# calculate assumption

```Economics 514
Macroeconomic Analysis
Due: November 13, 2008
Simulation Exercises
1. Consumption Function
Assume a household lives for an infinite period of time but begins with no financial
wealth. The intertemporal budget constraint is of the form:


Ct
Yt



t
t
t  0 (1  r )
t  0 (1  r )

Assume that the household maximizes
  u (C ) and   (1  r )  1 The permanent
t
t
t 0
income hypothesis suggests that temporary changes in output have relatively small
effects on consumption but permanent changes have 1- for-1 impacts on the level of
consumption. Use the assumption that r = .1.
Assume that output in period 0 will decay geometrically forever. That is, Y0 = 1 and
Yt 1    Yt . Beginning in period 0, and continuing through period 20, calculate Yt,
Ct, and St = Yt - Ct when ρ = 0, .5, and 1.
Under the permanent income hypothesis, consumption equals permanent income.
We can write human wealth as:


Yt
 tY0
  
W 


Y


0 

t
t
t  0 (1  r )
t  0 (1  r )
t 0  1  r 

T 1
t
  
1 

1 r
1 r 
 lim 

T 
1 r  
  
1 

 1 r 
We can write the permanent income as the annuity value of wealth

W 
t 0


Ct
C0
 1 


C


0 

t
t
(1  r )
t  0 (1  r )
t 0  1  r 
T 1
t
 1 
1 

1 r
r
1 r 
 C0  lim 
 C0 
 C0 
W
T 
r
1 r
 1 
1 

 1 r 
Therefore in this case we write C0 
ρ
0
.5
1
ρ=0
Y
t=0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
r
.1

1  r   1.1  
C
1/11
1/6
1
C
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
S
0.909091
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
-0.090909
ρ=0.5
Y
t=0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
1
0.5
0.25
0.125
0.0625
0.03125
0.015625
0.007813
0.003906
0.001953
0.000977
0.000488
0.000244
0.000122
6.1E-05
3.05E-05
1.53E-05
7.63E-06
3.81E-06
1.91E-06
C
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
S
0.833333
0.333333
0.083333
-0.041667
-0.104167
-0.135417
-0.151042
-0.158854
-0.16276
-0.164714
-0.16569
-0.166178
-0.166423
-0.166545
-0.166606
-0.166636
-0.166651
-0.166659
-0.166663
-0.166665
ρ=1
Y
t=0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
C
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
S
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2. Consumption Function
Assume a household lives for an infinite period of time, earns no income but begins
with financial wealth, FW0 = 1. The intertemporal budget constraint is of the form:

Ct
 FW0

t
t  0 (1  r )

Assume that the household maximizes
  u (C ) and   (1  r )  1 . Beginning in
t
t
t 0
period 0, and continuing through period 20, calculate Ct, and end of period wealth Bt
= (1+r)*Bt-1 + Yt – Ct when r = 0, .1, and 2.
t



Ct
C0
 1 
W 


C


0 

t
t
t  0 (1  r )
t  0 (1  r )
t 0  1  r 
T 1
 1 
1 

1 r
r
r
r
1 r 

 C0  lim
 C0 
 C0 
W
FW 
T 
r
1 r
1 r
1 r
 1 
1 

 1 r 
Therefore in this case we write C0 
R
.1
.2
2
r
1 r
C
1/11
1/6
2/3
Note that the original question said r = 2. I meant .2, either answer is acceptable.
t=0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
r = .1
B_t
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
0.909091
C
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
0.090909
r = .2
B_t
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
0.833333
C
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
0.166667
r =2
B_t
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
0.333333
C
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
0.666667
Empirical Work
3.
Real Cost of Capital and Investment
You want to check the sensitivity of investment in Japan to the real capital rental rate.
Use quarterly Japanese data available here in to construct a capital rental rate series.
Japanese Data
http://home.ust.hk/~davcook/jpndata.xls
a. Calculate inflation and investment goods inflation. Columns A and B contain
price indices for output, Pt, and investment goods, PtI. Calculate inflation for each
series as the continuous growth rate of each series.
I
 t  ln( Pt P )  tI  ln( Pt I )
Pt 1
t 1
b. Calculate the real interest rate. For each period, use the nominal interest rate in
Column E) as it and πt+1 from part a. rt = it - πt+1 (Note: This is a quarterly interest
rate. Interest rates are usually cited in annual terms. The annualized interest rate
is just 4 times the quarterly interest rate).
c. Calculate the relative price of investment goods. Calculate the relative price of
investment goods as the ratio of the investment index in column B) to the output
PI
price index in column A): ptI  ( t
)
Pt
d. Calculate the capital rental rate. Assume that the quarterly depreciation rate is δ =
.02. Use the above data to calculate the quarterly capital rental rate
Rt
  r    ( tI1   t 1 )  ptI . Compare this series with the series you would
Pt  t
get if you assumed that the real price of investment goods was always equal to pI
= 1: rt + δ. Calculate the average for both series over the period March-1971 to
December-2000.
e. Calculate the investment to capital ratio.
i.
