2 (7 points) Consider a modified version of the Gary Becker-style micro model of fertility choices, in which both men and women have child-care responsibilities. Suppose that the man in the household has Tm hours of time to devote to market work and/or child care, and that he must contribute cm hours per child in care time. His wage in market work is wm. The woman has Tw hours available for market work and/or child care; she must contribute cw hours per child in child care. Her wage is ww. The household has no other sources of income. In addition to parental time, each child requires p in goods such as food and clothing. The household's utility function is U = Na X 1a , with a being a preference parameter such that 0 < a < 1.

(a) Algebraically derive and graph the budget constraint and the time constraints for this modified version of the model. Please provide detailed labels in your graph and add explanations.

(b) What is the desired (utility-maximizing) number of children N and other consumption X ? (Your solution should be in symbols.) Make sure to state clearly any assumptions

that you are making.

(c) Show in a carefully labeled graph how the budget constraint would change if the man's wage were to increase from wm to w 0 m. (d) What role does the man's child-care obligation cm play in the position of this new budget constraint (the constraint with the higher wage w 0 m) relative to the position of the old constraint (the one with wm)? In particular, what would happen if the man's wage rises to w 0 m but cm = 0, in which case the man would have no child-care time obligation at all. Please explain and show this special case in a graph.