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The Theory of Economic Growth:
The Solow Growth Model
Reading: DeLong/Olney:
Macroeconomics; McGraw-Hill;
2006; Chapter 4
4-1
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Questions
• What are the causes of long-run
economic growth?
• What is the “efficiency of labor”?
• What is an economy’s “capital
intensity”?
• What is an economy’s “balancedgrowth path”?
• What can we say about convergence
to the “balanced growth path”?
4-2
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Questions
• How important is faster labor-growth
as a drag on economic growth?
• How important is a high saving rate
as a cause of economic growth?
• How important is technological and
organizational progress for economic
growth?
4-3
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Long-Run Economic Growth
• We classify the factors that generate
differences in productive potentials
into two broad groups
– differences in the efficiency of labor
• how technology is deployed and organization
is used
– differences in capital intensity
• how much current production has been set
aside to produce useful machines, buildings,
and infrastructure
4-6
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
The Efficiency of Labor
• The efficiency of labor has risen for
two reasons
– advances in technology
– advances in organization
• Economists are good at analyzing the
consequences of advances in
technology but they have less to say
about their sources
4-7
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Capital Intensity
• There is a direct relationship between
capital-intensity and productivity
– a more capital-intensive economy will be
a richer and more productive economy
4-8
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Standard Growth Model
• Also called the Solow growth model
• Consists of
– variables
– behavioral relationships
– equilibrium conditions
• The key variable is labor productivity
– output per worker (Y/L)
4-9
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Solow Growth Model
• Balanced-growth equilibrium
– the capital intensity of the economy (K/Y)
is stable, but (K/L) and (Y/L) grow
– the economy’s capital stock and level of
real GDP are growing at the same rate
– the economy’s capital-output ratio is
constant
4-10
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Solow Growth Model
• The balanced-growth path:
– if the economy is on its balanced-growth
path, the present value and future values
of output per worker will continue to
follow the balanced-growth path
– if the economy is not yet on its balancedgrowth path, it will head towards it
4-11
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
The Production Function
• The production function tells us how
the average worker’s productivity
(Y/L) is related to the efficiency of
labor (E) and the amount of capital at
the average worker’s disposal (K/L)
(Y/L)  F[(K/L), E]
• Cobb-Douglas production function
(Y/L)  (K/L)  (E)1-
4-12
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The Production Function
(Y/L)  (K/L)  (E)1-
•  measures how fast diminishing
marginal returns to investment set in
– the smaller the value of , the faster
diminishing returns are occurring
4-13
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Figure 4.2 - The Cobb-Douglas Production
Function for Different Values of 
4-14
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The Production Function
(Y/L)  (K/L)  (E)1-
• The value of the efficiency of labor (E)
tells us about the placement of the
production function
– a higher level of E means that more
output per worker is produced for each
possible value of the capital stock per
worker
4-15
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Figure 4.3 - The Cobb-Douglas Production
Function for Different Values of E
4-16
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The Production Function:
Example
• E = $10,000
•  = 0.3
• K/L = $125,000

Y K 
1 
0.3
0.7
   E   (125,000) (10,000)  $21,334
L L
4-17
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The Production Function:
Example
• If K/L rises to $250,000

Y K 
1 
0. 3
0 .7
   E   (250,000) (10,000)  $26,265
L L
– the first $125,000 of K/L increased Y/L
from $0 to $21,334
– the second $125,000 of K/L increased Y/L
from $21,334 to $26,265
4-18
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Saving, Investment, and
Capital Accumulation
• The net flow of saving is equal to the
amount of investment
• Remember from National Accounts
that real GDP (Y) can be divided into
four parts
– consumption (C)
– investment (I)
– government purchases (G)
– net exports (NX = GX - IM)
4-19
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Saving, Investment, and
Capital Accumulation
C  I  G  GX  IM  Y
C  I  (G  T )  GX  IM  Y  T
I  ( Y  T  C)  (G  T )  (IM  GX )
4-20
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Saving, Investment, and
Capital Accumulation
I  ( Y  T  C)  (G  T )  (IM  GX )
• The right-hand side shows the three
pieces of total saving
– household saving (SH)
– government saving (SG)
– foreign saving (SF)
I  SH  SG  SF
4-21
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Saving, Investment, and
Capital Accumulation
• Let’s assume that total saving is a
constant fraction (s) of real GDP
S S S
s
Y
• Therefore, it must be true that
H
G
F
I  sY
4-22
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Saving, Investment, and
Capital Accumulation
• We will refer to s as the economy’s
saving rate
– we will assume that it will remain at its
current value as we look far into the
future
– s measures the flow of saving and the
share of total production that is invested
and used to increase the capital stock
4-23
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Saving, Investment, and
Capital Accumulation
• The capital stock is not constant
• We will let
– K0 will mean the capital stock at some
initial year
– K2003 will mean the capital stock in 2003
– Kt will mean the capital stock in the
current year
– Kt+1 will mean the capital stock next year
– Kt-1 will mean the capital stock last year
4-24
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Saving, Investment, and
Capital Accumulation
• Investment will make the capital stock
tend to grow
• Depreciation makes the capital stock
tend to shrink
– the depreciation rate is assumed to be
constant and equal to 
4-25
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Saving, Investment, and
Capital Accumulation
• Next year’s capital will be
Kt 1  Kt  investment  depreciation
K t 1  K t  sYt  K t
• The capital stock is constant when
sYt  K t
Kt s

