# Chapter 7 learning objectives by phf13063

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Chapter 7 learning objectives
Learn the closed economy Solow model
See how a country’s standard of living
depends on its saving and population
growth rates
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
In aggregate terms: Y = F (K, L )
Define: y = Y/L = output per worker
k = K/L = capital per worker
Assume constant returns to scale:
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)         where f(k) = F (k, 1)

CHAPTER 7   Economic Growth I                   slide 15

2
The production function
Output per
worker, y
f(k)

MPK =f(k +1) – f(k)
1

Note: this production function
Note: this production function
exhibits diminishing MPK.
exhibits diminishing MPK.

Capital per
worker, k
CHAPTER 7   Economic Growth I                        slide 16

The national income identity

Y=C+I             (remember, no G )

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

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

Consumption function: c = (1–s)y
(per worker)

CHAPTER 7   Economic Growth I                  slide 18

Saving and investment
saving (per worker) = y – c
= y – (1–s)y
=  sy
National income identity is y = c + i
Rearrange to get: i = y – c = sy
(investment = saving, like in chap. 3!)

Using the results above,
i = sy = sf(k)
CHAPTER 7   Economic Growth I                  slide 19

4
Output, consumption, and investment

Output per                                  f(k)
worker, y

c1
y1                      sf(k)

i1

k1             Capital per
worker, k
CHAPTER 7   Economic Growth I                       slide 20

Depreciation

Depreciation          δ = the rate of depreciation
δ = the rate of depreciation
per worker, δk          = the fraction of the capital stock
= the fraction of the capital stock
that wears out each period
that wears out each period

δk

δ
1

Capital per
worker, k
CHAPTER 7   Economic Growth I                       slide 21

5
Capital accumulation

The basic idea:
The basic idea:
Investment makes
Investment makes
the capital stock bigger,
the capital stock bigger,
depreciation makes it smaller.
depreciation makes it 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

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The equation of motion for k

Δk = s f(k) – δk
the Solow model’s central equation
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.,
income per person: y = f(k)
consump. per person: c = (1–s) f(k)

CHAPTER 7    Economic Growth I                slide 24

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

CHAPTER 7    Economic Growth I                slide 25

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Investment
and                                   δk
depreciation
sf(k)

k*       Capital per
worker, k
CHAPTER 7   Economic Growth I                     slide 26

Δk = sf(k) − δk
Investment
and                                   δk
depreciation
sf(k)

Δk
investment

depreciation

k1         k*       Capital per
worker, k
CHAPTER 7   Economic Growth I                     slide 27

8

Δk = sf(k) − δk
Investment
and                                        δk
depreciation
sf(k)

Δk

k1 k2      k*           Capital per
worker, k
CHAPTER 7    Economic Growth I                         slide 29

Δk = sf(k) − δk
Investment
and                                        δk
depreciation
sf(k)

Δk
investment
depreciation

k2     k*           Capital per
worker, k
CHAPTER 7    Economic Growth I                         slide 30

9

Δk = sf(k) − δk
Investment
and                                δk
depreciation
sf(k)

Δk

k2 k3 k*   Capital per
worker, k
CHAPTER 7   Economic Growth I                  slide 32

Δk = sf(k) − δk
Investment
and                                δk
depreciation
sf(k)
Summary:
Summary:
As long as k < k**,
As long as k < k ,
investment will exceed
investment will exceed
depreciation,
depreciation,
and k will continue to
and k will continue to
grow toward k**.
grow toward k .

k3 k*    Capital per
worker, k
CHAPTER 7   Economic Growth I                  slide 33

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Now you try:
Draw the Solow model diagram,
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

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A numerical example, cont.

Assume:
s = 0.3
δ = 0.1
initial value of k = 4.0

CHAPTER 7    Economic Growth I                         slide 36

A Numerical Example
Assumptions:         y = k ; s = 0.3; δ = 0.1; initial k = 4.0
Year
Year         kk         yy       cc       ii       δk
δk     Dk
Dk
11        4.000
4.000      2.000
2.000   1.400
1.400   0.600
0.600   0.400
0.400   0.200
0.200
22         4.200
4.200      2.049
2.049   1.435
1.435   0.615
0.615   0.420
0.420   0.195
0.195
33         4.395
4.395      2.096
2.096   1.467
1.467   0.629
0.629   0.440
0.440   0.189
0.189

CHAPTER 7    Economic Growth I                         slide 37

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A Numerical Example
Assumptions:         y = k ; s = 0.3; δ = 0.1; initial k = 4.0
Year
Year         kk         yy       cc       ii       δk
δk     Dk
Dk
11        4.000
4.000      2.000
2.000   1.400
1.400   0.600
0.600   0.400
0.400   0.200
0.200
22         4.200
4.200      2.049
2.049   1.435
1.435   0.615
0.615   0.420
0.420   0.195
0.195
33         4.395
4.395      2.096
2.096   1.467
1.467   0.629
0.629   0.440
0.440   0.189
0.189
44        4.584
4.584      2.141
2.141   1.499
1.499   0.642
0.642   0.458
0.458   0.184
0.184
……
10
10        5.602
5.602      2.367
2.367   1.657
1.657   0.710
0.710   0.560
0.560   0.150
0.150
……
25
25        7.351
7.351      2.706
2.706   1.894
1.894   0.812
0.812   0.732
0.732   0.080
0.080
……
100
100        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
CHAPTER 7    Economic Growth I                         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 *       eq'n of motion with Δk = 0
0.3 k * = 0.1k *         using assumed values
k*
3=       = 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                                       dk
depreciation                                   s2 f(k)

s1 f(k)

k
k 1*    k 2*
CHAPTER 7   Economic Growth I                         slide 41

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Prediction:
Higher s ⇒ higher k*.

