Chapter 7 learning objectives
ƒ Learn the closed economy Solow model
ƒ See how a country’s standard of living
depends on its saving and population
growth rates
ƒ Learn how to use the “Golden Rule”
to find the optimal savings rate and capital
stock
CHAPTER 7
Economic Growth I
slide 1
How Solow model is different from
Chapter 3’s model
1. K is no longer fixed:
investment causes it to grow,
depreciation causes it to shrink.
2. L is no longer fixed:
population growth causes it to grow.
3. The consumption function is simpler.
CHAPTER 7
Economic Growth I
slide 13
1
How Solow model is different from
Chapter 3’s model
4. No G or T
(only to simplify presentation;
we can still do fiscal policy experiments)
5. Cosmetic differences.
CHAPTER 7
Economic Growth I
slide 14
The production function
ƒ In aggregate terms: Y = F (K, L )
ƒ Define: y = Y/L = output per worker
k = K/L = capital per worker
ƒ Assume constant returns to scale:
zY = F (zK, zL ) for any z > 0
ƒ Pick z = 1/L. Then
Y/L = F (K/L , 1)
y = F (k, 1)
y = f(k)
where f(k) = F (k, 1)
CHAPTER 7
Economic Growth I
slide 15
2
The production function
Output per
worker, y
f(k)
1
MPK =f(k +1) – f(k)
Note:
Note: this
this production
production function
function
exhibits
diminishing
MPK.
exhibits diminishing MPK.
Capital per
worker, k
CHAPTER 7
Economic Growth I
slide 16
The national income identity
ƒ Y=C+I
(remember, no G )
ƒ In “per worker” terms:
y=c+i
where c = C/L and i = I/L
CHAPTER 7
Economic Growth I
slide 17
3
The consumption function
ƒ s = the saving rate,
the fraction of income that is saved
(s is an exogenous parameter)
Note: s is the only lowercase variable
that is not equal to
its uppercase version divided by L
ƒ Consumption function: c = (1–s)y
(per worker)
CHAPTER 7
Economic Growth I
slide 18
Saving and investment
ƒ saving (per worker) = y – c
= y – (1–s)y
=
sy
ƒ National income identity is y = c + i
Rearrange to get: i = y – c = sy
(investment = saving, like in chap. 3!)
ƒ Using the results above,
i = sy = sf(k)
CHAPTER 7
Economic Growth I
slide 19
4
Output, consumption, and investment
Output per
worker, y
f(k)
c1
sf(k)
y1
i1
Capital per
worker, k
k1
CHAPTER 7
Economic Growth I
slide 20
Depreciation
Depreciation
per worker, δk
δδ == the
the rate
rate of
of depreciation
depreciation
== the
the fraction
fraction of
of the
the capital
capital stock
stock
that
wears
out
each
period
that wears out each period
δk
1
δ
Capital per
worker, k
CHAPTER 7
Economic Growth I
slide 21
5
Capital accumulation
The
Thebasic
basicidea:
idea:
Investment
Investmentmakes
makes
the
thecapital
capitalstock
stockbigger,
bigger,
depreciation
depreciationmakes
makesititsmaller.
smaller.
CHAPTER 7
Economic Growth I
slide 22
Capital accumulation
Change in capital stock = investment – depreciation
Δk
=
i
–
δk
Since i = sf(k) , this becomes:
Δk = s f(k) – δk
CHAPTER 7
Economic Growth I
slide 23
6
The equation of motion for k
Δk = s f(k) – δk
ƒ the Solow model’s central equation
ƒ Determines behavior of capital over time…
ƒ …which, in turn, determines behavior of
all of the other endogenous variables
because they all depend on k.
E.g.,
income per person: y = f(k)
consump. per person: c = (1–s) f(k)
CHAPTER 7
Economic Growth I
slide 24
The steady state
Δk = s f(k) – δk
If investment is just enough to cover depreciation
[sf(k) = δk ],
then capital per worker will remain constant:
Δk = 0.
This constant value, denoted k*, is called the
steady state capital stock.
