Econ 387L: Macro II
Spring 2006, University of Texas
Instructor: Dean Corbae
Problem Set #3 Due 2/7/06
Consider a stochastic growth model with the following speci
fi
cation.
•
Preferences:
U
(
C
t
, H
t
, λ
t
) =
³
C
t
+
λ
t
h
(1
−
H
t
)
1
−
η
−
1
1
−
η
i´
1
−
ψ
−
1
1
−
ψ
where
λ
t
= (1
−
γ
)
−
1
λ
t
−
1
ε
t
, λ
−
1
= 1
.
(1)
and
log
ε
t
is i.i.d.
N
(0
, σ
ε
)
.
Agents discount the future at rate
β <
1
.
•
Technology:
Y
t
=
K
θ
t
¡
(1 +
g
)
t
H
t
¢
1
−
θ
(2)
where
K
0
is given,
1
≥
H
t
≥
0
.
•
Information: Households must choose
H
t
before knowing the shock to preferences
λ
t
but choose
K
t
+1
after its realization.
1) Derive the stochastic Euler equation for the savings choice.
2) Derive the equation describing the labor/leisure choice.
3) Along a possible balanced growth path, assume that
ε
t
is constant and equal to its
expected value of 1. Under what conditions on parameters might a balanced growth path
where
K
t
, C
t
, Y
t
all grow at rate
κ
while
H
t
remains constant exist? Is the set empty?
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 Spring '07
 CORBAE
 Utility, ht, balanced growth path

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