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8/13/2019 Lecture4-Solow Growth Model
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Simplified Solow Growth Model
Consumers: Consume a constant fraction of GDP and own
all the capital in the economy
Not modeling:
Unemployment (everyone always works)
Lifecycle (no children, students or retirees)
Within-country income inequality
Consumers described by one equation:I = s Y
where s, a number between 0 and 1, is the
fraction of output that gets invested.
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Simplified Solow Growth Model
Firms: Use the capital to produce output
Not modeling:
Labor markets (searching for workers)
Finance (borrowing to take on projects)
Executive compensation
Firms described by one equation:
Y = A Kwhere Yis GDP,A is productivity
and Kis the capital stock
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Simplified Solow Growth ModelEquilibrium:
All output is used either in investment or
consumption (no trade, no government):
Y = C + I
How the stock of capital changes over time:
K = I + (1- )Kwhere Kis the capital stock next year,
Kis the capital stock this year,
Iis investment this year, and
is the depreciation rate
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Simplified Solow Growth ModelSo the entire model is described by four equations:
Households: I = s Y
Firms: Y = A K0.3
Capital Accumulation: K = I + (1- )K GDP: Y = C + I
Rearranging terms:
I = s Y= s A K0.3
K = I + (1- )K = s A K0.3+ (1- )K
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How does the capital stockchange over time?
K
K
K= K
How are capital this year, and
capital next year related?
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How does the capital stockchange over time?
K
K
K= K
K = s A K0.3+ (1- )K
The equation above tells you
how much capital there will benext year
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How does the capital stockchange over time?
K
K
K= K
K = s A K0.3+ (1- )K
Suppose the economy starts
with some low capital level K0
K0
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How does the capital stockchange over time?
K
K
K= K
K = s A K0.3+ (1- )K
Then the equation says that
next years capital stock will
be K1
K0
K1
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How does the capital stockchange over time?
K
K
K= K
K = s A K0.3+ (1- )K
Using the red 45 degree line
as a reference, we can find
K1on the horizontal axis.
K0
K1
K1
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How does the capital stockchange over time?
K
K
K= K
K = s A K0.3+ (1- )K
Then we can find K2
K0
K1
K1
K2
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How does the capital stockchange over time?
K
K
K= K
K = s A K0.3+ (1- )K
Repeating these steps, we
can find the capital stock in
any future year
K0
K1
K1
K2
8/13/2019 Lecture4-Solow Growth Model
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How does the capital stockchange over time?
K
K
K= K
K = s A K0.3+ (1- )K
Repeating these steps, we
can find the capital stock in
any future year
K0
K1
K1
K2
K2
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How does the capital stockchange over time?
K
K
K= K
K = s A K0.3+ (1- )K
Repeating these steps, we
can find the capital stock in
any future year
K0
K1
K1
K2
K2
K3
8/13/2019 Lecture4-Solow Growth Model
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How does the capital stockchange over time?
K
K
K= K
K = s A K0.3+ (1- )K
Repeating these steps, we
can find the capital stock in
any future year
K0
K1
K1
K2
K2
K3
K3
8/13/2019 Lecture4-Solow Growth Model
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How does the capital stockchange over time?
K
K
K= K
K = s A K0.3+ (1- )K
Repeating these steps, we
can find the capital stock in
any future year
K0
K1
K1
K2
K2
K3
K3
K4
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How does the capital stockchange over time?
K
K
K= K
K = s A K0.3+ (1- )K
Notice that the capital stock
is approaching the point
where the two lines meet
K0
K1
K1
K2
K2
K3
.
K10
K10
.
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How does the capital stockchange over time?
K
K
K= K
K = s A K0.3+ (1- )K
The point where the two
lines meet is the steady
state level of capital. Once
the economy is at this level,
the capital level does not
change.
K*
K*
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Some Things to Notice
The further the economy starts below the steadystate level of capital, the faster the economy
initially grows
Mankiw refers to this as the catch-up effect
This is due to the effect of diminishing returns
The amount of extra output from each
additional unit of capital goes down as the
capital stock gets larger Growth slows over time until the capital stock
reaches the steady state level
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Further Remarks
This model can be made more complicated (andrealistic) by makingAgrow over time
The mechanics of the model are the same, except
instead of reaching a steady state, the capital
stock grows at the same rate as productivity
This is called a balanced growth path
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Savings and Productivity
What happens if the savings rate of the countrychanges?
Increase sfrom its initial level to a higher level
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Increase in the Savings Rate
K
K
K= K
K = s A K0.3+ (1- )K
Suppose the economy is in
a steady state with savings
rate s.
K*
K*
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Increase in the Savings Rate
K
K
K= K
K = s A K0.3+ (1- )K
Then the savings rate
increases to s.
K*
K*
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Increase in the Savings Rate
K
K
K= K
K = s A K0.3+ (1- )K
Now capital accumulates
according to the new
equation with the higher
savings rate
K1
K*
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Increase in the Savings Rate
K
K
K= K
K = s A K0.3+ (1- )K
And we proceed
exactly like before.
K1
K*
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Increase in the Savings Rate
K
K
K= K
K = s A K0.3+ (1- )K
Eventually a new,
higher steady state
capital stock is
reached.
K0
K0 K*
K*
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Savings and Productivity
If the savings rate is increased, the economytransitions to a higher steady state capital level
What happens if instead productivity is
increased?
Same thing. Income goes up, so consumers have more to
invest, which increases the capital stock.
How are they different?
Higher savings: Decreases consumption
today, increases it in the future
Higher productivity: Increases consumption
both today and in the future
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Savings and Productivity
Back to what Solow found: Savings rates (even historical) have little
relationship to relative wealth
Apparently the wealth of countries that are now
rich is notbecause of long term savings andinvestmentper se
That is, clearly the fact that rich countries are
rich is partly because they have more capital.
BUT they have more capital becausethey have
high productivity.
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Savings and Productivity
This is an extremely important finding. Suggests that a long history of capital accumulation is
not necessary to be wealthy
If a country is able to increase its productivity, capital
will catch up quite quickly This shifted the emphasis in the study of promoting
development in low income countries awayfrom trying
to send them capital, and towardtrying to make their
economies more efficient
How do you do that?
Perhaps the most important open question in social
science.