PART III Growth Theory: The Economy in
the Very Long Run
Economic Growth I: Capital
Accumulation and Population Growth
Chapter 8 of Macroeconomics, 8th
edition, by N. Gregory Mankiw
ECO62 Udayan Roy
The Solow-Swan Model
• This is a theory of macroeconomic dynamics
• Using this theory,
– you can predict where an economy will be
tomorrow, the day after, and so on, if you know
where it is today
– you can predict the dynamic effects of changes in
• the saving rate
• the rate of population growth
• and other factors
Two productive resources and one
produced good
• There are two productive resources:
– Capital, K
– Labor, L
• These two productive resources are used to
produce one
– final good, Y
The Production Function
• The production function is an equation that
tells us how much of the final good is
produced with specified amounts of capital
and labor
Y = 5K L
• Y = F(K, L)
0.3 0.7
– Example: Y =
5K0.3L0.7
capital
0
1
2
3
4
labor
0
0
0
0
0
0
10
0
25.06
30.85
34.84
37.98
20
0
40.71
50.12
56.60
61.70
30
0
54.07
66.57
75.18
81.95
Constant returns to scale
• Y = F(K, L) = 5K0.3L0.7
– Note:
• if you double both K and
L, Y will also double
• if you triple both K and L,
Y will also triple
• … and so on
– This feature of the Y =
5K0.3L0.7 production
function is called
constant returns to
scale
Y = 5K0.3L0.7
labor
0
capital
0
0
1
0
2
0
3
0
4
0
10
0
25.06
30.85
34.84
37.98
20
0
40.71
50.12
56.60
61.70
The Solow-Swan model
assumes that production
functions obey constant
returns to scale
30
0
54.07
66.57
75.18
81.95
Constant returns to scale
• Definition: The production function F(K, L)
obeys constant returns to scale if and only if
– for any positive number z (that is, z > 0)
– F(zK, zL) = zF(K, L)
• Example: Suppose F(K, L) = 5K0.3L0.7.
– Then, for any z > 0, F(zK, zL) = 5(zK)0.3(zL)0.7 =
5z0.3K0.3z0.7L0.7 = 5z0.3 + 0.7K0.3L0.7 = z5K0.3L0.7 = zF(K, L)
At this point, you should be able to do
problem 1 (a) on page 232 of the textbook.
Constant returns to scale
• CRS requires F(zK, zL) = zF(K, L) for any z > 0
• Let z be 1/L.
• Then,
1  1
1
F  K , L   F (K , L)
L  L
L
 K  F (K , L) Y
F  ,1  

L
L
 L 
k = K/L denotes per
worker stock of capital
y = Y/L denotes
per worker
output
So, y = F(k, 1) .
From now on, F(k, 1) will
be denoted f(k), the per
worker production
function.
So, y = f(k) .
Per worker production function:
example
Y  F (K , L)  5K
Y

5K
0 .3
L
y  5k
L
0 .7

L
0 .3
0 .3
5K
L
0 .7
0 .3
5
1 0 .7
L
K
L
 f (k )
0 .3
0 .3
K 
 5 
 L 
0 .3
y = 5k0.3
14
12
10
At this point, you
should be able to do
problems 1 (b) and 3
(a) on pages 232 and
233 of the textbook.
8
This is what a typical per
worker production function
looks like: concave
6
4
2
0
0
5
10
15
20
25
k
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
y
0
5
6.156
6.952
7.579
8.103
8.559
8.964
9.330
9.666
9.976
10.266
10.537
10.793
11.036
11.267
11.487
11.698
11.900
12.095
12.282
The Cobb-Douglas Production
Function
• Y = F(K, L) = 5K0.3L0.7
– This production function is itself an instance of a
more general production function called the CobbDouglas Production Function
• Y = AKαL1 − α ,
• where A is any positive number (A > 0) and
• α is any positive fraction (1 > α > 0)
Per worker Cobb-Douglas production
function
• Y = AKαL1 − α implies y = Akα = f(k)

