Economic Growth I
Chapter Seven
Introduction
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Having analyzed the overall production, distribution, and
allocation of national income, we now consider the determinants
of long-run growth.
Stylized fact – in developed economies, output grows over time
(although irregular at times); the trend is upward
Different countries also enjoy very different standards of living in
terms of income per person; standard of living means what?
Our goal is to understand what causes these differences in
income over time and across countries.
What determines a country’s output at any point in time? So
where must the differences across countries come from?
Solow growth model – shows how saving, population growth, and
technological progress affect the level of an economy’s output
and its growth over time
International Differences in the Standard of
Living: 1999
Income and poverty in the world
selected countries, 2000
100
Madagascar
% of population
living on $2 per day or less
90
India
Nepal
Bangladesh
80
70
60
Botswana
Kenya
50
China
40
Peru
30
Mexico
Thailand
20
Brazil
10
0
$0
Chile
Russian
Federation
$5,000
$10,000
S. Korea
$15,000
Income per capita in dollars
$20,000
The Accumulation of Capital
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Starting with the production function, Y = F(K,L), what
are the 3 possible sources of long-run output growth:
1.
2.
3.
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•
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Increase in capital stock, K
Increase in labor force (population increase), L
Increase in technology; the production function F changes
Our analysis of economic growth considers all of these
factors, but focuses primarily on the determination of
the capital stock.
Assumption – there is no technological progress and no
growth in population; we relax these later
What is the fundamental difference between our analysis
of economic growth and our previous analysis of income
determination? Static vs. dynamic?
The Supply and Demand for Goods
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The Supply of Goods and the Production Function:
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Supply of goods depends on production function, Y = F(K,L)
F exhibits constant returns to scale, zY = F(zK,zL)
z = 1/L  Y/L = F(K/L,1)
The amount of output per worker, Y/L, is a function of the
amount of capital per worker, K/L.
Does the size of labor force affect the relationship between
output per worker and capital per worker?
Write all variables in per-worker terms: y = Y/L, k = K/L,
y=f(k)
Example: Y = (KL)1/2; y = f(k); f(k) = ?
The Production Function
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What does the
slope of this perworker production
function
represent?
MPK = f(k+1) –
f(k)
Why is it that as
the amount of
capital per worker
increases, the
production
function becomes
flatter?
When k is small
(large), is MPK
large (small)?
Why?
… The Supply and Demand for Goods
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The Demand for Goods and the Consumption Function:
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The demand for goods in the Solow model comes from
consumption and investment.
Y/L = C/L + I/L  y = c + i; output per worker is divided
between consumption per worker and investment per worker
The Solow model assumes that each year people save a
constant fraction s of their income and consume (1-s)
c = (1-s)y; what assumptions have we made thus far about
demand? G = T = NX = 0
What does this consumption function imply about
investment? y= (1-s)y + i  i = sy; investment equals
saving, what is adjusting to ensure these two equate?
For a given k, what determines per capita output? What
determines the allocation of output between consumption and
investment?
Growth in the Capital Stock and the
Steady State
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Capital stock is a key determinant of output, if capital
grows over time then so will output
2 forces that influence change in the capital stock:
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Investment – expenditure on new plant/equipment and
causes capital stock to rise
Depreciation – wearing out of old capital, causes capital to fall
i = sy  i = sf(k); investment per worker is a function of
capital stock per worker; Figure 7-2
What governs output? What governs output allocation?
We assume that a certain fraction  of the capital stock
wears out each year;  - depreciation rate
How much capital depreciates every year? k
Output, Consumption, and Investment
Depreciation
Capital accumulation
The basic idea:
Investment makes
the capital stock bigger,
depreciation makes it
smaller.
… Growth in the Capital Stock and the
Steady State
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The overall change in the capital stock is the net effect of
new investment and depreciation:
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k = i - k = sf(k) - k
Figure 7-4, The higher the capital stock the higher the
amount of output, investment, and depreciation
At what level of capital is investment = depreciation?
If the economy finds itself at this capital stock, k*, will the
capital stock continue to change? Why or why not?
The only investment being undertaken is replacement
investment.
At k*, k = 0, so k and y=f(k) are steady over time.
