shekels
The Gemara does observe that retirement age proportionately devaluates a man more than a woman, hence the Talmudic saying:
An older lady in the house is a treasure; an older man in the house is a waste of space.
Which sounds similar to the way one lady described having her retired husband home (I believe this is attributed to the Cosby show):
It's like having a grand piano sitting in your kitchen. It looks good, but it's always getting in the way!
I don't have a source to cite, but historically speaking, since the ancient world revolved entirely around how to continue living, i.e. earning enough money to feed yourself and your family, and since women didn't need to provide income for the family like the men had to, therefore the monetary value of a working man was worth more than that of women, children, and the old. He had more working potential.