1. Consider a household utility U = ln(X), where X is nal goods from home production that directly affect the utility. And for X to be produced, Y has to be used. Let th, tw be the time spent by husband and wife in the production of X, And X = Y (min{th, tw}) 1 be the production function for nal home production good X. And hh, and hw be the hours they spend in labor market, earning wh, and ww.

(a) Set up a maximization problem for this household, where they maximize household's utility, subject to the family's budget constraint, and time constraints, by choosing how to allocate th, tw, and how much Y to buy.

(b) Consider yourself as a head of this household. By looking at this production function. Assume wh = ww. Do you think you will make the husband work more or the wife?

(c) What if wh < ww? Do you think you will make the husband work more or the wife?

(d) BONUS POINT: Solve the problem, and show me what is the optimal time the husband and the wife spent in home production and working. (i.e. Solve for t h , t w, h h , h w)