1 
Y  AK L
y
Y


1 
AK L
L
L


1 
AK L

1 
L L

K 

y  A    Ak
 L 
In problem 1 on page 232 of the textbook,
you get Y = K1/2L1/2, which is the CobbDouglas production with A = 1 and α = ½.
In problem 3 on page 233, you get Y = K0.3L0.7,
which is the Cobb-Douglas production with A
= 1 and α = 0.3.
Per worker production function: graph
Income, consumption, saving,
investment
• Output = income
• The Solow-Swan model assumes that each
individual saves a constant fraction, s, of his or
her income
• Therefore, saving per worker = sy = sf(k)
• This saving becomes an addition to the
existing capital stock
• Consumption per worker is denoted c = y – sy
= (1 – s)y
Income, consumption, saving,
investment: graph
Depreciation
• But part of the existing capital stock wears out
• This is called depreciation
• The Solow-Swan model assumes that a
constant fraction, δ, of the existing capital
stock wears out in every period
• That is, an individual who currently has k units
of capital will lose δk units of capital though
depreciation (or, wear and tear)
Depreciation
• Although 0 < δ < 1 is the fraction of existing
capital that wears out every period, in some
cases—as in problem 1 (c) on page 219 of the
textbook—depreciation is expressed as a
percentage.
• In such cases, care must be taken to convert
the percentage value to a fraction
– For example, if depreciation is given as 5 percent,
you need to set δ = 5/100 = 0.05
Depreciation: graph
DYNAMICS
Dynamics: what time is it?
• We’ll attach a subscript to each variable to
denote what date we’re talking about
• For example, kt will denote the economy’s per
worker stock of capital on date t and kt+1 will
denote the per worker stock of capital on
date t + 1
How does per worker capital change?
• A worker has kt units of capital on date t
• He or she adds syt units of capital through saving
• and loses δkt units of capital through
depreciation
• So, each worker accumulates kt + syt − δkt units of
capital on date t + 1
• Does this mean kt+1 = kt + syt − δkt?
• Not quite!
Population Growth
• The Solow-Swan model assumes that each
individual has n kids in each period
• The kids become adult workers in the period
immediately after they are born
– and, like every other worker, have n kids of their
own
– and so on
Population Growth
• Let the growth rate of any variable x be
denoted xg. It is calculated as follows:
xg 
change in the value of x
initial
value of x

new value of x  old value of x
old value of x
• Therefore, the growth rate of the number of
workers, Lg, is:
Lg 
( n  1)  1
1
n
How does per worker capital change?
Dynamics: algebra
Dynamics: algebra
k t 1 
1
n 1

 k t  sAk

t
 k t

Now we are ready for
dynamics!
Dynamics: algebra
k t 1 
A = 10
α = 0.3
δ = 0.1
n = 0.2
k0 = 12
s = 0.2
1
n 1

 k t  sAk

t
 k t

t
0
1
2
yt = Aktα
12 21.07436
12.51239 21.34038
12.94102 21.55711
syt
4.214872
4.268076
4.311423
1.2
1.251239
1.294102
3
4
5
6
7
8
9
10
13.29862
13.59632
13.84373
14.04907
14.2193
14.36031
14.47702
14.57357
4.346823
4.375789
4.399526
4.419003
4.434999
4.448147
4.458962
4.467862
1.329862
1.359632
1.384373
1.404907
1.42193
1.436031
1.447702
1.457357
kt
21.73412
21.87895
21.99763
22.09501
22.17499
22.24074
22.29481
22.33931
δkt
Dynamics: algebra
k t 1 
A = 10
α = 0.3
δ = 0.1
n = 0.2
k0 = 12
s = 0.2
1
n 1

