Consider a household that lives for two periods and consumes 01 in period 1 and 02 in period 2. The household earns a wage (1 + 'r) W in period 2 and can save or borrow S which yields (1 + r) S in period 2, where r is the net interest rate. The household does not earn a wage in period 1. The household chooses consumption in period one and two (01,02) and savings 8 to maximize U (01) + BU (02) subject to its budget constraint. Assume that utility is constant relative risk aversion: (a) Write down the budget constraint for period one and the budget constraint for period two. Combine them into a lifetime budget constraint. (b) Derive the optimality condition for household savings. (c) Dene the intertemporal elasticity of substitution (7 = 1/7. Solve for 01 and 02 as a function of r, a, 3, and W. ((1) Describe the effect of an increase in W on rstperiod consumption 01 and second period consumption 02. Provide economic intuition for this result. (e) Compute the elasticity of the ratio of 01 and 02 with respect to the interest rate 1 + 'r (the elasticity is dened as W) Does 02/01 increase or decrease in response to an increase in r? Explain with economic intuition (hint: you should mention income effects and substitution effects). How does the magnitude of your answer demand on the intertemporal elasticity of substitution or? (f) Will 01 rise or fall when the interest rate 1 + 'r rises? How does your answer depend on 0'? Explain with economic intuition. Use your answer to explain why 01 is completely insensitive to the interest rate 'r with log utility U (C) = log (0)