Thus, k* is called the steady-state level of capital.
Investment, Depreciation, and the
Steady State
… Growth in the Capital Stock and the
Steady State
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The steady state level of capital is significant for two
reasons:
1.
2.
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An economy at the steady state will stay there.
An economy not at the steady state will eventually go there
regardless of the level of capital with which the economy
begins.
The steady state represents the long-run equilibrium of
the economy.
Suppose economy starts with k1 < k*, why will the
capital stock rise?
Suppose economy start with k2 > k*, why will the capital
stock fall?
Approaching the Steady State: A
Numerical Example
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Production Function: Y = (KL)1/2
Derive the per-worker production function f(k).
Assume 30% of output is saved and the capital stock
depreciates at a rate of 10%.
Assume the economy starts off with 4 units of capital per
worker.
See Excel Worksheet
Over time what level of capital stock, output, consumption,
investment, and depreciation does the economy approach?
Is there another way to derive the steady-state without so
many calculations?
Case Study: The Miracle of Japanese
and German Growth
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Japan and Germany experienced rapid economic growth
following World War II.
The war destroyed a large portion of their capital stocks
(plants, equipment, heavy machinery).
Between 1948 and 1972 real GDP per capita grew at 8.2%
per year in Japan and 5.7% per year in Germany while the
U.S. experienced a meager 2.2% per year in comparison.
Does this make any sense from the standpoint of the
Solow growth model? What happens to output after a
collapse in the capital stock? What happens to saving and
investment? Should output begin to grow at a faster rate?
Why or why not?
How Saving Affects Growth
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Low levels of initial capital is not the only thing that affects
the rate of economic growth; the fraction of output
devoted to saving/investment affects economic growth
Consider an increase in the saving rate from s1 to s2:
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What happens to the investment schedule?
At the initial saving rate s1, and the initial capital stock k1*,
the amount of investment just offsets what?
What happens immediately after the saving rate rises?
Where will the new steady-state end up?
The Solow model shows that the saving rate is a key
determinant of the steady-state capital stock. If the
saving rate is high (low), the economy will have a large
(small) capital stock and a high (low) level of output.
… How Saving Affects Growth
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What does the
Solow model
say about the
relationship
between
saving and
growth?
Is the
relationship
permanent or
temporary?
Can we more
fully explain
the impressive
performance of
Japan and
Germany after
WWII?
Case Study: Saving and Investment
Around the World
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Revisit: why are
some countries rich
and some poor?
What answer does
the Solow model
provide?
Does international
data support this
theoretical result?
The data clearly
show a positive
relationship between
the fraction of
output devoted to
investment and the
level of per capital
income.
The Golden Rule Level of Capital
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The Solow model shows how the rate of saving and
investment determines the long-run levels of capital and
income. Is higher saving always a good thing since it
always leads to higher income?
What amount of capital accumulation is optimal from the
standpoint of economic well-being?
Assume that we can set our nation’s savings rate, what
rate should we choose? What should be our goal?
Policymakers should aim for a savings rate that delivers a
steady state with the highest level of consumption
possible.
The steady-state value of k that maximizes consumption is
called the Golden Rule level of capital, kg.
Comparing Steady States
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Where is the golden rule
level of capital?
Steady-state
consumption: y = c + i
c=y–i
Substitute steady-state
values for output and
investment: c* = f(k*) k*
Increase in steady-state
capital has two opposing
effects, what are they?
Steady-state consumption
is the gap between output
and depreciation
(investment)
kg is the capital level that
maximizes s.s.
consumption
… Comparing Steady States
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What happens to output and depreciation when we
increase the capital stock when capital is below the golden
rule level? (Figure 7-7)
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What happens when the capital stock is above the golden
rule level?
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Do the relative slopes of the production function and the
depreciation schedule tell us anything?
What does this imply about consumption?
Again, do the relative slopes give us any information?
What happens to consumption?
At the golden rule level of capital what is the relationship
between the slopes of the production function and the
depreciation schedule?
… Comparing Steady States
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Because the 2 slopes
are equal at kg, the
golden rule is
described by: MPK =

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Suppose s.s. capital
is k* and we are
considering
increasing capital to
k*+1:
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How much extra
output is produced?
