1. Household Behavior (2pts) We start with the household side of the model. We saw in the data that the investment rate in the economy is stable in the long run. Our goal in solving the household side of the model is to show that household choose to invest a constant share of income. Households have utility u(C, I) over consumption C and investment I. The household has income Y which it spends on C and I. Therefore the household's maximization problem is: max u(C, I) C,I s.t. YCI (a) Write the Lagrangian for the household's problem (b) Derive the first order conditions (c) Combine the first order conditions to eliminate the Lagrange multiplier. (Hint: If you're looking around for the relative price, it's one!) (d) Suppose that the household's utility function is u(C,I) = C-I, with 0 < < 1. Rewrite your answer to (c) using this functional form. (e) You now have two equations (the budget constraint and yoru answer from d) and two unknowns (C and I). Use these equations to show that investment I is always the share of income Y.