1. Consider a household utility U(XP, XC) = X0.5 P X0.5 C, where XP is parent's home production and XC is children's home production. Assume the wife spends hw hours at work, and the husband spends hh hours working with hourly wage ww, and wh. And the child does not work. (i.e. Time spends on home production for wife and husband are denoted by tw, and th). Assume that XP is produced directly by YP, where YP is the parent's consumption of goods. And XC is produced by YCt 0.5 h t 0.5 w, where YC is the goods used in child's home production.

a) Please derive the relationship between husband's time spends at working and that of wife's.

b) Looking at the child's part of the home production: XC = YCt 0.5 h t 0.5 w . is telling you this household gets utility from goods spent on the kid and also together with parents' time. Can you think of any example for XC, YC, th, tw in real life? (Please explain why.)

c) Related to b), do you think th, and tw are substitutes or complements? (Please explain why.)