Yt 
4-26
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Saving, Investment, and
Capital Accumulation
• Suppose that the economy has no
labor force growth and no growth in
the efficiency of labor
– if K/Y < s/, depreciation is less than
investment so K and K/Y will grow until
K/Y = s/
– if K/Y > s/, depreciation is greater than
investment so K and K/Y will fall until K/Y
= s/
4-27
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Saving, Investment, and
Capital Accumulation
• Thus, if the economy has no labor
force growth and no growth in the
efficiency of labor, the equilibrium
condition of this growth model is
Kt s

Yt 
4-28
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Saving, Investment, and
Capital Accumulation
• Remember that, in this particular
case, we are assuming that
– the economy’s labor force is constant
– the economy’s capital stock is constant
– there are no changes in the efficiency of
labor
• Thus, equilibrium output per worker is
constant
• Now we will complicate the model
4-30
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Adding in Labor Force and
Labor Efficiency Growth
• 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  (1.02)  10,000,000  10,200,000
4-31
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Figure 4.6 - Constant Labor-Force Growth
(at n = 2% per Year)
4-32
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Adding in Labor Force and
Labor Efficiency Growth
• Assume that the efficiency of labor (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  (1.015)  $10,000  $10,150
4-33
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.7 - Efficiency-of-Labor
Growth at g = 1.5% per Year
4-34
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The Balanced-Growth CapitalOutput Ratio
• When we assumed that the labor
force and efficiency were both
constant (so that n and g are equal to
0), the equilibrium condition was
K s

Y 
• Since s and  are constant, K/Y is
constant
4-35
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The Balanced-Growth CapitalOutput Ratio
• If we assume that the labor force and
efficiency grow at n and g, the
equilibrium condition still requires that
K/Y is constant
– the economy is in balanced growth
– output per worker is growing at the same
rate as the capital stock per worker
• both growing at the same rate as the
efficiency of labor
4-36
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
The Balanced-Growth CapitalOutput Ratio
• The economy will be in balanced
growth equilibrium when
K
s

Y (n  g  )
• This is the balanced-growth
equilibrium capital-output ratio
4-37
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Balanced-Growth Output
per Worker
• Suppose that the economy is on its
balanced-growth path
– K/Y is equal to its balanced-growth
equilibrium value
• Let’s calculate the level of output per
worker (Y/L)
4-40
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Balanced-Growth Output
per Worker
• Begin with the capital-output ratio
version of the production function
Yt  K t 
  
L t  Yt 
4-41

1 
Et 
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Balanced-Growth Output
per Worker
• Since the economy is on its balancedgrowth path

