MACROECONOMICS
Chapter 7
Economic Growth I:
Capital Accumulation and
Population Growth
Solow Growth Model
Real GDP in US is 5X its level 50 years
ago; per capita real GDP is 3X.
 In some poor countries, real GDP per
person is only 2-5% of US.
 Using the production function with only K,
L, and Θ, Robert Solow developed a very
abstract theory to capture growth.

2
Solow Model

To keep the analysis as simple as
possible, we will pretend that G=0, NX=0.


After we develop the model, we can see how
changing these parameters will affect the
results.
First, the contribution of capital to growth
and the importance of savings to capital
accumulation will be discussed.
3
Solow Model
The impact of increasing available labor
(dubbed population growth) will be
discussed after the basic model is
understood.
 The impact of technology change will be
the subject of the next chapter.

4
Accumulation of Capital
If L and Θ are fixed, the only factor of
production that will bring about growth of Y
is K.
 We will use the same approach of supply
of Y and demand for Y we used before, to
determine how much K will increase each
period.

5
Supply of Y
Y  F ( K , L)
zY  F ( zK , zL )
Let
1
z
L
Y
K 
 F  ,1
L
L 
y  f (k )
The production function is the familiar one with
constant returns to scale. The little trick of
defining z allows us to show the output as per
worker real GDP and the input as capital per
worker (also called capital-labor ratio).
The lower case depiction of the production
function, therefore, says that per worker output
depends on capital per worker.
6
Basic Rule of Derivatives
When x changes by one unit, by how many units will Z change?
Z  Ax r y s
dZ
s
r 1
 Ay rx
dx
Negative exponent means reciprocal.
7
Marginal Product of Capital
1
4
Y  20K L
3
4
Y
K
 20 
L
L
y  20k
MPK 
1
4
The MPK will be decreasing as
capital increases because as K
goes up the denominator
increases.
1
4
dY
dK
3
1
1
dY
4
4
 5L K
dK
dY
L
 5 
dK
K
dy
Using an arbitrary Cobb-Douglas
MPK 
function, we can see how the
dK
production function can be

dy
 1
presented in terms of GDP per
 20 1
dK
worker.
 4
L
3
4
Likewise, when capital-labor ratio
(k) increases, the marginal
product of k decreases.
The exercise here shows that
MPK=MPk and it doesn’t matter
If one uses Y or y.

1
 1  4 1
 4  K


dy
 1
 5 1 3
dK
 4 4
L K
MPk  5k
5
MPk  3




1
1
4
k4
8
The Shape of Prod. Fn.
y
Y
MPK
MPk
1
1
K
k
9
Demand for Y
The total output (GDP) is divided between C, I, G, and NX. For
simplicity, pretend that G=0 and NX=0. Then, Y = C + I
Let’s show this equation as per worker:
Y C I
 
L L L
y  ci
S
s
Y
C  (1  s )Y
c  (1  s ) y
Output per worker (y) is determined by capital
per worker (k). Given k, we know what y will be.
The output per worker (y) will be divided between
consumption per worker and investment per
worker according to the size of savings rate (s).
The higher the savings rate, more of the output
will be used for investment and less for
consumption.
y  (1  s ) y  i
sy  i
SI
10
Relationship of i and y
y
f(k)
y1
y=f(k) and i=sy
which is the
same as i=sf(k)
Consumption
per worker, c
sf(k)
y=c+i
Invesment
per worker, i
k1
k
What happens if s rises?
11
Depreciation
Some capital stock is used up.
 Some capital stock becomes obsolete.
 Some capital stock is broken.
 Collectively, let’s say, in general a certain
percentage of the capital will be lost per
year: δK.

12
Capital Accumulation
Investments add to the capital stock.
 Depreciation subtracts from the capital
stock.
 Net capital accumulation, ΔK, then,
must be I – δK.
 Per worker: Δk = i – δk
 Alternately, Δk = sf(k) – δk

