Problem Set 11
Econ 702, Spring 2004
Prepared by Ahu Gemici
May 3, 2004
Problem 1 (Romer Endogenous Growth)
Deﬁne the decentralized problem
in Romer’s endogenous growth model that we went over in class. Get a formula
for the balanced growth path in equilibrium and compare to the social planner’s
problem (Look at Problem 5 in Homework 10 for the social planner’s problem).
Solution
:
From the FOC of the ﬁrm in the ﬁnal goods sector, we have:
q
t
(
i
) = (1

α
)
L
α
1
t
x
t
(
i
)

α
(1)
w
t
=
αL
1
t
α

1
A
t
x
t
(
i
)
1

α
(2)
and since
x
t
=
K
t
ηA
t
, (2) becomes,
w
t
=
αL
1
t
α

1
A
t
(
K
t
ηA
t
)
1

α
(3)
From the FOC of the ﬁrm in the intermediate good sector:
(1

α
)
2
L
α
1
t
(
K
t
ηA
t
)

α
=
R
t
η
(4)
p
P
t
=
w
t
ζA
t
(5)
Also, in equilibrium, total proﬁt a patent generates will be equal to the
price of it so that the zero proﬁt condition for the intermediate goods sector is
satisﬁed.
p
P
t
=
∞
X
τ
=
t
π
t
(
i
)
(
R
)
τ

t
(6)
Now write down the consumer’s problem (
δ
= 1):
max
c
t
,L
1
t
,L
2
t
,k
t
+1
∞
X
t
=0
β
t
±
c
1

σ
t

1
1

σ
²
(7)
1
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c
t
+
k
t
+1
=
R
t
k
t
+
w
t
(
L
1
t
+
L
2
t
)
From the FOC to the consumer’s problem, we get:
(
c
t
+1
c
t
)
σ
=
βR
t
(8)
On the balanced growth path:
K
t
+1
=
γK
t
A
t
+1
=
γA
t
c
t
+1
=
γc
t
L
1
t
=
L
1
L
2
t
=
L
2
w
t
+1
=
γw
t
x
t
=
x
=
K
t
ηA
t
R
t
=
R π
t
(
i
) =
π p
P
t
=
p
P
q
t
(
i
) =
q
Use these above BGP conditions to write,
From (8),
γ
σ
=
βR
(9)
From (4),
(1

α
)
2
L
α
1
(
K
t
ηA
t
)

α
=
Rη
(10)
From (5) and (3),
p
P
=
αL
α

1
1
(
K
t
ηA
t
)
1

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 Spring '08
 Staff
 Economics, Trigraph, gross domestic product, Endogenous growth theory, Exogenous growth model

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