question 3e please

3. Consider a small open economy with a single household who receives utility from consumption of a final good in two periods. The household has h = 100 units of time which it devotes fully to working because it does not value leisure. Household preferences are given by U(C1, C2) = C;75C.,25 Labour is the only input in production and the final good is produced with the following technologies in 1 each period: Y1 = 21N Y2 = z2N2 The government is spending Gt and imposing a lump-sum tax equal to Tt on the household in period te {1, 2}. The household and government can borrow and lend at gross real interest rate R. (3.A.) (13 points) Derive the household's optimal consumption demand function in each period and its optimal savings function in period one as a function of the gross real interest rate, the productivity parameters, and the tax levels: CP(R, 21, 22,T1, T2), CZ (R, 21, 22, T1, T2), and S? (R, 21, 22, T1, T2). Suppose we have the following initial values: 21 = 6 22 = 4 G1 = 30 G2 = 30 Ti = 20. The closed economy (autarkic) equilibrium gross interest rate equals R* = 1.947 (this is the endogenous interest rate that clears the lending market within the country). In the open economy, the household and government can borrow and lend with each other and the rest of the world at a fixed gross interest rate equal to R= 1.2. (3.B.) (10 points) Complete Table 1. Provide economic intuition for the direction of the change in household utility in moving from the closed economy to the open economy. Table 1: Closed Versus Open Equilibrium Variables Closed Economy Open Economy 1 T2 2 Ci 3 C2 4 SP 5 sc 6 TB1 7 Utility 523.89 (3.C.) (10 points) Treating R as a parameter for this question for the open economy, derive the country's national savings function as a function of R and graph that function for values of R between 1.0 and 3.0. For the remaining questions, set R = 1.2. (3.D.) (17 points) Complete Table 2 for the values of the 21, 22, G1, G2, and T1 given in each column of the table. Note that the first column of Table 2 is the same as the second column of Table 1. (3.E.) (10 points) Using your results from Columns 1-3, describe and provide economic intuition for the direction of the change in the variables in Rows 2-4 when the country gets hit separately with (i.) a transitory productivity shock and (ii.) an anticipated productivity shock. Using your results from Row 7, explain whether or not our example economy is consistent with the following statement: If a country's trade deficit decreases, this is good for the country. (3.F.) (5 points) Using your results from Columns 1 and 4, explain how the variables in the table demonstrate that the Ricardian Equivalence Theorem holds in this economy. (3.G.) (10 points) Using your results from Columns 1, 5, and 6, calculate the household's marginal propensity to consume (mpc) out of first-period disposable income for (i.) a transitory fall in government spending and for (ii.) a permanent fall in government spending, where mpci AY-T) Provide economic intuition to explain why one of these marginal propensities to consume is larger than the other. C, 21 = 6 22 = 4 21 = 6 22 = 4 Table 2: Open Economy Equilibrium Variables 21 = 6 21 = 3 21 = 6 21 = 6 22 = 4 22 = 4 22 = 2 22 = 4 G1 = 30 G1 = 30 G1 = 30 Gi = 30 G1 = 25 G2 = 30 G2 = 30 G2 = 30 G2 = 30 G2 = 30 Ti = 20 Ti = 20 Ti = 20 T = 40 Ti = 10 Column 1 Column 2 Column 3 Column 4 Column 5 = G1 = 25 G2 = 25 Ti = 10 Column 6 Row 1 2 Variable T2 C C2 SP SC 3 4 5 6 7 TB1 Utility 3. Consider a small open economy with a single household who receives utility from consumption of a final good in two periods. The household has h = 100 units of time which it devotes fully to working because it does not value leisure. Household preferences are given by U(C1, C2) = C;75C.,25 Labour is the only input in production and the final good is produced with the following technologies in 1 each period: Y1 = 21N Y2 = z2N2 The government is spending Gt and imposing a lump-sum tax equal to Tt on the household in period te {1, 2}. The household and government can borrow and lend at gross real interest rate R. (3.A.) (13 points) Derive the household's optimal consumption demand function in each period and its optimal savings function in period one as a function of the gross real interest rate, the productivity parameters, and the tax levels: CP(R, 21, 22,T1, T2), CZ (R, 21, 22, T1, T2), and S? (R, 21, 22, T1, T2). Suppose we have the following initial values: 21 = 6 22 = 4 G1 = 30 G2 = 30 Ti = 20. The closed economy (autarkic) equilibrium gross interest rate equals R* = 1.947 (this is the endogenous interest rate that clears the lending market within the country). In the open economy, the household and government can borrow and lend with each other and the rest of the world at a fixed gross interest rate equal to R= 1.2. (3.B.) (10 points) Complete Table 1. Provide economic intuition for the direction of the change in household utility in moving from the closed economy to the open economy. Table 1: Closed Versus Open Equilibrium Variables Closed Economy Open Economy 1 T2 2 Ci 3 C2 4 SP 5 sc 6 TB1 7 Utility 523.89 (3.C.) (10 points) Treating R as a parameter for this question for the open economy, derive the country's national savings function as a function of R and graph that function for values of R between 1.0 and 3.0. For the remaining questions, set R = 1.2. (3.D.) (17 points) Complete Table 2 for the values of the 21, 22, G1, G2, and T1 given in each column of the table. Note that the first column of Table 2 is the same as the second column of Table 1. (3.E.) (10 points) Using your results from Columns 1-3, describe and provide economic intuition for the direction of the change in the variables in Rows 2-4 when the country gets hit separately with (i.) a transitory productivity shock and (ii.) an anticipated productivity shock. Using your results from Row 7, explain whether or not our example economy is consistent with the following statement: If a country's trade deficit decreases, this is good for the country. (3.F.) (5 points) Using your results from Columns 1 and 4, explain how the variables in the table demonstrate that the Ricardian Equivalence Theorem holds in this economy. (3.G.) (10 points) Using your results from Columns 1, 5, and 6, calculate the household's marginal propensity to consume (mpc) out of first-period disposable income for (i.) a transitory fall in government spending and for (ii.) a permanent fall in government spending, where mpci AY-T) Provide economic intuition to explain why one of these marginal propensities to consume is larger than the other. C, 21 = 6 22 = 4 21 = 6 22 = 4 Table 2: Open Economy Equilibrium Variables 21 = 6 21 = 3 21 = 6 21 = 6 22 = 4 22 = 4 22 = 2 22 = 4 G1 = 30 G1 = 30 G1 = 30 Gi = 30 G1 = 25 G2 = 30 G2 = 30 G2 = 30 G2 = 30 G2 = 30 Ti = 20 Ti = 20 Ti = 20 T = 40 Ti = 10 Column 1 Column 2 Column 3 Column 4 Column 5 = G1 = 25 G2 = 25 Ti = 10 Column 6 Row 1 2 Variable T2 C C2 SP SC 3 4 5 6 7 TB1 Utility