Macroeconomic Model Building
Fall 2017
Problem Set 1
N7TVH3
Due: 18. Sept. 17:20
Note: If we ask for an equation, do not substitute the value of exogenous variables! Discussion is encouraged, however, you have to solve the problem set individually. In line with the Code of Conduct of Corvinus University of Budapest, accepting solutions from others is an act of academic misconduct and will be penalized by a grade of zero to the assignment. If your logic is correct but miscalculate the result, you still get half the points.
1. (3.1 points) An economy of interest operates for two periods. We know that the rst period exogenous income of the representative household is 538, while the second period income is 110. The representative household can freely save or borrow with respect to
its budget constraints. The savings holder is willing to lend or accept deposits at an p
interest rate of 5 percent. The one period utility function is Ut = Ct. The lifetime utility function is additively separable, and the impatience parameter equals 0.91.
(a) Formulate the lifetime utility function of the representative household. (0.1 point)
(b) What is the equation of the rst period budget constraint? (0.1 point)
(c) What is the equation of the second period budget constraint? (0.1 point)
(d) Formulate the lifetime budget constraint. (0.1 point)
(e) Formulate the Lagrangian for the household's problem. (0.1 point)
(f) Take the partial derivatives of the Lagrangian with respect to the decision variables. (0.3 points)
(g) Derive the Euler equation. (0.1 point)
(h) What are the goods market clearing conditions? (0.2 points)
(i) What are the assets market clearing conditions? (0.2 points)
(j) Solve for all the endogenous variables,that is, for consumpton and the savings holder's demand for goods in both periods, the interest rate and the savings of the household between the two periods. (0.6 points).
(k) Now suppose that the economy faces a change in the interest rate. The new interest rate is 4 percent. Calculate the optimal consumption in both periods with the new interest rate. How did the consumption change in period one? What about period two? (0.2 points)
(l) Now suppose that an other economy (Country B) is characterized by the exact same equations. The only di erence is that the rst period exogenous income of the representative household is 110, while the second period income is 538. Calculate the optimal consumption in both periods with the original, 5 percent interest rate. (0.2 points)
Fall 2017
Problem Set 1
N7TVH3
Due: 18. Sept. 17:20
Note: If we ask for an equation, do not substitute the value of exogenous variables! Discussion is encouraged, however, you have to solve the problem set individually. In line with the Code of Conduct of Corvinus University of Budapest, accepting solutions from others is an act of academic misconduct and will be penalized by a grade of zero to the assignment. If your logic is correct but miscalculate the result, you still get half the points.
1. (3.1 points) An economy of interest operates for two periods. We know that the rst period exogenous income of the representative household is 538, while the second period income is 110. The representative household can freely save or borrow with respect to
its budget constraints. The savings holder is willing to lend or accept deposits at an p
interest rate of 5 percent. The one period utility function is Ut = Ct. The lifetime utility function is additively separable, and the impatience parameter equals 0.91.
(a) Formulate the lifetime utility function of the representative household. (0.1 point)
(b) What is the equation of the rst period budget constraint? (0.1 point)
(c) What is the equation of the second period budget constraint? (0.1 point)
(d) Formulate the lifetime budget constraint. (0.1 point)
(e) Formulate the Lagrangian for the household's problem. (0.1 point)
(f) Take the partial derivatives of the Lagrangian with respect to the decision variables. (0.3 points)
(g) Derive the Euler equation. (0.1 point)
(h) What are the goods market clearing conditions? (0.2 points)
(i) What are the assets market clearing conditions? (0.2 points)
(j) Solve for all the endogenous variables,that is, for consumpton and the savings holder's demand for goods in both periods, the interest rate and the savings of the household between the two periods. (0.6 points).
(k) Now suppose that the economy faces a change in the interest rate. The new interest rate is 4 percent. Calculate the optimal consumption in both periods with the new interest rate. How did the consumption change in period one? What about period two? (0.2 points)
(l) Now suppose that an other economy (Country B) is characterized by the exact same equations. The only di erence is that the rst period exogenous income of the representative household is 110, while the second period income is 538. Calculate the optimal consumption in both periods with the original, 5 percent interest rate. (0.2 points)
Macroeconomic Model Building – Problem Set 1 – Page 2 of 3 Due: 18. Sept. 17:20
(m) What would be the optimal consumption in Country B if the interest rate changed to 4 percent? Would the rst period consumption increase or decrease? How about the second period? (0.2 points)
(n) Now that we have studied the e ect of the change of the interest rate in both economies, it is time to compare the two. Does the interest rate change have the same e ect on the consumption in both economies? What does this imply regarding the magnitude of the income e ect and the magnitude of the substitution e ect? Come up with a utility function, which would cause the two economies to react di erently to the interest rate change! (0.6 points)
2. (2.2 points) In our economy of interest, there are two di erent households. The high income (A) household receives an exogenous income of 197 in period 1 and 157 in period
2. The low income household (B) receives 54 in period 1 and 96 in period 2. High income households represent 29% of the economy. The lifetime utility of high income households is given by
UA(C1A; C2A) = ln(C1A) + A ln(C2A)
and the lifetime utility of the low income household is
UA(C1B; C2B) = ln(C1B) + B ln(C2B)
where A = 0:94 and B = 0:96. Households are free to save or borrow at an interest rate agreed by both parties.
