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Why growth matters
 Data on infant mortality rates:
 20% in the poorest 1/5 of all countries
 0.4% in the richest 1/5
 In Pakistan, 85% of people live on less than $2/day.
 One-fourth of the poorest countries have had
Chapter 7: Economic Growth I:
Capital Accumulation and Population
Growth
famines during the past 3 decades.
 Poverty is associated with oppression of women
and minorities.
Economic growth raises living standards and
reduces poverty….
CHAPTER 1
0
The Science of Macroeconomics
Income and poverty in the world
Madagascar
% of population
living on $2 p
per day or less
90
growth – even by a tiny amount – will have huge
effects on living standards in the long run.
Bangladesh
70
60
Botswana
Kenya
50
China
Peru
40
Mexico
30
Thailand
20
Brazil
10
Chile
Russian
Federation
annual
growth rate of
income per
capita
…25 years
…50 years
…100 years
2.0%
64.0%
169.2%
624.5%
2.5%
85.4%
243.7%
1,081.4%
S. Korea
0
$0
1
 Anything that effects the long-run rate of economic
India
Nepal
80
Economic Growth I
Why growth matters
selected countries, 2000
100
CHAPTER 7
$5,000
$10,000
$15,000
$20,000
Income per capita in dollars
CHAPTER 7
Why growth matters
percentage
t
increase
i
in
i
standard of living after…
Economic Growth I
3
The lessons of growth theory
…can make a positive difference in the lives of
hundreds of millions of people.
 If the annual growth rate of U.S. real GDP per
capita had been just one-tenth of one percent
higher during the 1990s, the U.S. would have
generated an additional $496 billion of income
during
g that decade.
These lessons help us
 understand why poor
countries are poor
 design policies that
can help them grow
 learn how our own
growth rate is affected
by shocks and our
government’s policies
CHAPTER 7
Economic Growth I
4
CHAPTER 7
Economic Growth I
5
1
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How Solow model is different from
Chapter 3’s model
The Solow model
 due to Robert Solow,
1. K is no longer fixed:
won Nobel Prize for contributions to
the study of economic growth
investment causes it to grow,
depreciation causes it to shrink
 a major paradigm:
 widely used in policy making
 benchmark against which most
2. L is no longer
g fixed:
population growth causes it to grow
3. the consumption function is simpler
recent growth theories are compared
 looks at the determinants of economic growth
and the standard of living in the long run
CHAPTER 7
Economic Growth I
6
How Solow model is different from
Chapter 3’s model
CHAPTER 7
Economic Growth I
7
The production function
 In aggregate terms: Y = F (K, L)
4. no G or T
 Define: y = Y/L = output per worker
(only to simplify presentation;
we can still do fiscal policy experiments)
k = K/L = capital per worker
 Assume constant returns to scale:
5. cosmetic differences
zY = F (zK, zL ) for any z > 0
 Pick z = 1/L. Then
Y/L = F (K/L, 1)
y = F (k, 1)
y = f(k)
CHAPTER 7
Economic Growth I
8
The production function
CHAPTER 7
where f(k) = F(k, 1)
Economic Growth I
9
The national income identity
 Y=C+I
(remember, no G )
 In “per worker” terms:
Output per
worker, y
f(k)
y=c+i
where c = C/L and i = I /L
MPK = f(k +1) – f(k)
1
Note: this production function
exhibits diminishing MPK.
Capital per
worker, k
CHAPTER 7
Economic Growth I
10
CHAPTER 7
Economic Growth I
11
2
9/30/2013
The consumption function
Saving and investment
 saving (per worker)
 s = the saving rate,
= y – (1–s)y
= sy
the fraction of income that is saved
(s is an exogenous parameter)
 National income identity is y = c + i
Note: s is the onlyy lowercase variable that
is not equal to
its uppercase version divided by L
Rearrange to get: i = y – c = sy
(investment = saving, like in chap. 3!)
 Consumption function: c = (1–s)y
 Using the results above,
(per worker)
CHAPTER 7
= y – c
i = sy = sf(k)
Economic Growth I
12
Output, consumption, and investment
Output per
worker, y
CHAPTER 7
Economic Growth I
13
Depreciation
Depreciation
per worker, k
f(k)
 = the rate of depreciation
= the fraction of the capital stock
that wears out each period
k
c1
sf(k)
y1

1
i1
Capital per
worker, k
k1
CHAPTER 7
Economic Growth I
Capital per
worker, k
14
 The Solow model’s central equation
= investment – depreciation
=
i
–
k
 Determines behavior of capital over time…
 …which, in turn, determines behavior of
all of the other endogenous variables
because they all depend on k. E.g.,
Since i = sf(k) , this becomes:
income per person: y = f(k)
k = s f(k) – k
Economic Growth I
15
k = s f(k) – k
The basic idea: Investment increases the capital
stock, depreciation reduces it.
