Download Homework 2 Answer Key

Homework Assignment #2
Economics 215
Intermediate Macroeconomics
Assigned: Thursday, March 18, 2004
Due: Friday, April 2, 2004
1. Assume that the liquidity preference of an economy is a negative function of the
nominal interest rate.
M
.2
 L( i ) 


P  GDP
i
Calculate the velocity and the amount of money that would be demanded if the
nominal interest rate were 10% and the nominal GDP were HK$1.1 trillion
dollars. How much additional money would the central bank need to supply to
reduce the nominal interest rate to 5% holding the nominal GDP to be constant?
Velocity is the ratio of nominal GDP to the money supply or
1  P  GDP  i  .1  0.632  M  0.632*1.1  0.696 or HK$696
L
.2
.2
M
billion. Under what money supply would the interest rate be 5%.
P  GDP
1.1
M

 .984 or 984 billion. The increase in money supply
i
.05
.2
.2
necessary to reduce the interest rate to 5% is 388 billion.
2. Assume the same liquidity preference curve as in problem 1 and that real GDP
growth is equal to the real interest rate.
a. Calculate the nominal interest rate and the share of real GDP which could
be collected as real seignorage at long term money growth rates equal to
5%, 10% and 25%.
b. Real seignorage is
M t  M t 1 M t  M t 1 M t 1 M t
g M Mt
gM
gM
.2






 Lt  GDPt 
  GDP
M
M
M
Pt
M t 1
M t Pt 1  g
Pt 1  g
1 g
i
In the long run, the nominal interest rate is the money growth rate minus
the real output growth rate plus the real interest rate. If the latter two are
equal, the nominal interest rate is equal to the money growth rate.
gM
gM
.2
M t  M t 1
GDP 


 .2  . Bond prices are the face
M
Pt
1 g M
g M 1 g
value, normalized to 100, divided by the nominal interest rate raised to the
power of the number of years of the maturity of a bond.
Money Growth
Seignorage Share Bond Prices
5.00%
4.26% 61.39133
10.00%
5.75% 38.55433
25.00%
8.00% 10.73742 Calculate
100
(1  i)T
price of a 10 year bond under each of these monetary growth rates.
PB 
the
3. You are given the current yield curve for Exchange Fund bonds. Calculate the
markets expectation of the 1-year interest rate in each of the next two years using
the Expectations Theory of the Term Structure.
Table 1
Maturity (yrs)
Exchange Fund Bills Yield
T= 1
2
3
0.39%
1.00%
1.94%
Under the expectations theory a bond is the average of the interest rates over its
life.
it ,T 
T  it ,T
it ,1  it 1,1  ...  it T 1,1
it ,1  it 1,1  ...  it T 1,1  it T ,1

T 1
  it ,1  it 1,1  ...  it T 1,1  , (T  1)  it ,T 1   it ,1  it 1,1  ...  it T 1,1  it T ,1  
T
, it ,T 1 
(T  1)  it ,T 1  T  it ,T   it ,1  it 1,1  ...  it T 1,1  it T    it ,1  it 1,1  ...  it T 1,1   it T ,1
T  1  2  it ,2  it ,1  it 1,1  2  .01  .0039  .0161
T  2  3  it ,3  2  it ,2  it  2,1  3  .0194  2  .01  .0382
4. Assume that the permanent income hypothesis is true. Calculate the wealth effect
of a $1 increase in financial wealth on consumption if the real interest rate were
5% and the average life profile were 25 years (i.e. calculate the annuity value of
$1 with r = .05 and T = 25). Calculate the wealth effect if the real interest rate
were 5% and the average life profile were 10 years (r = .05 and T = 10). Calculate
the wealth effect if the real interest rate were 10% and the average life profile was
25 years. (r = .10 and T = 25).
If the permanent income theory is true, then the present discounted value of the
consumption stream should be equal to the sum of human and financial wealth. If
households plan to have perfectly smooth consumption overtime, they will consume a
fraction of their wealth. That fraction is based on the number of years in their life and the
real interest rate.
C
C
C
C
C


 .... 
W  F 
2
3
1  r (1  r ) (1  r )
(1  r )T
1
1  xT 1
1  ( 11 r )T 1
, C  (1  x  x 2  ....xT )  C 
C
W  F 
1 r
1 x
1  ( 11 r )
1  ( 11 r )
C
 [W  F ]
1  ( 11 r )T 1
1  ( 11 r )
r  .05, T  25 
 0.067573769
1  ( 11 r )T 1
1  ( 11 r )
r  .05, T  10 
 0.12333769
1  ( 11 r )T 1
1  ( 11 r )
r  .10, T  25 
 0.100152793
1  ( 11 r )T 1
x
5. A household earns $100 in income today in period t = 0 and this grows at a 5%
rate until period t+T = T = 5. After period 5, the household dies. Calculate the
present value of lifetime income if the real interest rate is 5%. Calculate
consumption for this agent if the permanent income theory was true.
The present value of lifetime income is discounted using the interest rate.
Y3
Y5
Y
Y2
Y4
W  Y0  1 



2
3
4
(1  r ) (1  r ) (1  r ) (1  r ) (1  r )5
Income grows at a constant rate so Yt = (1+g)t *Y0 = (1.05)t * 100.
(1  g )Y0 (1  g ) 2 Y0 (1  g )3 Y0 (1  g ) 4 Y0 (1  g )5 Y0
W  Y0 




