Macroeconomic Analysis 2003
Golden Rule of Saving and Capital
Accumulation
Lecture 3
1
Saving and Investment Ratios Across the World 1984
60.00
40.00
Series1
SI Global
I/Y = 0.5966(S/Y) - 5.9608
R2 = 0.1297
Saving Ratio
20.00
0.00
0.00
10.00
20.00
30.00
40.00
50.00
60.00
-20.00
-40.00
-60.00
Investment ratio
Lecture 3
2
How does a higher saving rate affect the level of output
in the steady state?
y  k
2 2
y2
y1
S  sy  sk
2
2
2
i  n   k
2
2
High saving country
S  sy  sk
1
1
Low saving country
Saving rate
affects level of
income but not
the growth rates
k2
k1
Lecture 3
k
K
L
3
How does the technological advancement affect the per capita
capital and per capita output in the steady state?
y  k
2 2
y2
Advanced Technology
i  n   k
2
2
S  sy  sk
2
2
2
Primitive Technology
y  k
1 1
y1
S  sy  sk
1
1
k2
k1
Lecture 3
k
K
L
4
Golden Rule for Saving and Capital Accumulation
y  k
y
Y
L
MPK  n  
i  n  k
k 1    n
C-max
S  sy  sk
Kg
Lecture 3
Kss
k
K
L
5
Golden Rule of Saving
What is the saving rate that maximises consumption?
c  y i
c  y    n k
Max c  y    n k
c
 k 1    n   0
dk
k 1    n 
MPK    n 
(1)
(2)
(3)
(4)
(5)
(6)
Purpose of all economic activities is consumption. The
capital stock where the slope of production function
(marginal product of capital) equals the slope of the
investment requirement line (depreciation plus the growth
rate of the population) as given in (6).
Lecture 3
6
A Numerical Example for the Golden Rule of Saving
Take 11 different saving rates, s = 0, 0.1, 0.2, ……., 1.0.
And capital stock like : 1, 4, 9, 16, 25, 36, 49, 64, 81
Y  0 .5 K L
Assume
 in per capita terms y
  0.05 ,   0.5 , A=0.5
 0.5k 0.5
Then calculate consumption, mpk and mpk –d as following
s-rate
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
k
1.00
4.00
9.00
16.00
25.00
36.00
49.00
64.00
81.00
y
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
dk
0.05
0.2
0.45
0.8
1.25
1.8
2.45
3.2
4.05
c
Saving
s
0.45
0.8
1.05
1.2
1.25
1.2
1.05
0.8
0.45
0.05
0.2
0.45
0.8
1.25
1.8
2.45
3.2
4.05
mpk
mpk-d
0.2500
0.1250
0.0833
0.0625
0.0500
0.0417
0.0357
0.0313
0.0278
0.2000
0.0750
0.0333
0.0125
0.0000
-0.0083
-0.0143
-0.0188
-0.0222
For this parametric configuration 50 percent saving rate maximises consumption.
Lecture 3
7
How High Should be the Saving Rate?
Saving Rate that Maximises Consumption
C-max = 1.25
y = 0.5*k0.5
y=2.5
k = 25
s1
s2
s*=1.25
Lecture 3
s4
s5
Saving rate
8
Lecture 3
ly
Lu Jap
xe
a
m n
b
Ne o u
rg
th
Ne erla
nd
w
Ze s
al
an
No d
rw
Po a y
rtu
ga
l
Sp
a
S w in
Sw ed
Un itz en
ite er l
a
d
Ki nd
Un ng
ite d om
d
St
at
es
Ita
lia
Au
st
r
B e ia
lg
iu
m
Ca
na
De da
nm
ar
Fi k
nl
an
Fr d
an
c
G
er e
m
an
G y
re
ec
Ic e
el
an
d
Ire
la
nd
Au
st
ra
Investment Ratio in OECD Countries:Average 1980-2000
35.00
30.00
25.00
20.00
15.00
10.00
5.00
0.00
9
0.00
Netherlands
Luxembourg
Japan
10
United States
United Kingdom
Switzerland
Sweden
Spain
Portugal
Norway
New Zealand
Lecture 3
Italy
Ireland
Iceland
Greece
Germany
France
Finland
Denmark
Canada
Belgium
Austria
Australia
Variation in the Saving Ratio Across OECD Countries: Average 1980-2000
35.00
30.00
25.00
20.00
15.00
10.00
5.00
Average Saving Rate in Growth Disaster Countries (1980-2000)
10.00
Series1
5.00
Za
m
bi
a
Se
ne
ga
Si
l
er
ra
Le
on
Ve
e
ne
zu
el
a,
RB
Ni
ge
r
Ha
iti
G
ha
na
M
ad
ag
as
ca
r
Ni
ca
ra
gu
a
-5.00
Ch
ad
CA
FR
0.00
-10.00
-15.00
-20.00
Lecture 3
11
Investment Ration in Growth Miracle Countries: Average 1980-2000
45.00
40.00
35.00
30.00
25.00
20.00
15.00
10.00
5.00
0.00
China
Hong
Kong,
China
Ireland
Korea,
Rep.
Japan
Lecture 3
Malta
Portugal
Singapore
Thailand
12
Lecture 3
13
Lecture 3
14
Optimal Saving and Consumption in Two Period Model
Intertemporal budget constraint:
w2
C2
100
  w1 
 C1 
 100 
 195.23
1 r
1 r
1.05
Optimal consumptions

