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5
The Solow Growth Model
5.1
Models and Assumptions
•
What is a model? A mathematical description of the economy.
•
Why do we need a model? The world is too complex to describe it in
every detail.
•
What makes a model successful? When it is simple but eFective in de
scribing and predicting how the world works.
•
A model relies on simplifying assumptions. These assumptions drive the
conclusions of the model. When analyzing a model it is crucial to spell
out the assumptions underlying the model.
•
Realism may not a the property of a good assumption.
67
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Basic Assumptions of the Solow Model
1. Continuous time.
2. Single good produced with a constant technology.
3. No government or international trade.
4. All factors of production are fully employed.
5. Labor force grows at constant rate
n
=
˙
L
L
.
6. Initial values for capital,
K
0
and labor,
L
0
given.
68
Production Function
•
Neoclassical (CobbDouglas) aggregate production function:
Y
(
t
)=
F
[
K
(
t
)
,L
(
t
)] =
K
(
t
)
α
L
(
t
)
1

α
•
To save on notation write:
Y
=
AK
α
L
1

α
•
Constant returns to scale:
F
(
λK, λL
)=
λ F
(
K,L
)=
λA K
α
L
1

α
•
Inputs are essential:
F
(0
,
0) =
F
(
K,
0) =
F
(0
,L
)=0
69
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Marginal productivities are positive:
∂F
∂K
=
αAK
α

1
L
1

α
>
0
∂F
∂L
= (1

α
)
AK
α
L

α
>
0
•
Marginal productivities are decreasing,
∂
2
F
∂K
2
=(
α

1)
αA K
α

2
L
1

α
<
0
∂
2
F
∂L
2
=

α
(1

α
)
AK
α
L

α

1
<
0
70
Per Worker Terms
•
Defne
x
=
X
L
as a per worker variable. Then
y
=
Y
L
=
AK
α
L
1

α
L
=
A
±
K
L
²
a
±
L
L
²
1

α
=
Ak
α
•
Per worker production Function has decreasing returns to scale.
71
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View Full Document Capital Accumulation
•
Capital accumulation equation:
˙
K
=
sY

δK
•
Important additional assumptions:
1. Constant saving rate (very specifc preFerences: no
r
)
2. Constant depreciation rate
72
•
Dividing by
K
in the capital accumu equation:
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This note was uploaded on 12/06/2010 for the course ECON 3020 taught by Professor Williamson during the Spring '10 term at FSU.
 Spring '10
 Williamson
 Macroeconomics

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