CHAPTER 7
Economic Growth I
The equation of motion for k
Capital accumulation
Change in capital stock
k
CHAPTER 7
consumption per person: c = (1–s) f(k)
16
CHAPTER 7
Economic Growth I
17
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9/30/2013
The steady state
The steady state
k = s f(k) – k
Investment
and
depreciation
If investment is just enough to cover depreciation
[sf(k) = k ],
then capital per worker will remain constant:
k = 0.
k
sf(k)
This occurs at one value of k, denoted k*,
called the steady state capital stock.
k*
CHAPTER 7
Economic Growth I
18
Moving toward the steady state
Investment
and
depreciation
CHAPTER 7
Capital per
worker, k
Economic Growth I
19
Moving toward the steady state
k = sf(k)  k
Investment
and
depreciation
k
k = sf(k)  k
k
sf(k)
sf(k)
k
investment
k
depreciation
k1
CHAPTER 7
k*
k1 k2
Capital per
worker, k
Economic Growth I
20
k = sf(k)  k
Economic Growth I
Investment
and
depreciation
k
21
k = sf(k)  k
k
sf(k)
sf(k)
k
investment
Capital per
worker, k
Moving toward the steady state
Moving toward the steady state
Investment
and
depreciation
CHAPTER 7
k*
k
depreciation
k2
CHAPTER 7
Economic Growth I
k*
k2
Capital per
worker, k
22
CHAPTER 7
Economic Growth I
k*
Capital per
worker, k
23
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Moving toward the steady state
Investment
and
depreciation
Moving toward the steady state
k = sf(k)  k
Investment
and
depreciation
k
sf(k)
CHAPTER 7
k
k3 k*
Capital per
worker, k
Economic Growth I
sf(k)
Summary:
As long as k < k*,
investment will exceed
depreciation,
and k will continue to
grow toward k*.
k
k2 k3 k*
k = sf(k)  k
24
NOW YOU TRY:
CHAPTER 7
Capital per
worker, k
Economic Growth I
25
A numerical example
Approaching k* from above
Production function (aggregate):
Draw the Solow model diagram,
labeling the steady state k*.
Y  F (K , L )  K  L  K 1 / 2L1 / 2
On the horizontal axis, pick a value greater than k*
f the
for
th economy’s
’ initial
i iti l capital
it l stock.
t k L
Label
b l it k1.
To derive the per-worker production function,
divide through by L:
1/2
Y K 1 / 2L1 / 2  K 

 
L
L
L 
Show what happens to k over time.
Does k move toward the steady state or
away from it?
Then substitute y = Y/L and k = K/L to get
y  f (k )  k 1 / 2
CHAPTER 7
Economic Growth I
27
Approaching the steady state:
A numerical example, cont.
A numerical example
Assumptions:
Assume:
y  k ; s  0.3;   0.1; initial k  4.0
Dk
Year
k
y
c
i
dk
1
4.000
2.000
1.400
0.600
0.400
0.200
  = 0.1
2
4.200
2.049
1.435
0.615
0.420
0.195
 initial value of k = 4.0
3
4 395
4.395
2 096
2.096
1 467
1.467
0 629
0.629
0 440
0.440
0 189
0.189
4.584
2.141
1.499
0.642
0.458
0.184
5.602
2.367
1.657
0.710
0.560
0.150
7.351
2.706
1.894
0.812
0.732
0.080
8.962
2.994
2.096
0.898
0.896
0.002
9.000
3.000 I 2.100
Economic
Growth
0.900
0.900
0.000
 s = 0.3
CHAPTER 7
Economic Growth I
28
4
…
10
…
25
…
100
…
¥
CHAPTER
7
29
5
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NOW YOU TRY:
ANSWERS:
Solve for the Steady State
Solve for the Steady State
k  0
Continue to assume
s = 0.3,  = 0.1, and y = k 1/2
Use the equation
q
of motion
k = s f(k)  k
to solve for the steady-state values of k, y, and c.
def. of steady state
s f (k *)   k *
eq'n of motion with k  0
0.3 k *  0.1k *
using
g assumed values
k*
 k*
k*
3
Solve to get: k *  9
and y *  k *  3
Finally, c *  (1  s )y *  0.7  3  2.1
An increase in the saving rate
Prediction:
An increase in the saving rate raises investment…
…causing k to grow toward a new steady state:
Investment
and
depreciation
 Higher s  higher k*.
 And since y = f(k) ,
dk
higher k*  higher y* .
s2 f(k)
 Thus, the Solow model predicts that countries
s1 f(k)
with higher rates of saving and investment
will have higher levels of capital and income per
worker in the long run.
CHAPTER 7
k 1*
Economic Growth I
k 2*
k
32
International evidence on investment rates
and income per person
CHAPTER 7
Economic Growth I
33
The Golden Rule: Introduction
 Different values of s lead to different steady states.
Income per 100,000
person in
2003
How do we know which is the “best” steady state?
(log scale)
 The “best” steady state has the highest possible
10,000
consumption per person: c* = (1–s) f(k*).
 An increase in s
 leads to higher k* and y*, which raises c*
 reduces consumption’s share of income (1–s),
1,000
which lowers c*.
 So, how do we find the s and k* that maximize c*?
