Lecture 3 1
Macroeconomic Analysis 2003
Golden Rule of Saving and Capital Accumulation
Lecture 3 2
Saving and Investment Ratios Across the World 1984
I/Y = 0.5966(S/Y) - 5.9608
R2 = 0.1297
-60.00
-40.00
-20.00
0.00
20.00
40.00
60.00
0.00 10.00 20.00 30.00 40.00 50.00 60.00
Investment ratio
Sav
ing
Rat
io
Series1
SI Global
Lecture 3 3
How does a higher saving rate affect the level of output in the steady state?
11
sksyS
22ky
1y
L
Kk
222
sksyS 22kni
2y
1k 2k
Low saving country
High saving country
Saving rate affects level of income but not
the growth rates
Lecture 3 4
How does the technological advancement affect the per capita capital and per capita output in the steady state?
11
sksyS
22ky
1y
L
Kk
222
sksyS
22kni
2y
1k 2k
Primitive Technology
Advanced Technology
11ky
Lecture 3 5
Golden Rule for Saving and Capital Accumulation
ky
L
Yy
L
Kk
sksyS
kni
KssKg
C-max
nMPK
nk 1
Lecture 3 6
Golden Rule of SavingW h a t i s t h e s a v i n g r a t e t h a t m a x i m i s e s c o n s u m p t i o n ?
iyc ( 1 ) knyc ( 2 )
knycMax ( 3 )
01
nkdk
c ( 4 )
nk 1 ( 5 ) nMPK ( 6 )
P u r p o s e o f a l l e c o n o m i c a c t i v i t i e s i s c o n s u m p t i o n . T h ec a p i t a l s t o c k w h e r e t h e s l o p e o f p r o d u c t i o n f u n c t i o n( m a r g i n a l p r o d u c t o f c a p i t a l ) e q u a l s t h e s l o p e o f t h ei n v e s t m e n t r e q u i r e m e n t l i n e ( d e p r e c i a t i o n p l u s t h e g r o w t hr a t e o f t h e p o p u l a t i o n ) a s g i v e n i n ( 6 ) .
Lecture 3 7
T a k e 1 1 d i f f e r e n t s a v i n g r a t e s , s = 0 , 0 . 1 , 0 . 2 , … … . , 1 . 0 .A n d c a p i t a l s t o c k l i k e : 1 , 4 , 9 , 1 6 , 2 5 , 3 6 , 4 9 , 6 4 , 8 1
LKY 5.0 i n p e r c a p i t a t e r m s5.05.0 ky
A s s u m e 05.0 , 5.0 , A = 0 . 5
T h e n c a l c u l a t e c o n s u m p t i o n , m p k a n d m p k – d a s f o l l o w i n gs - r a t e k y d k c S a v i n g
sm p k m p k - d
0 . 1 1 . 0 0 0 . 5 0 0 . 0 5 0 . 4 5 0 . 0 5 0 . 2 5 0 0 0 . 2 0 0 00 . 2 4 . 0 0 1 . 0 0 0 . 2 0 . 8 0 . 2 0 . 1 2 5 0 0 . 0 7 5 00 . 3 9 . 0 0 1 . 5 0 0 . 4 5 1 . 0 5 0 . 4 5 0 . 0 8 3 3 0 . 0 3 3 30 . 4 1 6 . 0 0 2 . 0 0 0 . 8 1 . 2 0 . 8 0 . 0 6 2 5 0 . 0 1 2 50 . 5 2 5 . 0 0 2 . 5 0 1 . 2 5 1 . 2 5 1 . 2 5 0 . 0 5 0 0 0 . 0 0 0 00 . 6 3 6 . 0 0 3 . 0 0 1 . 8 1 . 2 1 . 8 0 . 0 4 1 7 - 0 . 0 0 8 30 . 7 4 9 . 0 0 3 . 5 0 2 . 4 5 1 . 0 5 2 . 4 5 0 . 0 3 5 7 - 0 . 0 1 4 30 . 8 6 4 . 0 0 4 . 0 0 3 . 2 0 . 8 3 . 2 0 . 0 3 1 3 - 0 . 0 1 8 80 . 9 8 1 . 0 0 4 . 5 0 4 . 0 5 0 . 4 5 4 . 0 5 0 . 0 2 7 8 - 0 . 0 2 2 2
F o r t h i s p a r a m e t r i c c o n f i g u r a t i o n 5 0 p e r c e n t s a v i n g r a t e m a x i m i s e s c o n s u m p t i o n .