(1  r )
(1  r ) 2
(1  r )3
(1  r ) 4
(1  r )5
r  g  W  Y0  Y0  Y0  Y0  Y0  Y0  (T  1) *Y0  600
Smooth consumption over time
C
C
C
C
C
C




W
2
3
4
(1  r ) (1  r ) (1  r ) (1  r ) (1  r )5
implies
1  ( 11 r )
1  ( 11.05)
C

W

 W  0.188*600=112.581
1  ( 11 r )T 1
1  ( 11.05) 6
6. Permanent Income Hypothesis & Consumption Choice. A household will earn
real income in period 0, Y0 =70, and real income in period 1, Y1 =143. A
household starts with zero financial wealth and receive real income in two
periods, period 0 and period 1. Real income for each household is reported in
Table 2. The real interest rate is 10% (i.e. r = .10).
Table. After Taxes
Household
Zero Taxes
Temporary
Tax Increase
Future Tax
Increase
Permanent
Tax Increase
After Tax
Income Y0
70
50
70
50
After
r=.1 → W C0 =Annuity Y0 – C0
Tax
Value of W
Income
Y1
143
200
104.76
-34.76
143
180
94.29
-44.29
180
94.29
-24.29
160
83.81
-33.81
121
121
a. Calculate the present value of the household’s income stream, W.
Calculate the annuity value of W. Calculate first period consumption and
saving if the permanent income hypothesis is true.
Y
The real present value is W  Y0  1 . Since N = 1, then the annuity value
1 r
1
1
1
1
1  r  W  1  r  W  1  1.11 W  11 W  .524W
was
2
1
1
21
1   11.1
1
1

N 1
2
1  r 
1  r 
b. The government considers a number of tax plans. Which plan will have
the strongest impact on consumption?
i. In the first tax plan, they will collect a tax of 20 in period 0 leaving
the household with an after-tax income stream equal to Y0 =50 and
Y1 = 143. Calculate the present value and annuity value of this
after tax income stream. If the household consumes the annuity
value, what is consumption and saving?
ii. In the second tax plan, the government will collect a tax of 22 in
period 1 creating an after-tax income stream equal to Y0 =70 and
Y1 = 121. each household’s income stream. Calculate the present
value of after-tax income and consumption and savings in period 0.
iii. In the third tax plan, the government collects a tax of 20 in period
0 and a tax of 22 in period 1 creating an after tax income stream of
Y0 = 50 and Y1 =121. Calculate the present value of after-tax
income and consumption and savings in period 0.
See Table for solutions. A permanent increase in taxes has the strongest impact on
consumption because it has the strongest effect on lifetime income.
7. A firm has a Cobb-Douglas production function.
GDP  K ZL
The real interest rate is 5% and the depreciation rate of capital is also 5% (d=.05).
a. Calculate the average productivity of capital when the firm is at its optimal
capital stock. Calculate the capital-labor ratio when technology next
period is equal to Z = 1.
Assuming the price of output is equal to the price of new capital, the optimal
capital stock is achieved when the marginal product of capital is equal to the
real interest cost plus the depreciation cost MPK=r+δ = .10. Under the CobbDouglas production function, the average product of capital is proportional to
the marginal product of capital.
GDP
GDP 1
a
1 a
GDP   K   ZL   MPK  a

 a  MPK  1a  (r   )  2  .1  .2
K
K
In this case, the firm has a capital intensity parameter a = .5, so the average
product of capital is twice the marginal product of capital. The average
product of capital can be written as a function of the capital-labor ratio
GDP
K ZL
ZL
L
APK 


 Z
 Z L 
K
K
K K
K
K
L
K
 APK
Z
L
K
2
 APK
Z
K
L
Z
APK 2
 1 2  25
.2
b. Calculate the optimal capital stock when Z = 1 and L = 4. Calculate the
output level at the optimal capital stock. If the current capital stock is
equal to Kt =375, how much investment will you need to set next period’s
capital stock equal to the optimal level if Kt+1 = (1-d)Kt. + It.
We showed that the optimal capital stock is 25 times the amount of labor
which is equal to 100. If the capital stock is already 375, then we will have to
reduce the capital stock by selling some capital.
*
I t  K  (1   ) K  100  (.95)375  -256.25
c. Now assume that the technology level goes up by 21%, so Z = 1.21.
Calculate the optimal capital stock at the new technology level.
K Z
 1.21 2  30.25  K  121
L
APK 2
.2
Calculate the new level of output at the new technology level.
GDP  K ZL  121 1.21 4  111.1 2  24.2
What is the percentage increase in the optimal capital stock compared to the
answer in b. How much investment will you need to do to set next period’s
capital stock equal to the optimal level? In percentage terms, how does this
compare with the answer in section b?
The capital stock goes up by 21%. The output also goes up by 21%. The
amount of the investment that we need to do is -235.25.
8. The replacement cost of a firm’s capital stock is calculated to be worth $250
million. The firm has issued 1 million shares of stock, each of which is expected
to pay a $30 dividend next period. Calculate the q for this firm, if the required
return on the shares is 10% and there is zero expected growth in dividends in the
future. Should the firm add to its capital stock or subtract?
The dividend yield is equal to the required return minus the growth rate of dividends.
When the growth rate of dividend is zero, we can write
Dt 1
Dt 1 30
 r EQ  Pt EQ  EQ

 300
EQ
Pt
r
.1
If we assume that the firm has zero debt, then the q for this firm is market capitalization
divided by the replacement cost. Market capitalization is the share price times number of
shares which is $300 million implying q = 1.2. The firm should add capital since q is
greater than 1.