Consider a household utility U = In(X), where X is final goods from home production that directly affect the utility. And for X to be produced, Y has to be used. Let th, tu be the time spent by husband and wife in the production of X, And X = Ya(min{th, tw})l-a be the production function for final home production good X. And hh, and how be the hours they spend in labor market, earning Wh, and Ww. (a) For this family to produce X. From the production function, can you tell the relation- ship between time spent by husband and time spent by wife? (Are they complement, substitutes or neither?) (b) Give some examples of X. ( i.e. What are kind of home production goods can you think of that this production function is mimicking? Try to look at question (a) for hint.) (c) Please rite down the time constraints for both husband and wife. Given that they all have 1 as total time endowment. (Assuming besides working, they both devote their time into home production.) (d) Given the price of Y is p, And total family non-labor income is V. Please write down the family’s budget constraint. (e) 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. (f) Consider yourself as a head of this household. By looking at this production function. Assume wn = Ww. Do you think you will make the husband work more or the wife? (g) What if wn < Ww? Do you think you will make the husband work more or the wife? (h) 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;,,, hj, h)

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