Transcript Neo-Classical Growth Model

Neo-Classical Growth Model
Large Variations in Labor per Person
(www.ggdc.net)
Hours per Worker 2001
Taiwan
South Korea
Singapore
Hong Kong
Japan
USA
EU
0
500
1,000
1,500
2,000
2,500
3,000
Variation in Labor Force Participaton
Employment as a share of Population
52.00%
50.00%
48.00%
46.00%
44.00%
42.00%
40.00%
38.00%
Europe
U.S.A
Japan
Hong Kong
Singapore South Korea
Taiwan
Main Differences in Countries are Due
to Variation in Labor Productivity
GDP per Worker
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
Th
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Ta
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ap
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Si
Ph
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pp
in
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al
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sia
M
Ko
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a
In
do
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sia
Ho
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0
Plan
• Come up with separate theories governing
both labor productivity and hours worked.
• First, labor productivity.
Rule of Thumb
• Growth Rate Rule of Thumb
.
.
.
.
Xt
Z X Y
Z
Zt 
      ln Z   ln X   ln Y
Yt
Z X Y
Z
• Productivity growth is output growth minus
labor growth.
.
.
.
.
Yt
y Y L
y
yt        ln y   ln Y   ln L
Lt
y Y L
y
Agricultural Era
• Prior to about 1775 or so, GDP per capita
remained stagnant in virtually every country
in the world.
• There were many technological advances
during this period.
• Greater, productivity per unit of land tended
to go into increasing the population level.
GDP per capita through history
Year
-5000
-1000
1
1000
1500
1800
Population
5
50
170
265
425
900
GDP per Capita
130
160
135
165
175
250
Macroeconomics by J. Bradford DeLong, Chap. 5
Pre-Industrial Revolution
Source: Angus Madisson, Measuring the Chinese Economy
GDP per Capita
1200
1000
1990 US$
800
China
600
Europe
400
200
0
50AD
960AD
1280
1400
1820
Industrial Revolution Spreads to NA,
W, Europe and East Asia
GDP Per Capita
12000
10000
1990 US$
8000
UK
USA
6000
Germany
Japan
4000
2000
0
1700
1820
1870
1913
1950
Capital Productivity
Capital Productivity
0.63
0.62
0.61
0.6
0.59
0.58
0.57
0.56
0.55
0.54
USA
EU
Productivity Catch Up: Europe
Source: Groningen Growth & Development Center
U.S.A
% of
1950 USA
% of
2003 USA
Growth
Rate
12.00 100.0%
33.97 100.0%
2.00%
France
5.63
46.9%
37.75
111.1%
3.46%
Germany
4.36
36.3%
30.01
88.3%
3.95%
UK
7.49
62.4%
28.01
82.5%
2.91%
Spain
2.60
21.7%
22.21
65.4%
4.94%
1990 US$, Average Output per Hour (Y/L)
Productivity Catch Up:
Latin America
Source: Groningen Growth & Development Center
1950
U.S.A
12.00
% of
2003
USA
100.0% 33.97
% of
Growth
USA
Rate
100.0% 2.00%
Argentina
6.16
51.4%
10.57
31.1%
1.04%
Brazil
2.48
20.7%
7.81
23.0%
2.21%
Chili
4.66
38.9%
14.07
41.4%
2.12%
Mexico
3.56
29.7%
10.24
30.1%
2.03%
Productivity Catch Up: East Asia
Source: Groningen Growth & Development Center
1950
% of USA 2003
% of USA Growth
Rate
U.S.A
12.00
100.0%
33.97
100.0%
2.00%
Japan
2.30
19.2%
24.78
73.0%
4.57%
1973
% of USA 2003
% of USA
Hong
Kong
7.49
35.0%
22.28
65.6%
4.74%
Korea
3.64
17.0%
14.25
42.0%
5.93%
Singapore 6.80
31.8%
19.63
57.8%
4.61%
Taiwan
20.4%
18.77
55.2%
6.33%
4.37
Growth by Region
GDP 1950
ACNZUS
9,288
% of ACNZUS GDP 1998
100%
26,146
% of ACNZUS Growth Rate
100%
2.16%
W. EUROPE
4,594
