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140
Economic Dynamics
14.
f
1
(
x
) and
f
2
(
x
) are linearly dependent if and only if there exist constants
b
1
and
b
2
, not all zero, such that
b
1
f
1
(
x
)
+
b
2
f
2
(
x
)
=
0
for every
x
. If the set of functions is
not
linearly dependent, then
f
1
(
x
) and
f
2
(
x
) are linearly independent. Show that
y
1
=
Y
∗
and
y
2
=
tY
∗
are linearly independent.
15.
Given the following version of the Solow model with labour augmenting
technical progress
Y
t
=
F
(
K
t
,
A
t
L
t
)
K
t
+
1
=
K
t
+
δ
K
t
S
t
=
sY
t
I
t
=
S
t
L
t
+
1
−
L
t
L
t
=
n
A
t
=
γ
t
A
0
(i) show that
ˆ
k
t
+
1
=
(1
−
δ
)
ˆ
k
t
+
sf
(
ˆ
k
t
)
γ
(1
+
n
)
where
ˆ
k
is the capital/labour ratio
measured in ef±ciency units
, i.e.,
ˆ
k
=
(
K
/
AL
)
.
(ii) Approximate this result around
ˆ
k
∗
>
0
.
16.
Given the model
q
d
t
=
a
−
bp
t
b
>
0
q
s
t
=
c
+
dp
e
t
d
>
0
p
e
t
=
p
e
t
−
1
−
λ
(
p
t
−
1
−
p
e
t
−
1
)0
<λ<
1
(i) Show that price conforms to a Frstorder nonhomogeneous differ
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.
 Fall '11
 Dr.Gwartney
 Economics

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