How much extra
depreciation?
What is the net
effect on
consumption?
What should we do
if MPK- < 0? MPK >0?
… Comparing Steady States
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At the golden rule level of capital, the marginal product of
capital net of depreciation (MPK - ) equals zero.
Will the economy naturally gravitate towards the golden
rule steady-state level of capital?
If a policymaker wants a specific steady-state capital
stock, such as the golden rule, the appropriate savings
rate must be used to support it.
What happens when the saving rate is set below (above)
the one required to support the golden rule?
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What happens to the steady-state capital stock?
What happens to the steady-state consumption?
Finding the Golden Rule Steady State:
A Numerical Example
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Per-worker production function: y = k1/2,  = 0.1
The policymaker chooses s in order to maximize
consumption.
In the steady-state: sf(k*) = k*
k*/f(k*) = s/  k*/(k*)1/2 = s/0.1  k* = 100s2
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What happens to steady-state capital, output, and
depreciation as the savings rate climbs?
What happens to consumption? What is the golden rule
savings rate? What is net marginal product of capital?
Why does net marginal product of capital eventually become
zero?
Is there an easier way to find the golden rule level of capital,
consumption, and saving? Perhaps using calculus
The Transition to the Golden Rule
Steady State
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So far we have assumed that the policymaker can choose
any savings rate and the economy will jump directly to the
golden rule steady state; unrealistic assumption
Rather, suppose economy has reached a steady state other
than golden rule:
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What happens to consumption, investment, and capital when
the economy transitions between steady states?
Are there any undesirable consequences in the transition
process that might deter policymakers?
Consider two cases: the economy begins with more capital
than in the golden rule steady state, or with less
The two cases offer very different problems for
policymakers.
Starting With Too Much Capital
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With too much capital,
what should
policymaker do to
approach the golden
rule?
What happens
immediately following a
reduction in the
savings rate? Why?
What happens to c, y,
and i over time?
Is there anything
noticeable about the
path of consumption?
When k* > kg, reducing
saving is a good policy;
it increases
consumption at every
point in time
y
c
i
t0
time
Starting With Too Little Capital
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y
c
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i
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t0
time
With too little capital,
what should policymaker
do to approach the
golden rule?
What happens
immediately and over
time to y, c, and i?
Is there anything
noticeable about the path
of c?
Does there appear to be
any tradeoff between
current and future
economic well-being?
When k* < kg reaching
the golden rule requires
an initial reduction in
consumption which will
rise over time.
Population Growth
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Does the basic Solow model explain sustained
(permanent) growth in output?
To explain permanent growth in output we must augment
the basic model with population growth.
Assume population grows at a constant rate n:
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Example: n = .01  population grows at 1% per year
What now are the 3 forces acting on the stock of capital to
drive it towards a steady-state? How does population
growth specifically change capital per worker?
Derive: k = i – (+n)k; what do i, , and n do to k?
(+n)k – break even investment – the amount of
investment needed to keep the capital stock per worker
constant
The Steady State With Population
Growth
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Why does break
even investment
include the term
nk?
How does
population growth
reduce k as
opposed to
depreciation?
k* satisfies: k = 0
 sf(k) = (+n)k 
s/(+n) = k/f(k)
What happens if k
< (>) k*?
Once the economy
is in the steady
state, investment
has 2 purposes.
What are they?
The Effects of Population Growth
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1.
2.
3.
Population growth
alters the basic Solow
model 3 ways:
It explains sustained
economic growth, but
does it explain
sustained growth in
the standard of
living?
It gives another
reason for why some
countries are rich and
some are poor. How?
It affects the criterion
for determining the
golden rule level of
capital. What is the
new golden rule
condition?
Case Study: Population Growth
Around the World
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Does the Solow
model tell us
anything about
the correlation
between high
population growth
and low steadystate income per
worker?
Why would high
population growth
tend to
impoverish a
country?
Does the
international data
support this
theory?
Chapter Summary
The Solow growth model shows that, in the long run, a country’s
standard of living depends:
1.
•
positively on its saving rate.
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negatively on its population growth rate.
An increase in the saving rate leads to:
2.
3.
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higher output in the long run
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faster growth temporarily
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but not faster steady state growth.
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.