55.4
17,921
69%
LATIN
AMERICA
2,554
26.8
4,531
17%
ASIA
AFRICA
635
852
8.3
8.9
2,936
1,368
11%
5%
2.84%
1.19%
3.19%
0.99%
The World Economy, A Millienial Perspective by Angus Madisson
USA in Industrial Era
Y
• Average Productivity of Capital, K
shows no trend upward or downward.
• The shares of income devoted to capital
and labor show no trend.
• The average growth rate of output per
person has been positive and relatively
constant over time.
USA Factor Productivity 1980-2003
38
36
34
32
30
28
.62
26
.60
.58
.56
80
82
84
86
88
90
Capital Productivity
92
94
96
98
00
Labor Productivity
1999 US$
.64
Productivity Growth in Korea and
the USA
• Compare the post-war growth of labor
productivity of Korea and the US. The US
started out with much higher labor
productivity than the Korea. Both countries
have seen positive productivity growth,
Korea’s has been much faster.
Korean Labor Productivity goes from less than 10%
of USA to more than 40%.
Output per Hour
40.00
35.00
1999 US$
30.00
25.00
20.00
15.00
10.00
5.00
0.00
65 9 68 9 71 9 74 9 77 9 80 9 83 9 86 9 89 9 92 9 95 9 98
9
1
1
1
1
1
1
1
1
1
1
1
1
U.S.A
South Korea
Capital Productivity:
How do you measure capital
• Option: Count it. Take surveys of industries,
firms and households to find out the value
of the capital that they own.
– Problem: Expensive
• Use the perpetual inventory method, to
calculate the capital stock.
Capital Accumulation
• Capital is accumulated through investment
and is lost through depreciation.
Kt 1  Kt  I t  DNt
• Depreciation is not measured either. We
might assume a constant rate of
depreciation, δ.
Kt 1  (1   ) Kt  It
Perpetual Inventory Method
•
Steps
1. Estimate capital depreciation rate (usually
δ≈.08 for annual).
2. Guess initial capital stock (e.g. K1950 = I1950/
δ).
3. Solve recursively forward.
Relatively Stable Capital Productivity in
USA, decline in South Korea
1.2
1
0.8
U.S.A
0.6
South Korea
0.4
0.2
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
1977
1975
1973
1971
1969
1967
1965
0
Data
• Labor productivity is from All series derived from
this database need to be referred to as: “Groningen
Growth and Development Centre and The
Conference Board,
Total Economy Database, August 2004,
http://www.ggdc.net"
• Capital productivity data can be downloaded at
Center for International Comparisons at
University of Pennsylvania.
– URL Address: http://pwt.econ.upenn.edu/
Labor Force Growth
• Typically, we expect to see growth in the
labor force due to population growth.
• This has been true in the US and Korea.
• Note that the labor force growth rate has
been roughly constant over time in each
country (though faster in Korea).
Labor Force Growth
Workers (logged)
18.8
16.8
18.6
16.6
18.4
16.4
18.2
16.2
18.0
16.0
17.8
15.8
50
55
60
65
70
Korea
75
USA
80
85
90
•World Bank Global
Development
Database.
Objective
•
Construct an economic theory that is consistent
with these growth facts:
–
1.
2.
–
1.
In mature economies (like the USA)
Labor productivity grows at a roughly constant rate.
Capital productivity stays roughly constant.
In developing economies (like Korea)
Labor productivity grows faster than mature
economies.
2. Capital productivity shrinks over time.
Neo-classical Productivity Function
(see Branson, p.576)
• We assume constant returns to scale production
function
Yt  F (Kt , At Lt )
• Because of constant returns we can write this as a
productivity function
1
1
1
Yt   F ( Kt  , At Lt  )
Lt
Lt
Lt
Y
K
y

