We start with the household side of the model. Our goal in solving the household side of the model is to show that children are normal goods, and that with Cobb-Douglas preferences, households choose to spend a constant share of their income producing children. Households have utility u(C,B) over consumption C and children B. The household has income Y which it sp ends on C at cost 1, and B at cost . Therefore the household's maximization problem is: maxC,B u(C,B) s.t. Y = C + B (a) Write the Lagrangian for the household's problem and derive the rst order con- ditions (b) Supp ose that the household's utility function is u(C,B) = C1B, with 0 < < 1. Using this functional form, combine the first order conditions to eliminate the Lagrange multiplier. (c) You now have two equations (the budget constraint and your answer from b) and two unknowns (C and B). Use these equations to show that births B are always proportional to income Y .