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CHAPTER 4
The Theory of Economic Growth
4-1
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Questions
• What are the principal determinants
of long-run economic growth?
• What equilibrium condition is useful in
analyzing long-run growth?
• How quickly does an economy head
for its steady-state growth path?
4-2
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Questions
• What effect does faster population
growth have on long-run growth?
• What effect does a higher savings
rate have on long-run growth?
4-3
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Long-Run Economic
Growth...
• is the most important aspect of how
the economy performs
• can be accelerated by good economic
policies
• can be retarded by bad economic
policies
4-4
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Long-Run Economic Growth
• Policies and initial conditions affect
growth through two channels
– their impact on the level of technology
• multiplies the efficiency of labor
– their impact on the capital intensity of
the economy
• the stock of machines, equipment, and
buildings that the average worker has at his
or her disposal
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Technology
• leads to a higher efficiency of labor
– skills and education of the labor force
– ability of the labor force to handle
modern machines
– the efficiency with which the economy’s
businesses and markets function
• Economists are good at analyzing the
consequences of better technology
– have less to say about the sources
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Capital Intensity
• There is a direct relationship between
capital-intensity and productivity
• Two principal determinants
– investment effort
• the share of total production saved and
invested in order to increase the capital stock
– investment requirements
• how much of new investment is used to equip
new workers with the standard level of capital
or to replace worn-out or obsolete capital
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Standard Growth Model
• Also called the Solow model
• Steady-state balanced-growth
equilibrium
– the capital intensity of the economy is
stable
– the economy’s capital stock and level of
real GDP are growing at the same rate
– the economy’s capital-output ratio is
constant
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Standard Growth Model
• First component is the production
function
– tells us how the productive resources of
the economy can be used to produce and
determine the level of output
(Y/L)  F[(K/L), E]
• Cobb-Douglas production function
(Y/L)  (K/L)  (E)1-
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Standard Growth Model
(Y/L)  (K/L)  (E)1-
• Parameters of the model
– E is the efficiency of labor
• a higher level of E means that more output
per worker can be produced for each possible
value of the capital stock per worker
–  measures how fast diminishing
marginal returns to investment set in
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Standard Growth Model

1- 
(Y/L)  (K/L)  (E)
• 0<<1
– a level of  near zero means that the
extra amount of output made possible by
each additional unit of capital declines
very quickly as the capital stock
increases
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Figure 4.1 - The Cobb-Douglas Production
Function for Parameter  Near Zero
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Standard Growth Model
(Y/L)  (K/L)  (E)1-
• 0<<1
– a level of  near one means that the next
additional unit of capital makes possible
almost as large an increase in output as
the last unit of capital
– a level of  equal to one means that
changes in output are proportional to
changes in capital
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.2 - The Cobb-Douglas Production
Function for Parameter  Near 1
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Figure 4.3 - The Cobb-Douglas Production
Function is Flexible
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Standard Growth Model
• Growth in labor force (L)
– assume that L is growing at a constant
rate (n)
L t 1  (1  n)  L t
– if this year’s labor force is equal to 10
million and the growth rate is 2% per
year, next year’s labor force will be
L t 1  (1  0.02)  10  10.2 million
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.4 - Constant Proportional Labor-Force
Growth (at Rate n = 2 Percent per Year)
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Standard Growth Model
• Growth in the efficiency of labor (E)
– assume that E is growing at a constant
proportional rate (g)
Et 1  (1  g)  Et
– if this year’s efficiency of labor is $10,000
and the growth rate is 1.5% per year,
next year’s efficiency of labor will be
Et 1  (1  0.015)  $10,000  $10,150
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.5 - Constant Proportional Growth
in the Efficiency of Labor
(at Rate g = 1.5 Percent per Year)
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Standard Growth Model
• Saving and investment
– assume that a constant share of real GDP
is saved and invested each year (s)
– capital stock does not grow by full
amount of gross investment
•  represents the fraction of the capital stock
that wears out or is scrapped each year
Kt 1  Kt  (s  Yt ) - (δ  Kt )
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.6 - Changes in the Capital Stock
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Standard Growth Model
• An example of capital accumulation
– current real GDP = $8 trillion
– current capital stock = $24 trillion
– savings rate = 20%
– depreciation rate = 4%
Kt 1  Kt  (s  Yt ) - (δ  Kt )
Kt 1  $24  (0.2  $8) - (0.04  $24)
K t 1  $24.64 trillion
4-22
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Figure 4.7 - Additions to and Subtractions
from the Capital Stock
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Summary of Standard
Growth Model
• Three assumptions
– labor force grows at proportional rate n
– efficiency of labor grows at proportional
rate g
– there is a constant proportion of real GDP
saved and invested each year (s)
• The capital stock changes over time
due to investment and depreciation
• Cobb-Douglas production function
4-24
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Equilibrium in Standard
Growth Model
• Key economic variables in the growth
model are never constant
• Equilibrium occurs when the variables
are growing together at the same
proportional rate
• Steady-state balanced growth
– occurs when the capital-output ratio is
constant over time
4-25
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Three Mathematical Rules
• The proportional growth rate of a
product (PQ) is the sum of the
proportional growth rates of each P
and Q
• The proportional growth rate of a
quotient (E/Q) is the difference of the
proportional growth rates of the
dividend (E) and the divisor (Q)
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Three Mathematical Rules
• The proportional growth rate of a
quantity raised to an exponent (Qy) is
equal to the exponent (y) times the
growth rates of the quantity (Q)
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Growth of Capital per Worker
• The proportional growth rate of capital
per worker [g(kt)] is
(K t 1 /L t 1 )  (K t /L t )
g(k t ) 
(K t /L t )
• Since this is a growth rate of a
quotient, it will be equal to the growth
rate of capital minus the growth rate
of labor (n)
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.8 - Calculating the Proportional
Growth Rate of the Capital-per-Worker Ratio
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Growth of Capital per Worker
• The growth rate of capital stock is
K t 1  K t
Kt
• Next year’s capital stock is
Kt 1  Kt  (s  Yt ) - (δ  Kt )
• Making the substitution, we get
(K t  (s  Yt )  (δ  K t ))  K t s  Yt

