Download (I) Consumption and an Interest Tax

E C O N 2 1 5 I N T E R M E D I AT E M A C R O E C O N O M I C S
SOLUTION
TO
ASSIGNMENT 1
I.
Consumption and an Interest Tax
a. Y1=Q1+(1+r)B0=Q1
Inter-temporal budget constraint:
C1+C2/(1+r1)=Q1+Q2/(1+r1)
C1+C2/1.1=100+220/1.1
C1+C2/1.1=300
Graphically,
C2
330
300
C1
When C2=0,
C1+0=300  C1=300
When C1=100,
100+C2/1.1=300  C2=220
∴(100,220) is on the budget constraint.
When C1=0,
0+C2/1.1=300  C2=330
Permanent Income
Yp+Yp/1.1=300  Yp=157.1
 S1=100-157.1= -57.1 (i.e. borrowing)
b. After-tax permanent income
1.
Ypt+Ypt/(1+r1)=(Q1-T1)+(Q2-T2)/(1+r1)
Ypt(1+1/1.1)=(100-50)+220/1.1
Ypt=130.95
2.
Ypt=104.76
3.
Ypt=130.95
After-tax Consumption and Saving
1.
2.
3.
C1=130.95
S1=50-130.95= -80.95
C2=130.95
C1=104.76
C2=104.76
C1=130.95
C2=130.95
S2=89.05
S1=-54.76
S2=60.24
S1=-30.95
S2=34.05
All the three tax policies lead to reduction in consumption of both periods. The
temporary tax increase represents a temporary fall in disposable income, households
trying to maintain a smooth consumption path will borrow against the future during the
period of temporary high tax. Therefore, the saving in the first period is lowest among
the three. For the anticipated tax increase, people’s future disposable income is lowered.
People tend to reduce the borrowing so as to reduce the future debt burden and maintain
a stable consumption. A permanent increase in tax reduces the disposable income for all
period. So, the saving pattern would not be affected significantly.
II.
a.
Output Growth
Qt  K t Lt
a
a
1 a
1 a
Qt K t Lt
 a 1 a
Lt
Lt Lt
qt  k t  1  k t
a
a
MPK=aKta-1 Lt1-a
As 0<a<1, (a-1)<0. When K increases, Kta-1, and thus MPK, will decrease. When
capital increase, the output, and thus the investment increase more slowly. Therefore,
there must be some point where the investment level meets the need for capital
replacement, that is, the steady state. Graphically,
(n+d)k
ska
k
kss
And this is applicable to both Hong Kong and Costa Rica.
At Steady State of Hong Kong,
s HK k t  (n HK  d )k t
a
s HK
1 a
 kt
n HK  d
1
s
k t  ( HK ) 1 a
n HK  d
a
qt  (
s HK 1 a
)
n HK  d
Similarly, in Costa Rica,
a
qt  (
s CR 1 a
)
nCR  d
s
n
d
HK
0. 2
0. 02
0. 08
CR
0. 16
0. 03
0. 08
1. 407
1. 203
2. 000
1. 455
a
0. 33
0. 5
q
0. 7
5. 040
2. 397
0. 9
512. 000
29. 144
When a is large, a little increase in capital stock gives a significant increase in output.
Therefore, the little difference in saving rate and labor growth rate between Hong Kong
and Costa Rica is sufficient to make a great difference in output.
b.
Q Q/L
q

a
K K/L
k
Therefore, for all levels of L, MPK remains constant. Even if the population grows at
MPK  a
steady state, MPK would not change.
When r=0.05,
MPK=r+d=0.05+0.08=0.13
Q
K
1Q
0.13 
3K
Q
 0.39
K
MPK  a
If a=1/3,
At steady state,
sk a  (n  d )k
nd
s
i sk a
 
 sk a 1
k
k
nd
 s(
)nd
s
k a 1 
I
K
I

K

When r=0.05, a=0.33
Therefore, in Hong Kong,
Similarly, in Costa Rica,
0.03  0.08
sCR 
 0.282
.39 0.08
0.002
s HK 
 0.256
0.39
s
I
I /K nd


Q Q/K Q/K
When r=0.07,
MPK=0.07+0.08=0.15
 Q/K =0.15/0.33=0.45
Then, in Hong Kong,
s' HK 
0.02  0.08
 0.222
0.45
In Costa Rica,
0.03  0.08
 0.244
0.45
As interest ate increases, the cost of capital become higher. This in turn dicourages
s 'CR 
investment. Therefore, investment rates in both countries fall.
III. Deflators
Nominal GDP in HK dollars = 110 x 20 + 60 x 30 = 4000
1500
 0.6
2500
1000

 0 .4
2500
HK
wSHOES

HK
wHAIR
Pc=0.6(20/15)+0.4(30/20)
Pc=1.4
RGDP1996=2500
RGDP1997=4000/1.4=2857.14
Growth rate of RGDP=(2857.14-2500)/2500=0.143
π1997=(1.4-1)/1=0.4
The absolute purchasing power parity is hold for shoes but not for haircut is due to that
the former is tradable good while the latter is non-tradable. International competition
can drive to price of tradable goods to the same level. However, for non-tradable, such
competition cannot be held. And thus the price might vary across countries.
In Japan,
Pshoes=15/0.5=30
Nominal GDP=100 x 30 + 50 x 50 = 5500
Nominal GDP in HK$=5500x0.5=2750
From this measure (exchange rate conversion), the level of output is higher in Japan.
In Hong Kong,
In Japan,
PH/PS=20/15=1.33
PH/PS=50/30=1.67
Thus, the relative price of haircuts (non-tradable goods) is higher in Japan. Countries
with non-competitive non-tradable goods sectors will generally have their ouput level
exaggerated by exchange rate conversion.
3000
 0.545
5500
2500
JPN
wHAIR

 0.455
5500
INT
wSHOES
 .5 x.545  .5 x.6  0.573
JPN
wSHOES

INT
wHAIR
 .5 x.455  .5 x.4  0.427
Price of International Basket=0.573x(30/15)+0.427x(50/20)=2.214
PPP conversion
Japanese GDP in HK$ = 5500/2.214=2484.19
The actual baskets of good in Hong Kong and Japan are the same in 1996. Therefore,
the value of their nominal GDP should be close. The PPP conversion provides a better
method for international comparison that is more accurate that exchange rate
conversion.