 k t  sAk

t
 k t

t
0
1
2
yt = Aktα
12 21.07436
12.51239 21.34038
12.94102 21.55711
syt
4.214872
4.268076
4.311423
3
4
5
6
7
8
9
10
13.29862
13.59632
13.84373
14.04907
14.2193
14.36031
14.47702
14.57357
4.346823
At
this point, 1.329862
you should be able to
do
problems 11.359632
(d) and 3 (c) on
4.375789
pages
232 and1.384373
233 of the
4.399526
textbook.
give them a try.
4.419003 Please
1.404907
4.434999
1.42193
4.448147 1.436031
4.458962 1.447702
4.467862 1.457357
kt
21.73412
21.87895
21.99763
22.09501
22.17499
22.24074
22.29481
22.33931
δkt
1.2
1.251239
1.294102
Dynamics: algebra to graphs
k t 1 
n 1
k t 1  k t 
k t 1  k t 
k t  sy t   k t
n 1
 kt 
k t  sy t   k t
n 1
k t  sy t   k t  ( n  1) k t
n 1
 k t  k t 1  k t 
kt 
Although this is the basic Solow-Swan dynamic
equation, a simple modification will help us analyze
the theory graphically.
k t  sy t   k t
sy t   k t  nk t
n 1
sf ( k t )  (  n ) k t
n 1



n 1
n 1
kt
k t  sy t   k t  nk t  k t
n 1
sy t  (  n ) k t
n 1
Now we are ready for graphical analysis.
Dynamics: algebra to graphs
 k t  k t 1  k t 
This version of the Solow-Swan
equation will help us understand
the model graphically.
sf ( k t )  (  n ) k t
n 1
Dynamics: algebra to graphs
 k t  k t 1  k t 
sf ( k t )  (  n ) k t
n 1
This version of the Solow-Swan 2. The economy shrinks if and only if per
equation will help us understand worker saving and investment [sf(kt)] is
the model graphically.
less than (δ + n)kt.
1. The economy grows if and only if per 3. The economy is at a steady state if
and only if per worker saving and
worker saving and investment [sf(kt)]
investment [sf(kt)] is equal to (δ + n)kt.
exceeds (δ + n)kt.
4. (δ + n)kt is called break-even investment
Dynamics: graph
 k t  k t 1  k t 
sf ( k t )  (  n ) k t
n 1
kt 
sf ( k t )  (  n ) k t
n 1
Investment and
break-even
investment
(δ + n)kt
sf(kt)
k1
k*
Steady state
Capital per
worker, k
Moving toward the steady state
Investment
and
depreciation
kt 
sf ( k t )  (  n ) k t
n 1
(δ + n)kt
sf(kt)
k1
investment
Break-even investment
k1
k*
Steady state
Capital per
worker, k
Moving toward the steady state
Investment
and
depreciation
kt 
sf ( k t )  (  n ) k t
n 1
(δ + n)kt
sf(kt)
k1
k1 k2
k*
Steady state
Capital per
worker, k
Moving toward the steady state
Investment
and
depreciation
kt 
sf ( k t )  (  n ) k t
n 1
(δ + n)kt
sf(kt)
 k2
investment
Break-even
investment
k1 k2
k*
Steady state
Capital per
worker, k
Moving toward the steady state
Investment
and
depreciation
kt 
sf ( k t )  (  n ) k t
n 1
(δ + n)kt
sf(kt)
k2
k1 k2 k3 k*
Steady state
Capital per
worker, k
Moving toward the steady state
Investment
and
depreciation
kt 
sf ( k t )  (  n ) k t
n 1
(δ + n)kt
sf(kt)
As long as k < k*,
investment will
exceed break-even
investment,
and k will continue to
grow toward k*
k1 k2 k3 k*
Steady state
Capital per
worker, k
STEADY STATE
The steady state: algebra
• The economy eventually
reaches the steady state
• This happens when per
worker saving and
investment [sf(kt)] is
equal to break-even
investment [(δ + n)kt].
• For the Cobb-Douglas
case, this condition is:
kt 
sf ( k t )  (  n ) k t
n 1
sf ( k )  sAk
sA
 n

k
k


 (  n ) k
k
1 
1
 sA  1  
k 

  n
*
At this point, you should be able to do
problems 1 (c) and 3 (b) on pages 232 and 233
of the textbook. Please give them a try.
Steady State and Transitional
Dynamics: algebra
k t 1 
A=
α=
δ=
n=
k0 =
s=
1
n 1
10
0.3
0.1
0.2
12
0.2
k* =15.03185