A Numerical Example for the Golden Rule of Saving
Lecture 3 8
Saving rate
C-max = 1.25
Con
sum
ptio
n
s*=1.25
y=2.5
k = 25
y = 0.5*k0.5
s1 s2 s4 s5
How High Should be the Saving Rate? Saving Rate that Maximises Consumption
Lecture 3 9
Investment Ratio in OECD Countries:Average 1980-2000
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
Austra
lia
Austri
a
Belgium
Canada
Denmar
k
Finlan
d
Franc
e
Germ
any
Greec
e
Icela
nd
Irelan
dIta
ly
Japa
n
Luxe
mbo
urg
Nether
lands
New Zea
land
Norway
Portu
gal
Spain
Sweden
Switzer
land
United K
ingdom
United S
tate
s
Lecture 3 10
Variation in the Saving Ratio Across OECD Countries: Average 1980-2000
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00A
ustr
alia
Aus
tria
Bel
gium
Can
ada
Den
mar
k
Fin
land
Fra
nce
Ger
man
y
Gre
ece
Icel
and
Irel
and
Ital
y
Japa
n
Luxe
mbo
urg
Net
herla
nds
New
Zea
land
Nor
way
Por
tuga
l
Spa
in
Sw
eden
Sw
itzer
land
Uni
ted
Kin
gdom
Uni
ted
Sta
tes
Lecture 3 11
Average Saving Rate in Growth Disaster Countries (1980-2000)
-20.00
-15.00
-10.00
-5.00
0.00
5.00
10.00
CAFRChad
Ghana
Haiti
Mada
gasc
ar
Nicara
guaNige
r
Senega
l
Sierra
Leo
ne
Venezu
ela, R
B
Zambia
Series1
Lecture 3 12
Investment Ration in Growth Miracle Countries: Average 1980-2000
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
45.00
China HongKong,China
Ireland Korea,Rep.
Japan Malta Portugal Singapore Thailand
Lecture 3 13
Population by Age and Gender in the UK in 2000How Does it Affect Saving and Investment?
0500
1,0001,5002,0002,5003,0003,5004,0004,5005,000
in '0
00
femalemale
Lecture 3 14
Population by age and Gender in India in 2000What is the prospect of Saving and Investment?
0
20,000
40,000
60,000
80,000
100,000
120,000
140,000
in '0
00
FemaleMale
Lecture 3 15
Intertemporal budget constraint:
23.19505.1
100100
112
12
1
r
CC
r
ww
Optimal consumptions
11.10095.1
23.195
11
C
86.99)05.112.10095.011
112
rrCC
If the interest rate is 10 percent 9.190
1.1
100100
9.971
9.1901
C
3.102)1.19.9795.0112 rCC
Optimal Saving and Consumption in Two Period Model
Lecture 3 16
A Three Period Optimal Consumption-Saving Model
3ln
32ln
21ln
3,
2,
1CCCCCCUMax
S u b j e c t t o :
1 . 21
31
2121
31
21
r
W
r
WW
r
C
r
CC
2 . ( 1W , 2W , 3W ) = ( 1 2 0 , 1 2 0 0 , - 1 2 0 )3 . ( 1C 0 , 2C 0 , 3C 0 )
W h a t i s t h e o p t i m a l c o n s u m p t i o n i n e a c h p e r i o d (1
C , 2
C ,
3C ) i f 1
21 a n d 0
21 rr ?
Lecture 3 17
Four Optimisation Conditions
21
31
213ln32ln21ln,3,2,1 r
C
r
CCCCCCCCL F
o u r f i r s t o r d e r c o n d i t i o n s :
( 1 ) 01
1
1
CCL
1
1C
( 2 ) 01
2
2
1
rCCL
rCC
11
1
2
2
rC
C 1
21
2 rCC 1212
( 3 ) 02
113
3
1
r
CCL
211
1
3
3
rCC
213
1
3 rC
C 21
313rCC
( 4 ) 0
21
31
21
r
C
r
CCL
13121
CCC
Lecture 3 18
Consumption and Savings in Three Periods
O p t i m a l C o n s u m p t i o n i n p e r i o d 1 : 321 1
C ;
O p t i m a l C o n s u m p t i o n i n p e r i o d 2 :
321
121212
rrCC
O p t i m a l C o n s u m p t i o n i n p e r i o d 3 :
321
21321313
rrCC
O p t i m a l s a v i n g i n p e r i o d 1 : 280400120111
CWS
O p t i m a l s a v i n g i n p e r i o d 2 : 4001200222
CWS
O p t i m a l s a v i n g i n p e r i o d 3 : 400120333
CWS
Lecture 3 19
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 20
Factor Shares Across Production Sectors in the UK (ONS)
-
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Sh
are
Labour share
Capital share
Which sectors are hit hard when the labour income is taxed more heavily than the capital income?
Do workers in the Service sector lobby for Higher capital income tax?
Lecture 3 21
Agri Mining Manf EGW Construc
Distribution
Transport
Fin.Serv
Pub.Adm.
Ed/health
Other TotalGDP
Maxgrwth
Mingrwth
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
1996P 0.021 0.101 0.036 -0.003 0.094 -0.013 0.027 0.044 0.017 0.004 0.060 0.027 0.101 -0.013
Source: www.statistics.gov.uk
Which Sectors are important for a higher economic growth in the Yorkshire and
Humberside region? (annual growth rates)
Lecture 3 22
Sectoral compositon of Imports in UK and EU
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1 Agr
i
2 M
ininn
g
3 M
nfcs
4 Elec
tricit
y
5 Con
struc
tion
6 Dist
ribut
ion
7 Tr
ansp
ort
8 Fi
nSer
v
9 Edu
catio
n
10 O
ther
1 EU
2 UK
Why is the Manufacturing Sector So Important for Economic Growth?
Lecture 3 23
Sectoral Composition of Exports in the UK and the EU
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1 Agr
i
2 M
ininn
g
3 M
nfcs
4 Elec
tricit
y
5 Con
struc
tion
6 Dist
ribut
ion
7 Tr
ansp
ort
8 Fi
nSer
v
9 Edu
catio
n
10 O
ther
1 EU
2 UK
Lecture 3 24
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?