Department of Economics Columbia University Homework 7 Economics W3213 Intermediate Macroeconomics...

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Department of Economics Columbia University Homework 7 Economics W3213 Intermediate Macroeconomics 1. Mankiw, page XXXXXXXXXX, Problems and Applications 2 and 3. 2. Consider the consumption and saving decision of a two-period lived household with preferences described by the utility function ln C1 + ln C2 The household’s income in periods 1 and 2 are given by Y1 = 7 and Y2 = 10, respectively. The government levies lump-sum taxes in the amount of 2 in each period (that is, T1 = T2 = 2). Finally, the interest rate is 25 percent (or r = XXXXXXXXXXa) Calculate the present discounted value of the household’s lifetime disposable income. (b) Calculate the optimal level of consumption in period 1. (c) How much does the household save in period 1? (i.e., calculate S p 1 ). (d) Now assume that the government announces a tax cut of 2 in period 1 coupled with a future tax increase of 2.5 in period 2 (that is, ?T1 = -2 and ?T2 = 2.5). Calculate the optimal levels of consumption in periods 1 and 2 and the optimal level of private savings in period 1 that result after this announcement. Provide an intuitive explanation for your findings. (e) Now assume that, due to frictions in the financial market, the interest rate at which households save, denoted rs, is smaller than the interest rate at which households can borrow, denoted rb. Specifically, assume that rs is 12.5 percent, and that rb is 25 percent (i.e., rs = 0.125

Robert · Accepted Answer

Answer to 2: 
U(C1, C2) = LnC1 + LnC2 
Y1 = 7 and Y2 = 10 
T1 = T2 = 2 
Interest rate (r) = 0.25 
(a)  
Disposable income (Y1d) =  Y1 – T1 = 7 -2 = 5  
And disposable income (Y2d) = Y2-T2 = 10-2 = 8 
Present discounted value of the household’s lifetime disposable income = Y1d  + (Y2d/(1+r)) 
 =  5 + (8/(1+0.25)) = 11.4 
(b) 
Budget constraint: C1 + C2/(1+r) = 11.4 …………. (1) 
At the optimal point, C2/C1 = (1+r), implies C2 = C1*(1+r) …. (2) 
Substitute (2) in (1) 
C1 + (1+r)C1/(1+r) = 11.4, implies optimal consumption in period 1 i.e. C1 = 11.4/2 = 5.7 
Optimal consumption in period 2 i.e. C2 = C1*(1+r) = 5.7*(1.25)  = 7.125 
(c) 
Deriving Saving in period 1 (i.e. S1
P
) 
Disposable income in period 1 (Y1d) = 5 
From (b), optimal consumption in period 1 i.e. C1 = 5.7 
So saving in period 1 (i.e. S1
P
) = Disposable income in period 1 (Y1d) - optimal consumption in 
period 1 = 5-5.7 = -0.7 i.e. at the optimal consumption point, consumer is borrowing 0.7 in 
period 1.
(d) 
We are given  T1 = -2 and  T2 = 2.5 
So net tax in period 1 i.e. T1’ = T1 +  T1 = 2-2 = 0 and net tax in period 2 i.e. T2’ = T2 +  T2 = 
2+2.5 = 4.5 
Now disposable income in period 1 i.e. Y1d = Y1 – T1’ = 7-0 = 7 and disposable income in 
period2 i.e. Y2d = Y2 – T2’ = 10-4.5 = 5.5 
With these values, present discounted value of the household’s lifetime disposable income = Y1d  
+ (Y2d/(1+r)) = 7 + (5.5/(1+0.25)) = 11.4 
So budget constraint: C1 + C2/(1+r) = 11.4 …………. (3) 
At the optimal point, C2/C1 = (1+r), implies C2 = C1*(1+r) …. (4) 
Substitute (4) in (3) 
C1 + (1+r)C1/(1+r) = 11.4,

Department of Economics Columbia University Homework 7 Economics W3213 Intermediate Macroeconomics 1. Mankiw, page XXXXXXXXXX, Problems and Applications 2 and 3. 2. Consider the consumption and saving...

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