3. Consider a household that lives for two periods and consumes C in period 1 and C2 in period 2. The household earns a wage W in period one and can save A which yields (1+r)A in period 2 where r is the net interest rate. The household does not earn a wage in period 2. The household chooses consumption in period one and two (C1,C2) and savings A to maximize U(C1) + BU (C2) subject to the budget constraints. Assume that utility has the CRRA form: U(C) = C1-7 I-7 if 1 U(C) = In(C) if y=1 where > 0. (a) Write down the budget constraint for period one and period 2. Derive the lifetime budget constraint. (b) Let R (1+r). Derive the optimality condition for household savings. Show that this implies: C = (RB)C for =1/y. What is the elasticity of C2/C with respect to R? (c) Define the intertemporal elasticity of substitution = 1/7. Solve for C and C as function of r,o, and W. Compute the elasticity of C and C2 with respect to the first period wage W (the elasticity is defined as dlog(W) (d) Describe the effect of an increase in on first-period consumption C and second-period consumption C2. Provide economic intuition for this result. (e) Describe the effect of an increase in the interest rate r on first- and second-period consumption. Graph the budget set and how it changes in response to the increase in r. The budget set is the com- bination of C1 and C2 that are feasible to purchase. To graph this, put C2 on the vertical axis and C on the horizontal axis. Be clear on what determines the intercepts and the slope of the budget line. (f) Does first-period consumption increase or decrease in response to an increase in r? How does your answer depend on the value of the intertemporal-elasticity of substitution ? Explain.