δ
Kt
Kt
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Growth of Capital per Worker
• The proportional growth rate of capital
per worker is
s
g(k t ) 
δ n
(K t /L t )
• Let  represent the capital-output
ratio (K/Y)
g(k t )  (s/κ t )  δ  n
4-31
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Growth of Capital per Worker
g(k t )  (s/κ t )  δ  n
• All else equal, the rate of growth of
capital per worker will be lower
– the
(n)
– the
– the
– the
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higher the rate of labor force growth
higher the rate of depreciation ()
lower the rate of saving (s)
higher the capital-output ratio ()
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.9 - Capital-per-Worker Growth as a
Function of the Capital-Output Ratio
4-33
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Growth of Output per Worker
• With the Cobb-Douglas production
function, output per worker is
(Yt /L t )  (K t /L t )  (Et )1 
• The growth rate of output per worker
[g(yt)] will be
g(y t )  [  g(k t )]  (1  )  g
4-34
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.10 - Calculating the Growth Rate of
Output per Worker
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Growth of Output per Worker
g(y t )  [  g(k t )]  (1  )  g
• Since g(kt) is equal to [(s/t)--n], we
can substitute
g(y t )  [  {s/κ t - δ - n}]  (1  )  g
• and simplify to get
g(y t )  g  [  {s/κ t - (n  g  δ)}]
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Growth of Capital-Output Ratio
• Equilibrium occurs when the capitaloutput ratio (=K/Y) is constant
• The growth rate of the capital-output
ratio [g(t)] is equal to the difference
in the growth rate of capital [g(kt)]
and the growth rate of output [g(yt)]
g(κ t )  g(k t )  g(y t )
4-37
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Growth of Capital-Output Ratio
g(κ t )  g(k t )  g(y t )
• We can make substitutions for [g(kt)]
and [g(yt)]
g(κ t )  {s/κ t  δ  n}  [g    {s/κ t  (n  g  δ)}]
• and simplify to get
g(κ t )  (1  )  {s/κ t  (n  g  δ)}
4-38
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Growth of Capital-Output Ratio
g(κ t )  (1  )  {s/κ t  (n  g  δ)}
• The growth rate of the capital-output
ratio depends on the balance between
– investment requirements (n+g+)
– investment effort (s)
• All else equal, a higher investment
requirement will mean a lower growth
rate of the capital-output ratio
4-39
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.11 - Growth of the
Capital-Output Ratio
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Steady-State Growth
Equilibrium
• occurs when the capital-output ratio is
constant
g(κ t )  (1  )  {s/κ t  (n  g  δ)}
• If t>[s/(n+g+)]
– the capital-output ratio will be shrinking
• If t<[s/(n+g+)]
– the capital-output ratio will be growing
4-41
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.12 - Growth of the Capital-Output
Ratio as a Function of the Level of the
Capital-Output Ratio
4-42
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Steady-State Growth
Equilibrium
g(κ t )  (1  )  {s/κ t  (n  g  δ)}
• If t=[s/(n+g+)]
– the growth rate of the capital-output ratio
will be zero
– the capital-output ratio will be stable
(neither shrinking nor growing)
• *=[s/(n+g+)] is the equilibrium
level of the capital-output ratio
4-43
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.13 - Convergence of the CapitalOutput Ratio to Its Steady-State Value
4-44
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Steady-State Growth
Equilibrium
• When the capital-output ratio (t) is at
its steady state value (*)
– output per worker [g(yt)] is growing at
proportional rate g
– capital stock per worker is growing at the
same proportional rate g
– the economy wide capital stock is
growing at the proportional rate n+g
– real GDP is also growing at proportional
rate n+g
4-45
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Steady-State Growth Path
• When the capital-output ratio is at its
equilibrium value (*), the economy is
on its steady-state growth path
• From Chapter 3, we know that as long
as we are on the steady-state growth
path