Estimate the capital stock in the first quarter of 1970, t = 0. Assume in
R
that period, the marginal product of capital equals the average of t
Pt
Y1970:1 R
 . Use the information of output in
K1970:1 P
period 1970:1 from column C) along with the assumption that α = ⅓ to
calculate K0.
ii.
Calculate an investment series recursively. Starting from period 0, use
the investment series in column D) and the initial capital level
calculated in section ii) to estimate the capital stock in every period.
Kt+1 = (1- δ)Kt + It.
I
iii.
Calculate the investment to capital ratio in each period t K
t
Y
f. Calculate the output to capital ratio. ykt  ( t )
Kt
calculated in section d. 
g. Calculate the correlation between
It
Kt
&amp;
Rt
Pt
on the one hand and between
It
&amp; ykt over the period March-1971 to December-2000. Which seems to have
Kt
the stronger relationship?
http://home.ust.hk/~davcook/japandata2008ak.xls
Averages
Average
πt+1
πIt+1
r
p It
R/P
r+δ
I/K
R/P
Y/K
I/K
I/K
R/P
Y/K
R/P
0.007603
0.005296
0.004516
0.816935
0.022078
0.024516
0.836935
0.022078
0.044516
Y/K
1
-0.03201
1
0.771141
0.06783
1
The correlation between capital productivity and investment is very strong. The
relationship between the capital rental rate and investment is weak.
4. Precautionary Savings
Estimate the effect of volatility on savings rates using Chinese provincial data. The
following dataset has data on the average consumption and average disposable
income (both in current dollars).
Chinese Provincial Data
http:\\home.ust.hk\~davcook\ChinaPrecautionaryData.xls
a. Calculate the personal savings rate as
Disposable Incomet ,i  Consumption Expendituret ,i
st ,i 
. Calculate the average
Disposable Incomet ,i
from 2000 to 2003. s i 
1 2003
 st ,i
4 t  2000
level from the UN website. UN Main Aggregates Database. Download the series
“GDP, Implicit Price Deflators – National Currency”. Use the implicit price index
divided by 100 as the price level, P. To get constant dollar disposable income for
each year, YDt, divide the disposable income for a given year t, by Pt.
YDt ,i
c. Calculate the growth rate of real disposable income as gtYD
,i  ln(
YDt 1,i
).
Calculate this for each province for each year.
YD
d. Calculate the mean g i 
 iYD 
2003

t 1993


1 2003 YD
 gt ,i and standard deviation
11 t 1993
YD 2
gtYD
,i  g i
10
of real disposable income growth.
e. Calculate the correlation between the standard deviation of income growth and
the average savings rate.
See http:\\home.ust.hk\~davcook\ChinaPrecautionaryData2008AK.xls
Real Disposable
Average
Income
Standard
Growth
Growth
Deviation
Beijing
0.108649 0.041684
Tianjin
0.093695 0.038047
Hebei
0.077771 0.028354
Shanxi
0.087768 0.032846
Inner Mongolia
0.095334 0.036285
Liaoning
0.074128 0.023503
Jilin
0.086982 0.036311
Heilongjiang0.083018 0.028592
Shanghai 0.099522 0.070686
Jiangsu
0.088085 0.030953
Zhejiang
0.10171 0.041006
Anhui
0.074959 0.033784
Fujian
0.089102 0.027363
Jiangxi
0.09187 0.04253
Shandong 0.086446 0.027532
Henan
0.087572 0.035022
Hubei
0.078268 0.042346
Hunan
0.072872 0.048791
Guangdong 0.070275 0.040106
Guangxi
0.073767 0.058043
Hainan
0.058592 0.06376
Sichuan
0.069189 0.027381
Guizhou
0.067612 0.037958
Yunnan
0.073315 0.03858
Shaanxi
0.079985 0.030327
Gansu
0.073393 0.053883
Qinghai
0.074588 0.031945
Ningxia
0.070932 0.04352
Xinjiang
0.073145 0.036673
Average
2000-2003
0.195561
0.23376
0.24325
0.229084
0.225107
0.181447
0.193345
0.241641
0.246553
0.250955
0.250708
0.217767
0.265104
0.28813
0.257252
0.245308
0.174534
0.187275
0.203909
0.226042
0.232955
0.181427
0.213471
0.203621
0.15941
0.180062
0.19248
0.164615
0.214015
Correlation &lt; .02. There does not seem to be strong evidence for precautionary
savings as a source of high Chinese savings at the firm level.
```