 k t  sAk

t
 k t

1
 sA  1  
k 

  n
*
t
0
1
2
yt = Aktα
12 21.07436
12.51239 21.34038
12.94102 21.55711
syt
4.214872
4.268076
4.311423
1.2
1.251239
1.294102
3
4
5
6
7
8
9
10
13.29862
13.59632
13.84373
14.04907
14.2193
14.36031
14.47702
14.57357
4.346823
4.375789
4.399526
4.419003
4.434999
4.448147
4.458962
4.467862
1.329862
1.359632
1.384373
1.404907
1.42193
1.436031
1.447702
1.457357
kt
21.73412
21.87895
21.99763
22.09501
22.17499
22.24074
22.29481
22.33931
δkt
SOLOW-SWAN PREDICTIONS FOR
THE STEADY STATE
There is no growth!
• The Solow-Swan model predicts that
– Every economy will end up at the steady-state; in
the long run, the growth rate is zero!
• That is, k = k* and y = f(k*) = y*
– Growth is possible—temporarily!—only if the
economy’s per worker stock of capital is less than
the steady state per worker stock of capital (k < k*)
• If k < k*, the smaller the value of k, the faster the
growth of k and y
There is no growth!
From per worker to total
• We have seen that per worker capital, k, is
constant in the steady state
• Now recall that k = K/L
• and that L increases at the rate of n
• Therefore, in the steady state, the total stock
of capital, K, increases at the rate of n
From per worker to total
• We have seen that per worker income y = f(k)
is constant in the steady state
• Now recall that y = Y/L
• and that L increases at the rate of n
• Therefore, in the steady state, the total
income, Y, must also increase at the rate of n
• Similarly, although per worker saving and
investment, sy, is constant in the steady
state, total saving and investment, sY,
increases at the rate n
From per worker to per capita
• Recall that L = amount of labor employed
• Suppose P = amount of labor available
– This is the labor force
– But in the Solow-Swan model everybody is capable of
work, even new-born children
– So P can also be considered the population
• Suppose u = fraction of population that is not
engaged in production of the final good.
– u is assumed constant
• Then L = (1 – u)P or L/P = 1 – u
Per worker to per capita
Steady state: summary
Variable
Symbol
Steady state behavior
Capital per worker
k
Constant
Income per worker
y = f(k)
Constant
sy
Constant
Consumption per worker
c = (1 – s)y
Constant
Labor
L = (1 – u)P
Grows at rate n
Capital
K
Grows at rate n
Income
Y = F(K, L)
Grows at rate n
Saving and investment
sY
Grows at rate n
Population
P
Grows at rate n
Capital per capita
(1 – u)k
Constant
Income per capita
(1 – u)y
Constant
Saving and investment per capita
(1 – u)sy
Constant
Consumption per capita
(1 – u)c
Constant
Saving and investment per worker
SOLOW-SWAN PREDICTIONS FOR
CHANGES TO THE STEADY STATE
A sudden fall in capital per worker
• A sudden decrease in k could be caused by:
– Earthquake or war that destroys capital but not
people
– Immigration
– A decrease in u
• What does the Solow-Swan model say will be
the result of this?
A sudden fall in capital per worker
Investment
and
depreciation
kt 
sf ( k t )  (  n ) k t
n 1
(δ + n)kt
sf(kt)
At this point, you should be able to
do problem 2 on pages 232 and
233 of the textbook. Please give it
a try.
2. A gradual return to the steady state
k1
k*
1. A sudden decline
Capital per
worker, k
An increase in the saving rate
Investment
and
break-even
investment
(δ + n)kt
s2f(k)
s1f(k)
An increase in the saving
rate causes a temporary
spurt in growth. The
economy returns to a
steady state. But at the
new steady state, per
worker capital, output,
and saving are all higher.
Per worker consumption
is a bit trickier.
k1
*
k2
*
k
An increase in the saving rate
1. An increase in the saving rate raises investment …
2. … causing k to grow (toward a new steady state)
Investment
and
break-even
investment
(δ + n)kt
s2f(k)
s1f(k)
3. This raises steadystate per worker
output y* = f(k*) and
saving sy*.
4. The growth rate
begins at zero,
becomes positive for a
while, and eventually
returns to zero.
k1
*
k2
*
k
An increase in the saving rate
1. Recall that the saving rate is
a fraction between 0 and 1.
Investment
and
break-even
investment
2. What can we say about the
steady state levels of k, y, and c
when s = 0?
3. And when s = 1?
4. So, how is consumption
per worker, c, affected by
changes in s?
(δ + n)kt
f(k)
sf(k)
c*
f(k*)
sf(k*)
k*
k
Being the grasshopper is not good!
1. Recall that the saving rate is
a fraction between 0 and 1.
Investment
and
break-even
investment
2. What can we say about the
steady state levels of k, y, and c
when s = 0?
3. They are all zero!
(δ + n)kt
f(k)
sf(k) when s = 0
k*= 0
k
Being a miserly any is not good either!
1. Recall that the saving rate is
a fraction between 0 and 1.
2. What can we say about the
steady state levels of k, y, and c
when s = 1?
Investment
and
break-even
investment
(δ + n)kt
f(k) = sf(k)
c*= 0
f(k*)
sf(k*)
k*
k
Effect of saving on steady state
consumption
For the Cobb-Douglas case, it can
be shown that the Golden Rule
saving rate is equal to capital’s
share of all income, which is
approximately 30% or 0.30.
Steady state
consumption per
worker, c* = (1 – s)f(k*)
The US saving rate is well below
0.30. So, according to the SolowSwan model, if we save more we
will, in the long run, consume more
too!
Golden Rule
consumption
per worker
0
Golden Rule
Saving rate
1
Saving rate, s
A decrease in the saving rate
An increase in the saving rate
What do we get for thrift?
• In the long run, a higher rate of saving and
investment gives us
– A higher per worker income
– But not a faster rate of growth
• And consumption per worker, c* = (1 – s)y*,
may increase or decrease or stay unchanged
when s increases.
At this point, you should be able to do
problem 4 on page 233 of the textbook.
Please try it.
Whole lecture in one slide!
Solow-Swan Predictions Grid
kt+1, yt+1, Δkt k*, y*, i* = sy* c* = (1 – s)y*
kt
?
0
0
s
+
+
?
A
+
+
+
n
–
–
–
δ
–
–
–