s

(Yt /L t )  
n  g  δ 
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
1 
  


 1  
 Et  κ *
Et
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.14 - Calculating Steady-State
Output per Worker along the
Steady-State Growth Path
4-47
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Steady-State Growth Path
• An increase in the capital-output ratio
increases the capital stock directly
and indirectly
– extra output generated by new capital is
source for additional saving and
investment
• This leads to a multiplier effect of
anything that raises *
4-48
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.16 - The Growth Multiplier: The
Effect of Increasing the Capital-Output Ratio
on the Steady-State Output per Worker
4-49
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Steady-State Growth Path
• Let the growth multiplier () equal
[/1- ]
• Output per worker along the steadystate growth path will be
 Yt 
   κ *λ Et
 Lt 
4-50
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Steady-State Growth Path
• To calculate output per worker when
the economy is on its steady-state
growth path
– calculate the steady-state capital-output
ratio [*=s/(n+g+)]
– amplify the steady-state capital-output
ratio (*) by the growth multiplier
[=/(1- )]
– multiply by the current value of the
efficiency of labor (Et)
4-51
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.15 - Output per Worker on the
Steady-State Growth Path
4-52
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Reaching the Steady-State
Growth Path
• How long does it take for the capitaloutput ratio to adjust to its steadystate value (*)?
– an economy that is not on its steadystate growth path will close a fraction
[(1- )(n+g+)] of the gap between the
steady state value (*) and its current
value (t) in a year
4-53
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.17 - West German Convergence to
Its Steady-State Growth Path
4-54
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Labor Force Growth
• The faster the growth of the labor
force, the lower will be the economy’s
steady-state capital-output ratio
– the larger the share of current
investment that must go to equip new
workers with the capital they need
• A sudden, permanent increase in
labor force growth will lower output
per worker on the steady-state growth
path
4-55
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.18 - Labor Force Growth and
GDP-per-Worker Levels
4-56
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Figure 4.19 - Effects of a Rise in Population
Growth on the Economy’s Growth Path
4-57
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Increases in the
Depreciation Rate
• The higher the depreciation rate, the
lower will be the economy’s steadystate capital-output ratio
– existing capital stock wears out and must
be replaced more quickly
• An increase in the depreciation rate
will lower output per worker on the
steady-state growth path
4-58
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Productivity Growth
• The faster the growth rate of
productivity, the lower will be the
economy’s steady-state capital-output
ratio
– past investment will be small relative to
current output
• An increase in productivity growth will
raise output per worker along the
steady-state growth path
4-59
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Increases in the Saving Rate
• The higher the share of real GDP
devoted to saving and investment, the
higher will be the economy’s steadystate capital-output ratio
– more investment increases the amount of
new capital
• A higher saving rate also increases
output per worker along the steadystate growth path
4-60
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.20 - National Investment Shares
and GDP-per-Worker Levels
4-61
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Chapter Summary
• One principal force driving long-run
growth in output per worker is the set
of improvements in the efficiency of
labor springing from technological
progress
4-62
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter Summary
• A second principal force driving longrun growth in output per worker are
the increases in the capital stock
which the average worker has at his
or her disposal and which further
multiplies productivity
4-63
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter Summary
• An economy undergoing long-run
growth converges toward and settles
onto an equilibrium steady-state
growth path, in which the economy’s
capital-output ratio is constant
4-64
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter Summary
• The steady-state level of the capitaloutput ratio is equal to the economy’s
saving rate divided by the sum of its
labor force growth rate, labor
efficiency growth rate, and
depreciation rate
4-65
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.