CHAPTER 7
Economic Growth I
slide 25
7
The steady state
Investment
and
depreciation
δk
sf(k)
k*
CHAPTER 7
Capital per
worker, k
Economic Growth I
slide 26
Moving toward the steady state
Investment
and
depreciation
Δk = sf(k) − δk
δk
sf(k)
Δk
investment
depreciation
k1
CHAPTER 7
k*
Economic Growth I
Capital per
worker, k
slide 27
8
Moving toward the steady state
Investment
and
depreciation
Δk = sf(k) − δk
δk
sf(k)
Δk
k1 k2
CHAPTER 7
k*
Capital per
worker, k
Economic Growth I
slide 29
Moving toward the steady state
Investment
and
depreciation
Δk = sf(k) − δk
δk
sf(k)
Δk
investment
depreciation
k2
CHAPTER 7
k*
Economic Growth I
Capital per
worker, k
slide 30
9
Moving toward the steady state
Investment
and
depreciation
Δk = sf(k) − δk
δk
sf(k)
Δk
k2 k3 k*
CHAPTER 7
Capital per
worker, k
Economic Growth I
slide 32
Moving toward the steady state
Investment
and
depreciation
Δk = sf(k) − δk
δk
sf(k)
Summary:
Summary:
*
As
long
As long as
as kk << kk*,,
investment
investment will
will exceed
exceed
depreciation,
depreciation,
and
k
and k will
will continue
continue to
to
**.
grow
toward
k
grow toward k .
k3 k*
CHAPTER 7
Economic Growth I
Capital per
worker, k
slide 33
10
Now you try:
Draw the Solow model diagram,
labeling the steady state k*.
On the horizontal axis, pick a value greater
than k* for the economy’s initial capital
stock. Label it k1.
Show what happens to k over time.
Does k move toward the steady state or
away from it?
CHAPTER 7
Economic Growth I
slide 34
A numerical example
Production function (aggregate):
Y = F (K , L ) = K × L = K 1 / 2L1 / 2
To derive the per-worker production function,
divide through by L:
1/2
Y K 1 / 2L1 / 2 ⎛ K ⎞
=
=⎜ ⎟
L
L
⎝L ⎠
Then substitute y = Y/L and k = K/L to get
y = f (k ) = k 1 / 2
CHAPTER 7
Economic Growth I
slide 35
11
A numerical example, cont.
Assume:
ƒ s = 0.3
ƒ δ = 0.1
ƒ initial value of k = 4.0
CHAPTER 7
Economic Growth I
slide 36
Approaching the Steady State:
A Numerical Example
Assumptions:
Year
Year
11
22
33
kk
4.000
4.000
4.200
4.200
4.395
4.395
CHAPTER 7
y = k ; s = 0.3; δ = 0.1; initial k = 4.0
yy
2.000
2.000
2.049
2.049
2.096
2.096
cc
1.400
1.400
1.435
1.435
1.467
1.467
Economic Growth I
ii
0.600
0.600
0.615
0.615
0.629
0.629
δk
δk
0.400
0.400
0.420
0.420
0.440
0.440
Dk
Dk
0.200
0.200
0.195
0.195
0.189
0.189
slide 37
12
Approaching the Steady State:
A Numerical Example
Assumptions:
Year
Year
11
22
33
44
……
10
10
……
25
25
……
100
100
……
∞∞
kk
y = k ; s = 0.3; δ = 0.1; initial k = 4.0
yy
cc
ii
δk
δk
Dk
Dk
4.000
4.000
4.200
4.200
2.000
2.000
2.049
2.049
1.400
1.400
1.435
1.435
0.600
0.600
0.615
0.615
0.400
0.400
0.420
0.420
0.200
0.200
0.195
0.195
5.602
5.602
2.367
2.367
1.657
1.657
0.710
0.710
0.560
0.560
0.150
0.150
7.351
7.351
2.706
2.706
1.894
1.894
0.812
0.812
0.732
0.732
0.080
0.080
8.962
8.962
2.994
2.994
2.096
2.096
0.898
0.898
0.896
0.896
0.002
0.002
9.000
9.000
3.000
3.000
2.100
2.100
0.900
0.900
0.900
0.900
0.000
0.000
4.395
4.395
4.584
4.584
CHAPTER 7
2.096
2.096
2.141
2.141
1.467
1.467
1.499
1.499
0.629
0.629
0.642
0.642
0.440
0.440
0.458
0.458
Economic Growth I
0.189
0.189
0.184
0.184
slide 38
Exercise: solve for the steady state
Continue to assume
s = 0.3, δ = 0.1, and y = k 1/2
Use the equation of motion
Δk = s f(k) − δk
to solve for the steady-state values of
k, y, and c.