195.23
C1 

 100.11
1    1.95
C 2  C1 1  r   

1  r   0.95100.121.05)  99.86
1   
If the interest rate is 10 percent
100
  100 
 190.9
1.1
190.9
C1 
 97.9
1   
C2  C1 1  r   0.9597.91.1)  102.3
Lecture 3
15
A Three Period Optimal Consumption-Saving Model
Max U  C , C , C   ln C   ln C   ln C
1 2
2 3
3
 1 2 3
Subject to:
C
W
C
W
1. C  2  3  W  2  3
1 1 r 
1 1 r 
2
1 r 
1 r 2
2. ( W , W , W ) = (120, 1200, -120)
3. (C 0, C 0, C 0)
1
1
2
3
2
3
What is the optimal consumption in each period C
( 1 , C2 ,
C ) if     1 and r  r  0 ?
3
1
2
1 2
Lecture 3
16
Four Optimisation Conditions

L C ,C ,C ,
1 2 3


C 
C
3 F
2

 ln C   ln C   ln C     C 

1
2
2
3
3
1
1  r  1  r 2 


our first order conditions:
(1) L  1    0 
 1
C C
C
1
1
1

L   2    0
1
(2)
 2

C C 1  r 
C
C 1  r 
1
2
2
1
C
2   1  r  C  C  1 r 
2
C
2 1 2
1

1
L   3  
 0 3 
(3)

C
2
C C 
2
3 C11  r 
1
3 1  r 
1

C
2
3   1  r 2


C

C

1

r

3
C
3 1 3
1
(4)
L    C  C2  C3  0
1 1 r 

1 r 2
   C1   2C1   3C1
Lecture 3
17
Consumption and Savings in Three Periods
Optimal Consumption in period 1: C 1 