k

Lt then,
• Define t
Lt and t
t
t
yt
 F (kt , At )
Example: Cobb-Douglas
Yt
 (Kt )  ( At Lt )1
0   1
• Divide both sides by L
yt 
Yt
Lt

( Kt )  ( At Lt )1
Lt
( Kt )  ( At Lt )1

( Lt )  ( Lt )1
Kt  At Lt 1
(
) (
)  (kt )  ( At )1
Lt
Lt
• The labor productivity function is in capital per labor
unit and technology.
Marginal Product
• It is true (because of CRTS) that the marginal
effect of capital on output is equal to marginal
product of capital per labor unit on labor
productivity.
• Example: Cobb-Douglas
 1
Y
K
 1
1
  K ( AL)    
K
L
y
 1 1
 k A 
k
1
A
Labor Productivity Function
(Constant A)
y
f(k,A)
Slope = MPK=
Y
K
k
Diminishing Returns
• The production function has diminishing
returns to capital and so does the
productivity function have diminishing
returns to capital per labor unit.
 2Y
 2
1
0




1
K
(
AL
)


2
K
 2
K
    1  
L
A1 L
2

y
  2 1
    1 k A  L  2  L  0
k
Cobb-Douglas Capital Productivity
Function
• We can also write the capital productivity
function as a function of the capital labor
ratio.
• In the Cobb-Douglas case, average
productivity of capital

1
Yt
( Kt )  ( At Lt )1
yt ( Kt )  ( At Lt )
 

Kt
kt
Kt
( Kt )  ( Kt )1
Kt  At Lt 1
Yt yt


1
1


(
) (
)  (kt )  ( At ) 

Kt
Kt
K t
kt
Capital Productivity
• Capital productivity is a decreasing function
of the capital/labor ratio.
• Intuition: If you give more and more capital
to the same amount of workers, the output
that each machine will produce will go
down.
y
K
 2
1
   1 k ( A)  0
k
Capital Productivity Function
(Constant A)
Y
K
f (k , A)
k
k
Reading Supplement
•
•
The best intermediate macroeconomics text
on growth theory is Delong, Chapter 4.
The main difference between these notes
and Delong is that
1. what we call technology, At, Delong calls Et.
2. Delong emphasizes the importance of the capital output
ratio. We emphasize capital productivity which is the
inverse of the capital-output ratio.
Capital Accumulation
• The increase in the capital stock that occurs in
every period is gross investment, It, net of
depreciation, δKt.
Kt 1  Kt
It
Kt 1  (1   ) Kt  I t 


Kt
Kt
Growth Rates vs. Continuous
Growth Rates
• The growth rate over a period of time is written as the
difference in the variable over the period over its initial
start value. g X  X t  X t 1
t
X t 1
• When the change in the variable is not too large or the
length of the time period is not too long, the growth rate
is close to the continuous growth rate.
X
g    ln X t
X
X
t
Investment Rate
(see Branson, p. 579)
• Define the Investment rate,
st 
It
Yt
• The growth rate of capital is a function of
the investment rate, capital productivity and
the depreciation rate.
Yt
K
 st

Kt
Kt
Growth Rate of Capital Per Labor
Unit
• The growth rate of capital per labor unit is
equal to capital growth rate minus labor
growth rate
.
.
.
.
Yt
L

 s
 
k K L
Kt
L
k
K
L
Investment per Labor Unit
• We can define investment per labor unit as
It
Yt
 st
 st yt  st F (kt , At )
Lt
Lt
• Assume a constant investment rate
Investment per Worker Function
(Constant A)
y
sF(k,A)
k
Replacement Investment per Worker
• Assume you had a constant growth rate of labor
L
n
Lt
t
• If you invest just enough per worker to keep the
capital-labor ratio constant, you need to replace
depreciated capital and equip new workers. The
greater is k, the more replacement investment you
need to do.
rep (k )  (  n)k
Replacement Investment per Worker
Function (Constant s,n)
y,rep
sF(k,A)
(δ+n)k
k
Capital per Worker as an Engine of
Labor Productivity Growth.
• As long as investment per worker is greater
than replacement investment per worker, the
capital stock per worker will be growing.
• Holding A constant, this implies output per
worker will be growing.
Steady State Capital Stock
• Investment per worker is an increasing function the
capital per worker since it is proportional to output per
worker.
• However, both output per worker and investment per
worker are diminishing functions of k.
• Investment per worker increases with k at a nondiminishing rate.
• Implication: There is a steady state capital stock
where investment per worker is exactly equal to
replacement investment per worker.
Replacement Investment per Worker
Function (Constant s,n)
y,rep
sF(k,A)
(δ+n)k
k
k*
Steady State Capital per Worker
• Define steady-state k* will solve the function
sF (k , A)  (  n)k
*
*
• Cobb-Douglas Example
 
sF (k , Z )  s k
*
 
s
k*
(  n)
 1

 A1  (  n)k * 
 A1  1 
 
s
 A1  k *
(  n)
*
1
 s 
k 

(


n
)