CHAPTER 7
Economic Growth I
slide 39
13
Solution to exercise:
Δk = 0
def. of steady state
s f (k *) = δ k *
0.3 k * = 0.1k *
3=
eq'n of motion with Δk = 0
using assumed values
k*
= k *
k*
Solve to get: k * = 9 and y * = k * = 3
Finally, c * = (1 − s )y * = 0.7 × 3 = 2.1
CHAPTER 7
Economic Growth I
slide 40
An increase in the saving rate
An increase in the saving rate raises investment…
…causing the capital stock to grow toward a new steady state:
Investment
and
depreciation
dk
s2 f(k)
s1 f(k)
k 1*
CHAPTER 7
Economic Growth I
k 2*
k
slide 41
14
Prediction:
ƒ Higher s ⇒ higher k*.
ƒ And since y = f(k) ,
higher k* ⇒ higher y* .
ƒ Thus, the Solow model predicts that countries
with higher rates of saving and investment
will have higher levels of capital and income
per worker in the long run.
CHAPTER 7
Economic Growth I
slide 42
International Evidence on Investment
Rates and Income per Person
Income per
person in 1992
(logarithmic scale)
100,000
U.S.
10,000
Mexico
Egypt
Germany
Japan
Finland
Brazil
Pakistan
Ivory
Coast
U.K.
Israel
FranceItaly
Singapore
Peru
Indonesia
1,000
Zimbabwe
India
Chad
100
Canada
Denmark
0
Uganda
5
Kenya
Cameroon
10
15
20
25
30
35
40
Investment as percentage of output
(average 1960 –1992)
CHAPTER 7
Economic Growth I
slide 43
15
The Golden Rule: introduction
ƒ Different values of s lead to different steady states.
How do we know which is the “best” steady state?
ƒ Economic well- being depends on consumption,
so the “best” steady state has the highest possible
value of consumption per person: c* = (1–s) f(k*)
ƒ An increase in s
• leads to higher k* and y*, which may raise c*
• reduces consumption’s share of income (1–s),
which may lower c*
ƒ So, how do we find the s and k* that maximize c* ?
CHAPTER 7
Economic Growth I
slide 44
The Golden Rule Capital Stock
*
k gold
= the Golden Rule level of capital,
the steady state value of k
that maximizes consumption.
To find it, first express c* in terms of k*:
c*
=
y*
− i*
= f (k*) − i*
= f (k*) − δk*
CHAPTER 7
Economic Growth I
In general:
i = Δk + δk
In the steady state:
i* = δk*
because Δk = 0.
slide 45
16
The Golden Rule Capital Stock
steady state
output and
depreciation
Then, graph
f(k*) and δk*,
and look for the
point where the
gap between
them is biggest.
f(k*)
*
c gold
*
*
y gold
= f (k gold
)
CHAPTER 7
δk*
*
*
i gold
= δ k gold
*
k gold
steady-state
capital per
worker, k*
Economic Growth I
slide 46
The Golden Rule Capital Stock
c* = f(k*) − δk*
is biggest where
the slope of the
production func.
equals
the slope of the
depreciation line:
δk*
f(k*)
*
c gold
MPK = δ
*
k gold
CHAPTER 7
Economic Growth I
steady-state
capital per
worker, k*
slide 47
17
The transition to the
Golden Rule Steady State
ƒ The economy does NOT have a tendency to
move toward the Golden Rule steady state.
ƒ Achieving the Golden Rule requires that
policymakers adjust s.
ƒ This adjustment leads to a new steady state
with higher consumption.
ƒ But what happens to consumption
during the transition to the Golden Rule?
