Consider a household utility $U(X_P, X_C) = X_P^{0.5}X_C^{0.5}$, where $X_P$ is parent's home production and $X_C$ is children's home production. Assume the wife spends $h_w$ hours at working, and the husband spends $h_h$ hours working with hourly wage $w_w$, and $w_h$. And child does not work. (i.e. Time spends on home production for wife and husband are denoted by $t_w$, and $t_h$). Assume that $X_P$ is produced directly by $Y_P$, where $Y_P$ is parent's consumption from goods. And $X_C$ is produced by $Y_C t_h^{0.5} t_w^{0.5}$, where $Y_C$ is the goods used in child's home production.