13
Capital Accumulation
and Steady State
δk, i
δk
δk2>i
Δk*=i
sf(k)
δk1<i
k1
k*
k2 k (Capital per worker)
14
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
k
40.00
47.09
49.21
50.71
51.78
52.53
53.05
53.42
53.68
53.86
53.99
54.08
54.14
54.19
54.22
54.24
54.25
54.26
54.27
54.28
y
50.30
52.39
52.97
53.37
53.65
53.84
53.98
54.07
54.14
54.18
54.21
54.24
54.25
54.26
54.27
54.28
54.28
54.28
54.28
54.29
c
35.21
31.43
31.78
32.02
32.19
32.31
32.39
32.44
32.48
32.51
32.53
32.54
32.55
32.56
32.56
32.57
32.57
32.57
32.57
32.57
i
15.09
20.96
21.19
21.35
21.46
21.54
21.59
21.63
21.65
21.67
21.69
21.69
21.70
21.71
21.71
21.71
21.71
21.71
21.71
21.71
Dep
8.00
18.84
19.68
20.29
20.71
21.01
21.22
21.37
21.47
21.55
21.60
21.63
21.66
21.67
21.69
21.70
21.70
21.71
21.71
21.71
k acc
7.09
2.12
1.50
1.06
0.75
0.53
0.37
0.26
0.18
0.13
0.09
0.06
0.04
0.03
0.02
0.02
0.01
0.01
0.01
0.00
15
Calculating k*
At k*, sf(k)=δk or sf(k*)= δk*
 Likewise, s/δ=k*/f(k*)
 Suppose s=0.4, δ=0.2, f(k)=10k0.3333
 2 = k/10k0.3333
 8 = k2/10
 80 = k2
 k* ≈ 9

16
The Impact of War or
Natural Disaster
Inv; dep.
δk
sf(k)
k
k*
k
17
Sudden Drop in the
Savings Rate
Inv; dep.
δk
sf(k)
s’f(k)
k
k*
k
18
Drop in s
y  10k
1
3
1
3
y  10k
s  0.15
s  0.25
  0.15
sf ( k )  k
1
3
0.25(10k )  0.15k
1
3
2 .5
k k
.15
1
50 3
k k
3
3
2
16.6667  k
k  68.4
  0.15
sf ( k )  k
1
3
0.15(10k )  0.15k
1
3
10k  k
103 k  k 3
103  k 2
k  31.6
19
Test of Solow Model
Solow model says, ceteris paribus, higher
investment rates bring higher steady-state
capital and higher income per worker.
 How does one test this?
 What does Figure 7-6 show?

20
Golden Rule of Capital
What steady state level of capital per
worker is optimal?
 Define optimal as maximum consumption
per worker (well-being = consumption).
 The higher the s, the higher the k.
 But which k is the best one?

21
Which k Maximizes c?
y, i
f(k)
δk
s’>s’’>s’’’
s’f(k)
s’’f(k)
s’’’f(k)
k
22
Golden Rule k

Consumption per worker is at maximum
when the slope of δk is exactly equal to
the slope of f(k).
Slope of δk is δk/k = δ.
 Slope of f(k) is dy/dk. But dy/dk = dY/dK (see
slide #7)
 When δ = MPK, consumption per worker is
maximized.

23
Which k Maximizes c?
f(k)
y, i
δk
c
sf(k)
i
k
24
Example
If y = 10k0.25, and δ = 0.15, what is the golden rule k?
MPK = 2.5k-0.75
MPK = δ
2.5k-0.75 = 0.15
k0.75 = 16.6667
k = 16.66671.333
k ≈ 42.6
Compare with slide # 15!
25
Population Growth
Population growth rate is given as n.
 If the population growth is equal to labor
force growth, next year’s L will be (1+n)L.
 To distinguish this year’s L from next
year’s, let’s say Lt+1 = (1+n)Lt.
 For K/L to be constant, the growth rate of
K should also be n.


If y*=f(k*), then at equilibrium Y, K, and L all
grow at the rate of n.
26
Population Growth

At equilibrium, i had to just match
depreciation. Now that K has to also grow
at rate n to keep k constant, i has to
compensate for both depreciation and
required capital growth:
Δk = i – δk – nk
Δk = i – (δ + n)k
It has to be nk because growth rate of L
has to match growth rate of K.
27
Population Growth
The impact of n on the model is to make
the δk line steeper.
 The steady state will now be
0 = sf(k) - (δ+n)k
i = (δ+n)k

28
Slowing Population Growth
(δ+n2)k
(δ+n1)k
i
sf(k)
n2 < n1
k1
k2
k
k
29
Population Growth
In the steady state consumption per
worker is not increasing but GDP (Y) is
increasing at rate n.
 A higher n implies a lower k and a lower y.
Do higher n countries have lower per
capita incomes? Figure 7-13.
 Golden Rule capital is now MPK = δ+n

30
Is Population Growth a
Curse or a Blessing?
Malthus: resource constraint
 Kremer: innovation and technology.

31