CHAPTER 7
Economic Growth I
slide 48
Starting with too much capital
*
If k * > k gold
then increasing
c* requires a
fall in s.
y
In the transition
to the
Golden Rule,
consumption is
higher at all
points in time.
c
CHAPTER 7
i
t0
Economic Growth I
time
slide 49
18
Starting with too little capital
*
If k * < k gold
then increasing c*
requires an
increase in s.
y
Future generations c
enjoy higher
consumption,
but the current one
i
experiences
an initial drop
in consumption.
CHAPTER 7
t0
time
Economic Growth I
slide 50
Population Growth
ƒ Assume that the population--and labor force-grow at rate n. (n is exogenous)
ΔL
L
= n
ƒ EX: Suppose L = 1000 in year 1 and the
population is growing at 2%/year (n = 0.02).
Then ΔL = n L = 0.02 × 1000 = 20,
so L = 1020 in year 2.
CHAPTER 7
Economic Growth I
slide 51
19
Break-even investment
(δ + n)k = break-even investment,
the amount of investment necessary
to keep k constant.
Break-even investment includes:
ƒ δ k to replace capital as it wears out
ƒ n k to equip new workers with capital
(otherwise, k would fall as the existing
capital stock would be spread more thinly
over a larger population of workers)
CHAPTER 7
Economic Growth I
slide 52
The equation of motion for k
ƒ With population growth, the equation of
motion for k is
Δk = s f(k) − (δ + n) k
actual
investment
CHAPTER 7
Economic Growth I
break-even
investment
slide 53
20
The Solow Model diagram
Investment,
break-even
investment
Δk = s f(k) − (δ +n)k
(δ + n ) k
sf(k)
k*
CHAPTER 7
Capital per
worker, k
Economic Growth I
slide 54
The impact of population growth
Investment,
break-even
investment
( δ + n 2) k
( δ + n 1) k
An increase in n
causes an
increase in breakeven investment,
leading to a lower
steady-state level
of k.
sf(k)
k 2*
CHAPTER 7
Economic Growth I
k1* Capital per
worker, k
slide 55
21
Prediction:
ƒ Higher n ⇒ lower k*.
ƒ And since y = f(k) ,
lower k* ⇒ lower y* .
ƒ Thus, the Solow model predicts that
countries with higher population growth
rates will have lower levels of capital and
income per worker in the long run.
CHAPTER 7
Income per
person in 1992
(logarithmic scale)
Economic Growth I
slide 56
International Evidence on Population
Growth and Income per Person
100,000
Germany
U.S.
Denmark
Canada
Israel
10,000
U.K.
Italy
Finland
Japan
France
Mexico
Singapore
Egypt
Brazil
Pakistan
Indonesia
1,000
Ivory
Coast
Peru
Cameroon
Kenya
India
Zimbabwe
Chad
100
0
CHAPTER 7
1
2
Economic Growth I
Uganda
3
4
Population growth (percent per year)
(average 1960 –1992)
slide 57
22
The Golden Rule with Population Growth
To find the Golden Rule capital stock,
we again express c* in terms of k*:
c* =
y*
−
i*
= f (k* ) − (δ + n) k*
c* is maximized when
MPK = δ + n
or equivalently,
MPK − δ = n
CHAPTER 7
In
In the
the Golden
Golden
Rule
Steady
Rule Steady State,
State,
the
marginal
product
the marginal product of
of
capital
capital net
net of
of
depreciation
depreciation equals
equals the
the
population
population growth
growth rate.
rate.
Economic Growth I
slide 58
Chapter Summary
1. The Solow growth model shows that, in the
long run, a country’s standard of living depends
ƒ positively on its saving rate.
ƒ negatively on its population growth rate.
2. An increase in the saving rate leads to
ƒ higher output in the long run
ƒ faster growth temporarily
ƒ but not faster steady state growth.
CHAPTER 7
Economic Growth I
slide 59
23
Chapter Summary
3. If the economy has more capital than the
Golden Rule level, then reducing saving will
increase consumption at all points in time,
making all generations better off.
If the economy has less capital than the
Golden Rule level, then increasing saving will
increase consumption for future generations,
but reduce consumption for the present
generation.
CHAPTER 7
Economic Growth I
slide 60
CHAPTER 7
Economic Growth I
slide 61
24