*
1
1
A
Determinants of Long-term Capital
Productivity
• Investment Rates: When economies invest a high
percentage of their output, they can “support” a
high level of capital per worker.
– To maintain a steady level of capital per worker,
investment must be done in every period to replace
depreciated equipment and equip new workers.
– If investment levels are high, a high level of depreciated
capital can be replaced.
• A high ratio of capital to labor implies a low level
capital productivity and a relatively high level of
labor productivity.
Steady State Capital in Two Countries,
B and D , sB > sD
(Constant A,n)
sB F(k,A)
sD F(k,A)
(δ+n)k
k
kD*
kB*
Determinants of Long-term Capital
Productivity
• Labor Growth Rates: When economies have
high population growth, a large share of
investment must be used to equip new
workers with capital. This means that less
investment can be used to build up the
capital allocated to each worker.
• When the capital-labor ratio is low, capital
productivity is high and labor productivity
is low.
Steady State Capital in Two Countries,
D and B , nD > nB
(Constant A,s)
(δ+nB)k
(δ+nD)k
sF(k,A)
k
kD*
kB*
Growth
• If k < k*, investment per worker is greater
than replacement investment implying that
capital per worker and output per worker
are growing.
• If k > k*, investment per worker is less than
replacement investment needs and capital
per worker and output per worker must fall.
The End of Growth
• Capital cannot be the engine of labor productivity
growth because it has diminishing returns.
• An economy eventually develops a capital stock so
large that the amount of investment needed in
every period to replace depreciated capital is
greater than the extra investment capacity
generated by extra productivity.
• But, in the early industrializers, the UK USA, labor
productivity growth has continued for 200 years.
So, the theory is incomplete.
Long-Term Productivity Growth
35
30
25
20
USA
15
UK
10
5
0
UK
1870
1913
1950
USA
1973
1998
Phenomena we can understand.
1. We can understand why Korea has had a
higher growth rate than the USA. Korea had a
higher product of capital and thus Korean
investment had a bigger impact on output in
that country.
2. We can also explain why Korean capital
productivity fell over time. As Korea
accumulated capital, output per unit of capital
fell. Eventually, the high Korean investment
rate leads to lower capital productivity.
Phenomena we can’t understand
• We can’t explain why the US has maintained
constant labor productivity growth while still
maintaining a relatively constant level of output
per capital.
• The missing element is technology growth. Here
we assume that A is fixed. But we observe A
growing over time (as in the case of Ireland).
Exogenous Growth
Most closely follows Delong “Macroeconomics,” Section 4.3.
• We assume that the technology level grows
over time as a natural and costless byproduct of economic activity.
• Assume a constant growth rate of
technology: At 1  At  g A
At
Productivity and Rules of Thumb
•
Growth Rules of Thumb
1. If
.
.
.
X Y Z
X t  Yt  Zt      ln X   ln Y   ln Z
X Y Z
.
2. If
.
X
Y
X t  Yt      ln X   ln Y
X
Y

Therefore
.
.
.
X
Y
Z
X t  Yt  Zt       ln X   ln Y  ln Z
X
Y
Z


Growth Rate of Labor
Productivity
• The growth rate of productivity in the
Cobb-Douglas case is a weighted average of
capital per worker growth and technology
growth.
yt  (kt )  ( At )1
.
.
.
.
y
k
A
k
   (1   )    (1   ) g A
y
k
A
k
Labor Productivity Growth Rate
• This implies that the growth rate of output
and the growth rate of capital is a function
of the average productivity of capital.
 Yt

