Econ 387L: Macro II
Spring 2008, University of Texas
Instructor: Dean Corbae
Problem Set #7 Due 3/25/08
I. The
fi
rst problem introduces you to dynamic programming in a
fi
nite
T
horizon
stochastic growth model. Speci
fi
cally, let
T
= 1
(i.e. a two period model). Let
z
t
∈
Z
(a
fi
nite and time independent set) denote an exogenous technology shock in period
t
and
π
(
z
t
, z
t
−
1
)
denote the probability of being in state
z
t
conditional on being in state
z
t
−
1
.
Let capital holdings at the beginning of period
t
be denoted
k
t
∈
X
and the state variable
be denoted
s
t
= (
k
t
, z
t
)
∈
X
×
Z
. Let
y
(
s
t
) =
z
t
f
(
k
t
)
denote output in state
s
t
where
f
(0) = 0
, f
0
>
0
, f
00
<
0
, c
(
s
t
)
denote consumption in state
s
t
, and
k
t
+1
(
s
t
)
denote
capital chosen in state
s
t
used for production next period. Households start with
k
0
units of
capital. Households are expected utility maximizers with preferences that satisfy
u
0
(
c
)
>
0
,
u
00
(
c
)
<
0
,
discount the future at rate
β
, and have a uniform prior before period
0
of
π
(
z
i
0
,
∅
) =
μ
for all
z
i
0
∈
Z
.
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 Spring '07
 CORBAE
 Probability theory, Stochastic process, Markov chain, Convex function, horizon stochastic growth

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