4. (30 points) Suppose consumers have preferences over consumption in two time periods given by U(C,...

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4. (30 points) Suppose consumers have preferences over consumption in two time periods given by U(C, C) = u(c) + u(c). where u' (c) > 0 and u" (c) < 0. a) (2 points) Assume that consumers start with no initial wealth and earn income exogenously in each period given by w and respectively. Assume furthermore that consumers can save (or borrow) in the first period and earn (pay) interest at the rate r in the second period on savings. Write the consumer's period specific budget constraints given that they divide all their resources between consumption and savings in the first period and then consume all their resources in the second period (there should be one constraint for each period). b) (2 points) Now suppose that the government levies lump-sum taxes in each period equal to T in the first period and T in the second period. Write the new period specific budget constraints. c) (4 points) Substitute the budget constraints into the utility function U(C, C) and formally state the consumer's maximization problem (if you've made the correct substitutions, this should be stated as an unconstrained maximization problem with respect to a single variable). d) (6 points) Write the FOC for maximization and verify that the SOC holds. Once you have obtained the FOC, substitute c and c back into the functions so that your FOC is an optimal relationship between c and c. e) (10 points) For the remainder of the question, assume that (1 + r) = 1. Using the FOC you derived in part d), and the two budget constraints, you should now be able to derive a system of three linear equations and three unknowns, c,c, and s*. Solve for c,c, and S*. f) (6 points) Now we are going to verify a famous result known as Ricardian Equivalence" which essentially states that so long as the present value of taxes remains unchanged, the timing of taxes has no effect on consumption behaviour. To verify this, suppose a different tax policy is proposed such that taxes in the first period are equal to zero and second period taxes are equal to T such that = [ + - (i.e., the present value of taxes remains unchanged). Does this new tax policy have any effect on consumption in either period? What about savings? Show all your work (Hint: Compare your optimal equation for consumption with the original tax policy to the new one. All you have to do is set T 0 and T = T to obtain the new optimal equations). T 1+r T2 1+r = 4. (30 points) Suppose consumers have preferences over consumption in two time periods given by U(C, C) = u(c) + u(c). where u' (c) > 0 and u" (c) < 0. a) (2 points) Assume that consumers start with no initial wealth and earn income exogenously in each period given by w and respectively. Assume furthermore that consumers can save (or borrow) in the first period and earn (pay) interest at the rate r in the second period on savings. Write the consumer's period specific budget constraints given that they divide all their resources between consumption and savings in the first period and then consume all their resources in the second period (there should be one constraint for each period). b) (2 points) Now suppose that the government levies lump-sum taxes in each period equal to T in the first period and T in the second period. Write the new period specific budget constraints. c) (4 points) Substitute the budget constraints into the utility function U(C, C) and formally state the consumer's maximization problem (if you've made the correct substitutions, this should be stated as an unconstrained maximization problem with respect to a single variable). d) (6 points) Write the FOC for maximization and verify that the SOC holds. Once you have obtained the FOC, substitute c and c back into the functions so that your FOC is an optimal relationship between c and c. e) (10 points) For the remainder of the question, assume that (1 + r) = 1. Using the FOC you derived in part d), and the two budget constraints, you should now be able to derive a system of three linear equations and three unknowns, c,c, and s*. Solve for c,c, and S*. f) (6 points) Now we are going to verify a famous result known as Ricardian Equivalence" which essentially states that so long as the present value of taxes remains unchanged, the timing of taxes has no effect on consumption behaviour. To verify this, suppose a different tax policy is proposed such that taxes in the first period are equal to zero and second period taxes are equal to T such that = [ + - (i.e., the present value of taxes remains unchanged). Does this new tax policy have any effect on consumption in either period? What about savings? Show all your work (Hint: Compare your optimal equation for consumption with the original tax policy to the new one. All you have to do is set T 0 and T = T to obtain the new optimal equations). T 1+r T2 1+r =