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1
Solow growth model I
Assumption 1
There is no technology progress and no population growth.
Since we are talking about changes in GDP over time, we add time subscript on variables.
Y
t
is GDP at time
t
,
K
t
capital at time
t
and
L
t
labor at time
t
.
In this basic Solow growth model, we do not take into account of technology progress and
increases of labor input, either due to increases of the percentage of population working or
due to population growth. For simplicity, in the presentation we don’t distinguish between
the number of workers and population. Our focus is on capital accumulation.
Mathematically, our assumption means that every period we have the same technology,
Y
t
=
F
(
K
t
,L
t
), and the labor input is constant over time,
L
t
=
¯
L
.
Small letter
y
and
k
denote the output/GDP per capita and capital per capita. That is,
y
t
=
Y
t
L
t
and
k
t
=
K
t
L
t
. The per capita production function is deﬁned as the following:
y
t
=
f
(
k
)
≡
=
F
(
K,L
)
L
=
F
(
K
L
,
L
L
) =
F
(
k,
1)
Example: CobbDouglas production function
The per capita production function for the CobbDouglas production function,
Y
t
=
F
(
K
t
,L
t
) =
AK
α
t
L
1

α
t
, is
y
t
=
f
(
k
t
) =
F
(
k
t
,
1) =
Ak
α
t
1
1

α
=
Ak
α
t
One can also divided the production function by labor input directly get the per capita
production function:
y
t
=
Y
t
L
t
=
AK
α
t
L
1

α
t
L
t
=
A
(
K
t
L
t
)
α
=
Ak
α
t
.
Note that the marginal product of capital
MP
K
=
A
·
αK
α

1
L
α

1
=
A
·
α
(
K
L
)
α

1
=
A
·
αk
α

1
=
f
0
(
k
)
With a bit of fancy calculus, one can show that
MP
L
=
f
(
k
)

f
0
(
k
)
·
k
The law of diminishing marginal product of capital still applies. When capital per capita
increases, the marginal product of capital decreases and capital becomes less productive.
The simple model of GDP determination says that GDP per capita grows over time only
when capital per capita grows over time. Furthermore, the growth of GDP per capita is also
aﬀected by the marginal product of capital. Let Δ
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 Spring '10
 Williamson
 Macroeconomics

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