100
0
5
10
15
20
25
30
35
Investment as percentage of output
(average 1960-2003)
CHAPTER 7
Economic Growth I
35
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The Golden Rule capital stock
The Golden Rule capital stock
steady state
output and
depreciation
k gold  the Golden Rule level of capital,
*
the steady state value of k
that maximizes consumption.
Then, graph
f(k*) and k*,
look for the
point where
the gap between
them is biggest.
To find it, first express c* in terms of k*:
c*
CHAPTER 7
=
y*
 i*
= f (k*)
 i*
= f (k*)
 k*
In the steady state:
i* = k*
because k = 0.
Economic Growth I
36
37
 This adjustment leads to a new steady state with
*
c gold
higher consumption.
 But what happens to consumption
during the transition to the Golden Rule?
steady-state
capital per
worker, k*
38
Economic Growth I
39
Starting with too little capital
y
c
Future generations
enjoy higher
consumption,
but the current
one experiences
an initial drop
in consumption.
i
Economic Growth I
CHAPTER 7
*
If k *  k gold
then increasing c*
requires an
increase in s.
t0
CHAPTER 7
Economic Growth I
 Achieving the Golden Rule requires that
Starting with too much capital
In the transition to
the Golden Rule,
consumption is
higher at all points
in time.
steady-state
capital per
worker, k*
policymakers adjust s.
Economic Growth I
then increasing c*
requires a fall in s.
CHAPTER 7
*
k gold
move toward the Golden Rule steady state.
f(k*)
*
k gold
*
If k *  k gold
*
*
i gold
  k gold
*
 The economy does NOT have a tendency to
k*
MPK = 
CHAPTER 7
*
c gold
The transition to the
Golden Rule steady state
The Golden Rule capital stock
c* = f(k*)  k*
is biggest where the
slope of the
production function
equals
l
the slope of the
depreciation line:
f(k*)
y gold  f (k gold )
*
k*
time
40
CHAPTER 7
y
c
i
Economic Growth I
t0
time
41
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Population growth
Break-even investment
 Assume the population and labor force grow
 ( + n)k = break-even investment,
at rate n (exogenous):
the amount of investment necessary
to keep k constant.
L
 n
L
 Break-even investment includes:
  k to replace capital as it wears out
 n k to equip new workers with capital
 EX: Suppose L = 1,000 in year 1 and the
population is growing at 2% per year (n = 0.02).
 Then L = n L = 0.02  1,000 = 20,
(Otherwise, k would fall as the existing capital stock
is spread more thinly over a larger population of
workers.)
so L = 1,020 in year 2.
CHAPTER 7
Economic Growth I
42
The equation of motion for k
CHAPTER 7
Economic Growth I
43
The Solow model diagram
 With population growth,
Investment,
break-even
investment
the equation of motion for k is:
k = s f(k)  ( +n)k
( + n ) k
k = s f(k)  ( + n) k
actual
investment
sf(k)
break-even
investment
k*
CHAPTER 7
Economic Growth I
44
The impact of population growth
45
Prediction:
( +n2) k
 And since y = f(k) ,
( +n1) k
An increase in n
causes an
increase in breakeven investment,
leading to a lower
steady-state level
of k.
lower k*  lower y*.
sf(k)
 Thus,
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.
k2*
Economic Growth I
Economic Growth I
 Higher n  lower k*.
Investment,
break-even
investment
CHAPTER 7
CHAPTER 7
Capital per
worker, k
k1* Capital per
worker, k
46
CHAPTER 7
Economic Growth I
47
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The Golden Rule with population
growth
International evidence on population growth
and income per person
Income per 100,000
person in
2003
To find the Golden Rule capital stock,
express c* in terms of k*:
(log scale)
c* =
10,000
=
y*
f (k* )

i*
 ( + n) k*
c* is maximized when
MPK =  + n
1,000
100
0
1
2
3
4
5
or equivalently,
MPK   = n
Population growth
CHAPTER 7
(percent per year, average 1960-2003)
In the Golden
Rule steady state,
the marginal product
of capital net of
depreciation equals
the population
growth rate.
Economic Growth I
Alternative perspectives on population
growth
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
Malthus, world population has increased
sixfold, yet living standards are higher than ever.
 Malthus neglected the effects of technological
progress.
The Kremerian Model (1993)
 Posits that population growth contributes to
economic growth.
 More people = more geniuses, scientists &
engineers,
i
so ffaster
t technological
t h l i l progress.
 Evidence, from very long historical periods:
49
 As world pop. growth rate increased, so did rate
of growth in living standards
 Historically, regions with larger populations have
enjoyed faster growth.
CHAPTER 7
Economic Growth I
Chapter Summary
1. The Solow growth model shows that,
in the long run, a country’s standard of living
depends:
 positively on its saving rate
 negatively
ti l on its
it population
l ti growth
th rate
t
2. An increase in the saving rate leads to:
 higher output in the long run
 faster growth temporarily
 but not faster steady state growth
50
CHAPTER 7
Economic Growth I
51
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.
9