3. Capital Income and Savings Taxation Consider a 2 period model where individuals earn labor income Y = 400 from working in period 1 and do not work in period 2 (retirement). Individuals choose how much to consume in each period. Savings in period 1 earn an interest rater = 20%. Let C1 denote consumption in period 1 and C2 denote consumption in period 2. Suppose that individuals have a utility function U = In C1 + In C2. . (a) Set up the individual's lifetime utility maximization problem and solve for the optimal C1, C2, and S in an economy without taxes. Consumption in the second period is savings from the first period plus interest. Savings is just income from the first period minus consumption during the first period, so: C2 = (400 - C1) (1 + 0.2) The utility maximization problem is max In C1 + In C2 subject to the budget constraint. When the budget constraint is incorporated into the expression for C2, as shown, the maximization problem is max In C1 + In((400 - C1) (1.2)) = max In C1 + In(480 - 1.2C1). Solving, the first-order condition is au 1 1.2 480 - 1.2C1 = 0 480 - 1.2C1 = 1.2C1 Using the first-order condition, we get the following: C1 = 480/2.2 = 200. S = 400 - C1 = 200. C2 = S(1 + r) = 200(1.2) = 240