And since y = f(k) ,
higher k* ⇒ higher y* .

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

Denmark   Germany         Japan
U.S.

10,000                                                                                       Finland
Mexico                                     U.K.
Brazil                                Singapore
Israel
FranceItaly
Pakistan
Egypt          Ivory
Coast                  Peru

Indonesia
1,000
India                Zimbabwe
Kenya
Uganda

100
0            5       10           15           20           25         30         35           40
Investment as percentage of output
(average 1960 –1992)
CHAPTER 7                 Economic Growth I                                                                 slide 43

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The Golden Rule: introduction
How do we know which is the “best” steady state?
- eing depends on consumption,
Economic well b
so the “best” steady state has the highest possible
value of consumption per person: c* = (1–s) f(k*)
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*
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*
In general:
= f (k*) − i*                  i = Δk + δk
= f (k*) − δk*             In the steady state:
i* = δk*
because Δk = 0.

CHAPTER 7    Economic Growth I                          slide 45

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The Golden Rule Capital Stock
output and
depreciation                                 δk*
Then, graph
f(k*) and δk*,                                                f(k*)
and look for the
point where the
gap between
c gold
*
them is biggest.
i gold = δ k gold
*          *

y gold = f (k gold )
*           *
k gold
*
capital per
worker, k*
CHAPTER 7     Economic Growth I                              slide 46

The Golden Rule Capital Stock

c* = f(k*) − δk*                                            δk*
is biggest where
the slope of the                                              f(k*)
production func.
equals
the slope of the                    c gold
*

depreciation line:
MPK = δ
k gold
*
capital per
worker, k*
CHAPTER 7     Economic Growth I                              slide 47

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The transition to the
The economy does NOT have a tendency to
move toward the Golden Rule steady state.
Achieving the Golden Rule requires that
with higher consumption.
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          y
c* requires a
fall in s.
In the transition        c
to the
i
Golden Rule,
consumption is
higher at all
points in time.                    t0             time

CHAPTER 7      Economic Growth I               slide 49

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Starting with too little capital
If k * < k gold
*

then increasing c*
requires an             y
increase in s.
Future generations c
enjoy higher
consumption,
but the current one
i
experiences
an initial drop
in consumption.                   t0            time

CHAPTER 7     Economic Growth I                 slide 50

Population Growth
Assume that the population--and labor force--
grow at rate n. (n is exogenous)
ΔL
= n
L
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

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Break-even investment
(δ + n)k = break-even investment,
the amount of investment necessary
to keep k constant.

Break-even investment includes:
δ k to replace capital as it wears out
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
With population growth, the equation of
motion for k is

Δk = s f(k) − (δ + n) k

actual
investment                  break-even
investment

CHAPTER 7   Economic Growth I                slide 53

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The Solow Model diagram
Investment,
Δk = s f(k) − (δ +n)k
break-even
investment
(δ + n ) k

sf(k)

k*   Capital per
worker, k
CHAPTER 7     Economic Growth I                            slide 54

The impact of population growth
Investment,
break-even                    ( δ + n 2) k
investment
( δ + n 1) k
An increase in n
causes an                                                     sf(k)
increase in break-
even investment,
of k.

k 2*   k1* Capital per
worker, k
CHAPTER 7     Economic Growth I                            slide 55

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Prediction:
Higher n ⇒ lower k*.

And since y = f(k) ,
lower k* ⇒ lower y* .

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                  Economic Growth I                                                                  slide 56

International Evidence on Population
Income per               Growth and Income per Person
person in 1992
(logarithmic scale)
100,000

Germany
Denmark        U.S.
Israel
10,000                               Japan   Singapore           Mexico
U.K.
Finland   France
Italy
Egypt       Brazil

Pakistan         Ivory
Peru                         Coast
Indonesia
1,000                                                                         Cameroon
Kenya
India
Zimbabwe

100
0                        1                     2                      3                 4
Population growth (percent per year)
(average 1960 –1992)
CHAPTER 7                  Economic Growth I                                                                  slide 57

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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                    In the Golden
In the Golden
MPK = δ + n                     Rule Steady State,
the marginal product of
the marginal product of
or equivalently,                         capital net of
capital net of
MPK − δ = n                   depreciation equals the
depreciation equals the
population growth rate.
population growth rate.
CHAPTER 7   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
positively on its saving rate.
negatively on its population growth rate.

2. An increase in the saving rate leads to
higher output in the long run
faster growth temporarily
but not faster steady state growth.

CHAPTER 7   Economic Growth I                       slide 59

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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

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