1   2   3  ;
 1  r 
2
Optimal Consumption in period 2: C  C  1  r  
2 1 2
1     

2 3 

2


r

1


2
3
Optimal Consumption in period 3: C  C  1  r  
3 1 3
1     

2 3 

S  W  C  120  400  280
1 1 1
Optimal saving in period 2: S  W  C  1200  400 
2 2 2
Optimal saving in period 3: S  W  C  120  400 
3 3 3
Optimal saving in period 1:
Lecture 3
18
Policy Issues:
Tax, Saving and Consumption
• What is the impact in consumption and saving in the
above model
– If there is a 20 percent tax on interest income?
– If there is a 20 percent subsidy in it?
– What sort of tax system is better for increasing the
ratio of saving? Does a higher rate of VAT promote
saving or consumption?
– Does a higher rate of tax on labour income
encourage or discourage saving?
– Does a higher rate of tax on pension income
increase saving or consumption?
Lecture 3
19
Which sectors are hit hard when the labour income is
taxed more heavily than the capital income?
Factor Shares Across Production Sectors in the UK (ONS)
1.00
0.90
Labour share
Capital share
0.80
0.60
0.50
0.40
0.30
0.20
0.10
in
d
in
g
an
Ag
ric
ul
tu
re
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ua
ci
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rry
M
,g
a
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as
nu
g
fa
an
ct
d
ur
w
in
at
g
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W
su
ho
pp
C
le
o
ly
an
al
ns
e
t
sp
ru
an
or
ct
d
io
ta
re
n
nd
ta
il
co
tra
m
de
m
Fi
un
na
ic
at
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Pu
io
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n
tio
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bl
ic
er
n,
vi
ad
he
ce
m
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s
in
th
i
s
an
tra
d
tio
so
n
ci
al
O
w
th
or
er
k
Se
rv
ic
es
-
M
Share
0.70
Do workers in the
Service sector
lobby for
Higher capital
income tax?
Lecture 3
20
Which Sectors are important for a higher
economic growth in the Yorkshire and
Humberside region? (annual growth rates)
Agri
Mining Manf
EGW Constr Distrib Transp Fin.Se Pub.A Ed/hea Other Total Maxgr Mingr
uc
ution ort
rv
dm.
lth
GDP wth
wth
1983 -0.065 -0.130 0.056 0.124 0.129 0.104 0.091 0.136 0.144 0.072 0.058 0.072 0.144 -0.130
1984 0.336 -0.734 0.071 -0.181
0.105
0.126
0.160
0.106
0.111
0.126 0.052
0.040
0.336 -0.734
1985 -0.086 2.451 0.100 0.091
0.027
0.100
0.056
0.131
0.055
0.100 0.075
0.125
2.451 -0.086
1986 0.056 -0.029 0.128 0.053
0.048
0.120
0.089
0.170
0.053
0.122 0.146
0.109
0.170 -0.029
1987 0.033 -0.103 0.096 -0.003
0.226
0.082
0.082
0.052
0.058
0.096 0.113
0.077
0.226 -0.103
1988 -0.027 -0.116 0.104 -0.023
0.222
0.132
0.107
0.072
0.094
0.158 0.124
0.100
0.222 -0.116
1989 0.162 -0.030 0.097 -0.019
0.198
0.109
0.099
0.171
0.055
0.180 -0.081
0.113
0.198 -0.081
1990 0.100 -0.098 0.036 0.013
0.065
0.099
0.081
0.163
0.089
0.124 0.110
0.083
0.163 -0.098
1991 -0.014 0.116 -0.034 0.309 -0.088
0.079
0.056
0.046
0.044
0.145 -0.035
0.036
0.309 -0.088
1992 0.063 0.010 0.012 -0.044 -0.079
0.010
0.008
0.106
0.092
0.121 0.084
0.041
0.121 -0.079
1993 0.064 -0.265 0.057 -0.052
0.059
0.018
0.122
0.056
0.035
0.055 -0.005
0.043
0.122 -0.265
1994 -0.027 -0.412 0.041 -0.018
0.068
0.091
0.052
0.137
0.010
0.073 0.041
0.061
0.137 -0.412
1995 0.250 0.132 0.062 -0.080
0.007
0.092
0.072
0.062
0.038
0.041 0.102
0.061
0.250 -0.080
0.094 -0.013
0.027
0.044
0.017
0.004 0.060
0.027
0.101 -0.013
1996P 0.021 0.101 0.036 -0.003
Source: www.statistics.gov.uk
Lecture 3
21
Why is the Manufacturing Sector So
Important for Economic Growth?
Sectoral compositon of Imports in UK and EU
0.8
0.7
1 EU
2 UK
0.6
0.5
0.4
0.3
0.2
0.1
El
ec
tri
cit
5
y
C
on
st
ru
ct
io
n
6
D
ist
rib
ut
io
n
7
Tr
an
sp
or
t
8
Fi
nS
er
v
9
Ed
uc
at
io
n
10
O
th
er
4
nf
cs
M
3
M
2
1
Ag
ri
in
in
ng
0
Lecture 3
22
Sectoral Composition of Exports in the UK and the EU
0.8
0.7
0.6
1 EU
2 UK
0.5
0.4
0.3
0.2
0.1
r
th
e
O
10
Ed
uc
at
Fi
n
8
io
n
rt
Se
rv
9
7
Tr
a
ut
rib
6
D
is
t
ns
po
io
n
n
tru
ct
io
ci
ty
5
C
on
s
El
ec
tri
cs
4
M
nf
3
ng
M
in
in
2
1
Ag
ri
0
Lecture 3
23
Exercises
• Calculate the saving rate consistent with the
golden rule
• Study of Relative income levels across
regions in the UK
• Study of economic growth in Hulls and
Humberside
• Which are the leading sectors of Economic
Growth in the UK and in the Yorkshire and
Humberside?
Lecture 3
24