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New Notes on the Solow Growth Model
∗
Roberto Chang
September 2009
1
The Model
The
f
rstingredientofadynamicmodelisthedescriptionofthetimehorizon. In
the original Solow model, time is continuous and the horizon is in
f
nite. Without
loss of generality assume that time is indexed by
in
[0
∞
)
At each point in time, there is only one
f
nal good that is produced via the
aggregate production function
:
=
(
)
Here
and
denote output, labor productivity, capital, and labor,
and are functions of time (i.e. we should really write
(
)
and so on, but we
omit time arguments when not needed). The product
is called e
f
ective
labor.
The function
is has neoclassical properties:
•
is continuously di
f
erentiable, strictly increasing, strictly concave
•
has constant returns to scale
A key example is the
Cobb Douglas
function:
(
)=
(
)
1
−
0
1
Exercise:
Show that the Cobb Douglas function enjoys the neoclassical prop
erties listed above.
Once you assume the existence of the aggregate production function, it is
clear that the growth of output can be due to growth of
or
Solow assumed that
and
grow at exogenous rates:
˙
=
=
˙
=
∗
These are revised but still rough notes for Econ 504.
1
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View Full Document At time
=0
and
are given by history. Then one can describe the
value of
and
at each point in time:
(
)=
(0)
(
)=
(0)
So it remains to describe the motion of capital. Assume
(0)
is given by
history. The accumulation of capital must follow:
˙
=
−
where
denotes investment and
the depreciation rate of capital.
As usual, investment is assumed to equal savings. The key behavioral equa
tion of the Solow model is that savings equal a constant fraction of output,
so:
=
This completes the description of the model.
2
The Solution
A solution is a description of the values of
and
at each point in time.
Now, from the capital accumulation equation:
˙
=
−
=
(
)
−
=
(
1)
−
=
[
(
1)
−
]
Here we have de
f
ned
=
De
f
ne
(
)=
(
1)
the intensive pro
duction function. Now we can rewrite the previous equation as:
³
˙
´
=
(
)
−
But
=
implies
˙
=
˙
−
(
+
)
Comb
inethetwotoobta
in
:
˙
=
(
)
−
(
+
+
)
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This note was uploaded on 12/10/2010 for the course ECON 3101 taught by Professor Staff during the Spring '08 term at Minnesota.
 Spring '08
 Staff
 Economics

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