1. Consider a household utility U(XP, XC) = X^0.5 P X^0.5 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 write down the whole maximization problem this household is facing. [ i.e. What are the things that the household should be maximizing over, and what are the constraints that they are facing.]

b) Please write down the first-order condition with respect to hours wife spends at home.

c) Please write down the rst order condition with respect to hours the husband spends at home.