kt 
k t 1 
sAk t  (  n ) k t
n 1
1
n 1

 k t  sAk
1

t
 k t

 sA  1  
k 

  n
*
Effect of saving: evidence
Faster population growth
3. This reduces steadystate per worker
output y* = f(k*),
saving sy*, and
consumption (1 – s)y*.
4. The growth rate
begins at zero,
becomes negative for a
while, and eventually
5. The effect is the same if
returns to zero.
depreciation increases
What do we get for having fewer kids?
• In the long run, a lower rate of population
growth gives us
– A higher per worker income
– But not a faster rate of growth
At this point, you should be able to do
problem 6 on page 233 of the textbook.
Please try it.
Faster population growth: evidence
Alternative perspectives on population growth
The Malthusian Model (1798)
– Predicts population growth will outstrip the
Earth’s ability to produce food, leading to the
impoverishment of humanity.
– Since Malthus, world population has increased
sixfold, yet living standards are higher than ever.
– Malthus neglected the effects of technological
progress.
Alternative perspectives on population growth
The Kremerian Model (1993)
– Posits that population growth contributes to
economic growth.
– More people = more geniuses, scientists &
engineers, so faster technological progress.
– Evidence, from very long historical periods:
• As world pop. growth rate increased, so did rate of growth
in living standards
• Historically, regions with larger populations have enjoyed
faster growth.
Alternative perspectives on population
growth: Kremer
Robert Solow and Trevor Swan