y
A
   s
 (  n)   (1   ) g
y
 Kt

Implications
• If the growth rate of capital per worker is faster
than the growth rate of technology, the growth
rate of capital per worker will be higher than the
growth rate of labor productivity.
• This, in turn, will imply that capital productivity
(the ratio of output to capital) will fall.
• This will in turn imply that labor productivity
growth & capital per worker growth will slow
down.
Capital Productivity & Growth
gk
gy
gA
Y 
 K
SS
Y
K
Two Phases of Growth
• Transition Path – Emerging economy with high
capital productivity experiences capital-investment
led growth in which the growth rate of labor
productivity is increasing faster than the world
frontier of technology. Along the transition path,
capital is growing faster than output and capital
productivity is falling.
• During much of the post-war period, Korea was
on its transition path.
Two Phases of Growth pt. 2
• Balanced Growth Path – On the balanced growth
path, labor productivity and capital per worker are
each growing at the same rate as the world
technology frontier.
• During the post-war period, the US was on its
balanced growth path.
• All balanced growth paths should increase at the
same rate (the growth rate of world technology
frontier, gA). However the positions of labor
productivity on the growth path may be different.
Transition to Balanced Growth
Path
USA
Japan
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1976
1974
1972
1970
1968
1966
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1964
%
Growth Rate Labor Productivity (5 year averages)
Dynamics of Productivity:
High Growth in Labor Productivity, but slowing to
technology growth rate
y
yBG
Balanced Growth Path
Transition Path
time
Initially high capital productivity,
but shrinking to steady state level.
[y ]
k
y
[ ]SS
k
Transition Path
Balanced Growth Path
time
Balanced growth path capital
productivity
• We can solve for capital productivity and
labor productivity along the balanced growth
path.
g A    s[ y ]SS  (  n)   (1   ) g A
k


A
(


n

g
)
y
y
SS
SS
 s[ ]  (  n)  [ ] 
k
k
s
 (  n  g ) 


s


A
BG
t
y
 1
At
Capital Productivity and Labor
Productivity
• Holding technology constant, there is a
negative relationship between capital
productivity and labor productivity.
yt   kt  ( At )

1

 yt
1
 kt 
 Yt 
1
   ( At )  yt  

 yt 
 Kt 
 1
• If you have a high ratio of machines to
workers, capital productivity will be low
and worker productivity high and vice
versa.
At
Capital productivity
• Different countries are likely to have
different investment growth rates and labor
force growth rates. Thus, they will have
different capital productivity levels in
steady state.
• Note that capital productivity along the
balanced growth path does not depend on
the level of technology.
High Investment Rates, Low
Capital Productivity
Investment Rates
45
40
30
25
20
15
10
5
0
19
65
19
67
19
69
19
71
19
73
19
75
19
77
19
79
19
81
19
83
19
85
19
87
19
89
19
91
19
93
19
95
19
97
19
99
% of GDP
35
U.S.A
South Korea
Investment, Long-term Capital
Productivity & Growth
gk
gA
Y 
 K
SS
Y
K
y 
B,
s
A
A
s
>
 
k 
SS
SS
y  , A
B
  A =A
k 
B
yA
yB
time
How do real wages and real capital rental
rates behave over the long-term?
• Factor prices (under perfect competition and CobbDouglas) are proportional to average productivity.
• Real wages are proportional to labor productivity.
Along the transition path, real wages will grow faster
than technology but slower than real capital per worker.
Along the balanced growth path, real wages will grow
at the same rate as technology.
• Capital rental rates will fall along the transition path
due to diminishing returns to capital. Along the
balanced growth path capital rental rates will remain
constant as capital productivity is constant.
Forecasting the Long-term Growth
Path
1. Choose parameters. Typically n is the average population or labor
force growth. Then α ≈ ⅓-½, δ ≈.07-.10. We might set the growth rate of
efficiency as gA ≈ .01-.02,

 yt 1
At  yt   
 kt 
2.
Calculate the current technology level.
3.
Forecast the constant path of labor efficiency.
4.
Calculate the steady-state capital ratio and multiply by the
future efficiency level.

 (  n  g ) 


s


A
yt T

At T  1  g

1
At T
Z

T
 At
Gap From Balanced Growth Path
• Along the balanced growth path, output reaches
a constant relative to technology

A
yt
Yt
 (  n  g ) 
BG
BG
BG
t  
yt    At
 

s
At At Lt


• Define the GAPt as the % difference between a
countries productivity relative to technology and
the long run balanced growth path.
t 
GAPt 
BG

BG

1
Convergence
BG






• We can show that
t
  

approximation.
t
 t 
holds in
• Solving this differential equation from some initial
starting point ζ0
BG


 t   BG



 t
 t
0

e

GAP

e
 GAP0
t


BG
BG

 

