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Solow Growth Model
 Observation: Richer countries have more capital (more
machines, factories, etc.)
 Is this the cause or the result of their greater income?
 Two possibilities considered:
 Countries have more capital because they save a
greater part of their income
 Countries have more capital because their income is
higher
 The whole model is beyond the scope of this class, so we
will consider a greatly simplified version
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=sY
where s, a number between 0 and 1, is the
fraction of output that gets invested.
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 K0.3
where Y is GDP, A is productivity
and K is the capital stock
Simplified Solow Growth Model
Equilibrium:
 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- δ)K
where K’ is the capital stock next year,
K is the capital stock this year,
I is investment this year, and
δ is the depreciation rate
Simplified Solow Growth Model
So the entire model is described by four equations:




Households:
Firms:
Capital Accumulation:
GDP:
I=sY
Y = A K0.3
K’ = I + (1- δ)K
Y=C+I
Rearranging terms:
 I = s Y = s A K0.3
 K’ = I + (1- δ)K = s A K0.3 + (1- δ)K
How does the capital stock
change over time?
K’
How are capital this year, and
capital next year related?
K’= K
K
How does the capital stock
change over time?
K’
K’ = s A K0.3 + (1- δ)K
K’= K
The equation above tells you
how much capital there will be
next year
K
How does the capital stock
change over time?
K’
K’ = s A K0.3 + (1- δ)K
Suppose the economy starts
with some low capital level K0
K’= K
K0
K
How does the capital stock
change over time?
K’
K’ = s A K0.3 + (1- δ)K
K1
K’= K
K0
Then the equation says that
next year’s capital stock will
be K1
K
How does the capital stock
change over time?
K’
K’ = s A K0.3 + (1- δ)K
K1
K’= K
K0
K1
Using the red 45 degree line
as a reference, we can find
K1 on the horizontal axis.
K
How does the capital stock
change over time?
K’
K’ = s A K0.3 + (1- δ)K
K2
Then we can find K2
K1
K’= K
K0
K1
K
How does the capital stock
change over time?
K’
K’ = s A K0.3 + (1- δ)K
K2
K1
K’= K
K0
K1
Repeating these steps, we
can find the capital stock in
any future year
K
How does the capital stock
change over time?
K’
K’ = s A K0.3 + (1- δ)K
K2
Repeating these steps, we
can find the capital stock in
any future year
K1
K’= K
K0
K1
K2
K
How does the capital stock
change over time?
K’
K’ = s A K0.3 + (1- δ)K
K3
K2
Repeating these steps, we
can find the capital stock in
any future year
K1
K’= K
K0
K1
K2
K
How does the capital stock
change over time?
K’
K’ = s A K0.3 + (1- δ)K
K3
K2
Repeating these steps, we
can find the capital stock in
any future year
K1
K’= K
K0
K1
K2 K3
K
How does the capital stock
change over time?
K’
K’ = s A K0.3 + (1- δ)K
K4
K3
K2
Repeating these steps, we
can find the capital stock in
any future year
K1
K’= K
K0
K1
K2 K3
K
How does the capital stock
change over time?
K’
K10
….
K3
K2
K’ = s A K0.3 + (1- δ)K
Notice that the capital stock
is approaching the point
where the two lines meet
K1
K’= K
K0
K1
K2 …. K10
K
How does the capital stock
change over time?
K’
K*
K’ = s A K0.3 + (1- δ)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
K*
K