(a) Derive all Euler equations. (0.2 points)
(b) What are the intertemporal budget constraints? (0.2 point)
(c) What are the market clearing conditions? (0.4 points)
(d) Derive the value of the interest rate, consumption for both types in all periods and savings of both types b/w the rst two periods. (1.4 points, 0.2 each)
3. (1.9 points) In this exercise you will have to use MATLAB to solve the model. Please submit both your solution and your code for this problem.
Take a representative household that lives for two periods. The household receives
exogenous incomes of 84 and 128, respectively. The one-period utility function of the household is log(Ct). The lifetime utility function is additively separable. The impatience parameter of the household is 0.97. The household may deposit or borrow from the savings holder with an exogenous real interest rate of 0.09.
(a) Write down the Euler equation. (0.1 point)
(b) Formulate budget constraints and the intertemporal budget constraint. (0.3 points)
(c) What are the market clearing conditions? (0.4 points)
(d) Characterize the decision of the savings holder. (0.1 point)
(m) What would be the optimal consumption in Country B if the interest rate changed to 4 percent? Would the rst period consumption increase or decrease? How about the second period? (0.2 points)
(n) Now that we have studied the e ect of the change of the interest rate in both economies, it is time to compare the two. Does the interest rate change have the same e ect on the consumption in both economies? What does this imply regarding the magnitude of the income e ect and the magnitude of the substitution e ect? Come up with a utility function, which would cause the two economies to react di erently to the interest rate change! (0.6 points)
2. (2.2 points) In our economy of interest, there are two di erent households. The high income (A) household receives an exogenous income of 197 in period 1 and 157 in period
2. The low income household (B) receives 54 in period 1 and 96 in period 2. High income households represent 29% of the economy. The lifetime utility of high income households is given by
UA(C1A; C2A) = ln(C1A) + A ln(C2A)
and the lifetime utility of the low income household is
UA(C1B; C2B) = ln(C1B) + B ln(C2B)
where A = 0:94 and B = 0:96. Households are free to save or borrow at an interest rate agreed by both parties.
(a) Derive all Euler equations. (0.2 points)
(b) What are the intertemporal budget constraints? (0.2 point)
(c) What are the market clearing conditions? (0.4 points)
(d) Derive the value of the interest rate, consumption for both types in all periods and savings of both types b/w the rst two periods. (1.4 points, 0.2 each)
3. (1.9 points) In this exercise you will have to use MATLAB to solve the model. Please submit both your solution and your code for this problem.
Take a representative household that lives for two periods. The household receives
exogenous incomes of 84 and 128, respectively. The one-period utility function of the household is log(Ct). The lifetime utility function is additively separable. The impatience parameter of the household is 0.97. The household may deposit or borrow from the savings holder with an exogenous real interest rate of 0.09.
(a) Write down the Euler equation. (0.1 point)
(b) Formulate budget constraints and the intertemporal budget constraint. (0.3 points)
(c) What are the market clearing conditions? (0.4 points)
(d) Characterize the decision of the savings holder. (0.1 point)
Macroeconomic Model Building – Problem Set 1 – Page 3 of 3 Due: 18. Sept. 17:20
(e) Solve the system of equations using the f solve function in MATLAB. (1 point)
4. (2.8 points) In this exercise you will have to use MATLAB to solve the model. Please submit both your solution and your code for this problem.
Take a representative household that lives for three periods. The household receives
exogenous incomes of 68, 41 and 63 respectively. The one-period utility function of the p
household is Ct. The lifetime utility function is additively separable. The impatiance parameter of the household is 0.91. The household may deposit or borrow from the savings holder with an exogenous real interest rate of 0.06.
(a) Write down the Euler equations. (0.2 points)
(b) Formulate the budget constraints and the intertemporal budget constraint. (0.4 points)
(c) What are the market clearing conditions? (0.6 points)
(d) Characterize the decision of the savings holder. (0.2 points)
(e) Solve the system of equations using the f solve function in MATLAB. (1.4 points)
(f) 