Yt 
s

 
Lt  n  g   

1 
Et 
• Since s, n, g, , and  are all
constants, [s/(n+g+)]/1- is a
constant
4-42
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Balanced-Growth Output
per Worker
• Along the balanced-growth path,
output per worker is simply a constant
multiple of the efficiency of labor
• Over time, the efficiency of labor
grows at a constant rate g
– Y/L must be growing at the same
proportional rate
4-43
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.8 – Balanced Growth: Output per
Worker and the Efficiency of Labor
4-44
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Balanced-Growth Output
per Worker
• Capital intensity (K/Y) determines what
multiple Y/L is of the current labor
efficiency
– things that increase K/Y make the
balanced-growth equilibrium Y/L a higher
multiple of the efficiency of labor
– things that reduce K/Y make the balancedgrowth equilibrium Y/L a lower multiple of
the efficiency of labor
4-45
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Balanced-Growth Output
per Worker
• Changes in the capital intensity shift
the balanced-growth path up or down
• But the growth rate of Y/L along the
balanced-growth path is simply the
rate of growth of the efficiency of labor
– the material standard of living grows at
the same rate of labor efficiency
– changes in K/L alone will not accomplish
this
4-46
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.9 – Calculating Balanced-Growth Output
per Worker
Output
per Worker
[K/Y = s/(n+g+)]
Balanced-Growth
Y/K
current
output per
worker along
the balancedgrowth path
Y/L
(function of Et)
Capital Stock per Worker
4-47
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Figure 4.9 – Calculating Balanced-Growth Output
per Worker
Output
per Worker
Balanced-Growth
Y/K
Y/L will rise
Y/K’
Y/L
Anything that increases the balancedgrowth capital-output ratio will
rotate the capital-output line clockwise
Capital Stock per Worker
4-48
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Figure 4.9 – Calculating Balanced-Growth Output
per Worker
Output
per Worker
Y/K’
Balanced-Growth
Y/K
Y/L will fall
Y/L
Anything that decreases the balancedgrowth capital-output ratio will
rotate the capital-output line counterclockwise
Capital Stock per Worker
4-49
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Off the Balanced-Growth Path
• When K/Y > s/(n+g+)
– K/Y is falling because investment is
insufficient to keep K growing as fast as Y
• When K/Y < s/(n+g+)
– K/Y is rising because the growth in K
outruns the growth in Y
• K/Y and Y/L will converge to their
balanced-growth paths
4-50
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.10 - Convergence to a BalancedGrowth Capital-Output Ratio of 4
4-51
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How Fast the Economy Heads for
Its Balanced-Growth Path
• A fraction (1-)(n+g+) of the gap
between the economy’s current
position and its balanced-growth path
will be closed each year
– assume that this fraction is 4%
– according to the rule of 72,the economy
will move halfway to equilibrium in 72/4
or 18 years
– the convergence does not happen quickly
4-52
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Figure 4.11 – The Return of the West German
Economy to Its Balanced Growth Path
4-53
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The Labor Force Growth Rate
• The faster the growth of the labor
force, the lower will be the economy’s
balanced-growth K/Y ratio
– the larger the share of current
investment that must go to equip new
workers with the capital they need
• Thus, a sudden, permanent increase
in labor force growth will also lower
Y/L on the balanced-growth path
4-54
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Figure 4.12 – The Labor Force Growth
Rate Matters
4-55
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The Saving Rate and the Price
of Capital Goods
• The higher the share of real GDP
devoted to saving and gross
investment, the higher will be the
economy’s balanced-growth K/L ratio
– more investment increases the amount of
new capital
• A higher saving rate also increases
Y/L along the balanced-growth path
4-56
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Figure 4.13 - Investment Shares of Output
and Relative Prosperity
4-57
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Growth Rate of the Efficiency
of Labor
• An increase in g will reduce the
economy’s balanced-growth K/L ratio
– past investment will be small relative to
current output
• Changes in g change the growth rate
of Y/L along the balanced-growth path
– these effects are overwhelmed by the
direct effect of g on Y/L
4-58
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Growth Rate of the Efficiency
of Labor
• The growth rate of the standard of
living can change if and only if g
changes
– other factors can shift Y/L up but do not
permanently change the growth rate of
Y/L
4-59
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Table 4.2 – Effects of Increases in
Parameters on the Solow Growth Model
When there
is an
increase in
the
parameter…
4-60
The Effect on…
Equilibrium
K/Y
Level
of Y
Permanent Permanent
Level
Growth
Growth
of Y/L
Rate of Y
Rate of Y/L
s
Increases
Up
Up
n
Decreases
Up
Down Increases

Decreases
Down Down No change
No change
g
Decreases
Up
Increases
Up
No change
Increases
No change
No change
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
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
4-61
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Summary
• A second principal force driving longrun growth in output per worker is the
increases in capital intensity – the
ratio of the capital stock to output
4-62
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Summary
• The balanced-growth equilibrium in
the Solow growth model occurs when
the capital output ratio K/Y is constant
– when K/Y is constant, the capital stock
and real output are growing at the same
rate
4-63
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Summary
• The Cobb-Douglas production function
we use is Y/L = [K/L]E(1-)
– this is equivalent to Y/L = [K/Y]/1-(E)
– an increase in  makes the production
function steeper
– an increase in E makes the production
function shift up
4-64
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Summary
• In equilibrium, investment equals
saving: I = S = SH+SG+SF
– we assume S/Y = s = saving rate is
constant
4-65
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Summary
• The balanced-growth equilibrium value
of the capital output ratio K/Y is a
constant equal to the saving rate s
divided by the sum of the labor force
growth rate n, the labor efficiency
growth rate g, and the depreciation
rate 
– in balanced growth: K/Y = s/(n+g+)
4-66
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Summary
• If the economy’s actual value of K/Y is
initially greater than s/(n+g+), then
K/Y will fall until it reaches its
equilibrium value
• If the economy’s actual value of K/Y is
initially less than s/(n+g+), then K/Y
will rise until it reaches its equilibrium
value
4-67
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Summary
• It can take decades or generations for
K/Y to reach its balanced-growth
equilibrium value
4-68
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Summary
• An increase in the saving rate s, a
decrease in the labor force growth rate
n, or a decrease in the depreciation
rate  increases Y/L
– the growth rate of Y/L will accelerate as
the economy moves to its new higher
balanced-growth path
– once there, Y/L will grow at the same rate
as it did initially
4-69
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Summary
• In balanced-growth equilibrium, the
growth rate of output per worker
equals the growth rate of labor
efficiency g
– only increases in g can produce a lasting
increase in the growth rate of output per
worker
4-70
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.