• The larger is the gap, the faster the country will be
growing.
• The parameter λ is the rate at which the transition path
closes called the convergence rate.
Convergence Rate
• For the Solow model, we can solve for the
convergence rate as λ = (1-α) (n+gA+δ)
• A country converges because the diminishing
returns to capital fail to keep up with replacement
investment needs when a country is quickly
accumulating capital.
• When returns to capital diminish quickly (when α
is small) or when replacement investment costs
increase quickly (when (n+gA+δ) is large) an
economy will converge quickly.
Convergence Rate
• Numerical example: (α = ⅓, n = .01, gA =
.01, and δ = .1) implies (λ = .04)
• Half Life is the number of periods for the
gap to be cut in half. ehl  .5  hl  .7

• When λ = .04, the half life is about 16.5
years. After 48 years, only 1/8 of the initial
gap should remain.
Will poor countries catch up with
rich countries?
• If all countries share the same technology, At,
countries will converge to a balanced growth path
along which they will grow at the same rate.
• The position of that balanced growth path is
determined by capital productivity which is
determined in steady-state by investment and
population growth rates.
• If a poor country has the same s and n, it will
catch up with the rich countries!
Education Levels
•
•
•
So far, we have thought of technology as
knowledge which grows exogenously.
One source of knowledge is human capital
(education, experience, etc.)
Workers in different countries may have
large differences in this area.
Variations in Education per Worker
Average Education (25+)
14
12
10
8
6
4
2
0
Brazil
Korea
USA
Capital Productivity
Capital Productivity
19
65
19
67
19
69
19
71
19
73
19
75
19
77
19
79
19
81
19
83
19
85
19
87
19
89
19
91
19
93
19
95
19
97
19
99
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
USA
Brazil
19
65
19
67
19
69
19
71
19
73
19
75
19
77
19
79
19
81
19
83
19
85
19
87
19
89
19
91
19
93
19
95
19
97
19
99
20
01
1990 US$
Labor Productivity
Labor Productivity
35.00
30.00
25.00
20.00
15.00
10.00
5.00
0.00
USA
Brazil
Human Capital
• Rewrite production function as

1
Yt  K t ( At HLt )
• Define H as the human capital level of the

work force
 years
1
H e
• Productivity function is
yt  kt ( At )1 e  years
• Along balanced growth path,
 (  n  g ) 


s


A
BG
t
y
 1
e

 years
1
At
yearsA > yearsB,
, AA
=
AB
yA
yB
time
Parameterization of human capital
• In cross-sectional studies of individual
workers, it is typically found (in many
countries) that an additional year of
schooling means 8-10% higher pay.
ln yt   ln kt  (1   ) ln( At )    years
y
 ln y
y


years years
• Parameterize φ = .08-.010
Numerical Exercise
• Assume that Brazil, Korea, and US have
access to the same technology and have
roughly the same capital productivity level.
• Calculate relative labor productivity as
predicted by the Solow model. Brazil
0.540619
Korea
 y BZ . KR



 years BZ . KR


  e1

USA  

USA
 years
y  

1
e


0.866858
World Technology Frontier?
•
Two problems with modeling At as some common
level of technology available in the world which
drives the long-run growth path.
1. As a matter of theory, we haven’t really explained
long-term growth. Long-term growth occurs through
the costless (and exogenous magic of) technology.
2.
Some countries on the balanced growth path seem
to have grown at different rates (for example, US
labor productivity growth has been faster over the
20th century than British. There have been long
periods of relatively slow technology growth in
developed countries, such as the productivity
slowdown of 1975-1995.
Effects of Saving
• In a closed economy or in the long run, domestic
investment is financed with domestic savings. Ct = (1s)*Yt
• Along the BGP, consumption per person is written as


Ct
Lt
s
BGP
 (1  s ) y 
 (1  s ) 

A
POPt
POPt
n

g





1
At 
Lt
POPt
• Savings has counter-veiling effects on consumption.
First, it directly reduces the share of output devoted
toward consumption. Second, it increases output along
the balance growth path.
Golden Rule Saving Rate
• Which saving rate maximizes consumption


per capita.
2


1
GR
1
1


s


A
n

g




At 
Lt

 (1  s GR )
sGR
POPt
1
 

GR 1
s 
1


1


A
n

g




At 
Lt
POPt
2 1

 (1  s GR )
s GR  1

1
sGR

GR


s

GR
(1  s ) 1  
Productivity Slowdown