macroeconomics

please solve question 1.5

Redo the entire question (1.4) but now assume the real interest rate for borrowing is higher than that for saving (that is r > r, similar to (1.3)). Additionally, discuss the implication this situation may have on the marginal propensity to consume. Explain how your answers may differ if r = r.

This question is about private consumption. Consider a household that lives for two periods. In period j, the price level is P, and the real labour income is given by w, for j = {1,2}. The initial real wealth is given by A. The nominal interest rate is i and the inflation rate (CPI) is A = R - 1. The household has a constant relative risk aversion (CRRA) utility function which reads U (G.C2) = u(G) + Bu(C2) = c-1 c!-"- 1 -B 1 - 1-0 where e > 0,5 (0,1) is the subjective discount factor of the household and C; is consumption in period j for j = {1,2}.' Now consider the consumption choice of the household. (1.1) Firstly, demonstrate how we can obtain the lifetime budget constraint of the household consist- ing of nominal interest rate, price levels, real labour income, initial real wealth, and consump- tion. Secondly, transform the budget constraint you have just obtained into another budget con- straint consisting of the real interest rate, real labour income, initial real wealth, and consumption. Lastly, interpret. (1.2) Given that the household maximises lifetime utility subject to the lifetime budget constraint, we u'(G know that the optimal condition boils down to sw(C) = 1+r where r is the real interest rate. Illustrate this consumption bundle in the (C1, C2)-space, and then explain intuitively why this is the optimal choice for the household. For (1.3)-(1.5), you do not need to compute partial or total derivatives. Remark: As 8 1 we have that u (C1,C2) = InC + B in C using L'Hpital's Rule. 1 (1.3) Based on the setup in (1.2), but now assume that if the household wants to borrow money in period 1, they face a higher real interest rate of r' > r (and if they save in period 1, they face the same real interest rate r). Both r' and r are finite. Illustrate this situation in the (C1, C2)- space. Comparing to (1.2), what happens to the lifetime budget constraint? What happens to the optimal consumption bundle (particularly does it matter if the household is a net saver or a net borrower)? (1.4) Based on the setup in (1.2), but now assume there are an exogenous increase in the real labour income in period 1 and an exogenous reduction of the real labour income in period 2 of equal amount (after discounting). That is, Awi + 2 = 0 where Aw, is the change in real labour income in period je {1,2}. Illustrate this situation in the (C1, C2)-space. What happens to the lifetime budget constraint? What happens to the optimal consumption bundle? Explain. (1.5) Redo the entire question (1.4) but now assume the real interest rate for borrowing is higher than that for saving (that is r' > r, similar to (1.3)). Additionally, discuss the implication this situation may have on the marginal propensity to consume. Explain how your answers may differ if r' = r. This question is about private consumption. Consider a household that lives for two periods. In period j, the price level is P, and the real labour income is given by w, for j = {1,2}. The initial real wealth is given by A. The nominal interest rate is i and the inflation rate (CPI) is A = R - 1. The household has a constant relative risk aversion (CRRA) utility function which reads U (G.C2) = u(G) + Bu(C2) = c-1 c!-"- 1 -B 1 - 1-0 where e > 0,5 (0,1) is the subjective discount factor of the household and C; is consumption in period j for j = {1,2}.' Now consider the consumption choice of the household. (1.1) Firstly, demonstrate how we can obtain the lifetime budget constraint of the household consist- ing of nominal interest rate, price levels, real labour income, initial real wealth, and consump- tion. Secondly, transform the budget constraint you have just obtained into another budget con- straint consisting of the real interest rate, real labour income, initial real wealth, and consumption. Lastly, interpret. (1.2) Given that the household maximises lifetime utility subject to the lifetime budget constraint, we u'(G know that the optimal condition boils down to sw(C) = 1+r where r is the real interest rate. Illustrate this consumption bundle in the (C1, C2)-space, and then explain intuitively why this is the optimal choice for the household. For (1.3)-(1.5), you do not need to compute partial or total derivatives. Remark: As 8 1 we have that u (C1,C2) = InC + B in C using L'Hpital's Rule. 1 (1.3) Based on the setup in (1.2), but now assume that if the household wants to borrow money in period 1, they face a higher real interest rate of r' > r (and if they save in period 1, they face the same real interest rate r). Both r' and r are finite. Illustrate this situation in the (C1, C2)- space. Comparing to (1.2), what happens to the lifetime budget constraint? What happens to the optimal consumption bundle (particularly does it matter if the household is a net saver or a net borrower)? (1.4) Based on the setup in (1.2), but now assume there are an exogenous increase in the real labour income in period 1 and an exogenous reduction of the real labour income in period 2 of equal amount (after discounting). That is, Awi + 2 = 0 where Aw, is the change in real labour income in period je {1,2}. Illustrate this situation in the (C1, C2)-space. What happens to the lifetime budget constraint? What happens to the optimal consumption bundle? Explain. (1.5) Redo the entire question (1.4) but now assume the real interest rate for borrowing is higher than that for saving (that is r' > r, similar to (1.3)). Additionally, discuss the implication this situation may have on the marginal propensity to consume. Explain how your answers may differ if r' = r