E
CON
714: M
ACROECONOMIC
T
HEORY
II
TA: T
IM
L
EE
F
EBRUARY
5, 2010
Dynamic Programming
Let’s use the simple growth example from class:
W
(
k
0
) =
max
{
c
t
,
k
t
+
1
}
∞
t
=
0
∞
∑
t
=
0
β
t
log
(
c
t
)
s.t.
c
t
+
k
t
+
1
≤
Ak
α
t
∀
t
.
We already know from class how to use the “guess and verify" technique to solve for the
value function instead:
V
(
k
)
=
max
k
0
∈
[
0,
Ak
α
]
log
(
Ak
α

k
0
) +
β
V
(
k
0
)
,
with the guess of
E
+
F
log
k
, so that
E
=
(
1

β
)

1
log
(
1

αβ
)
A
+
αβ
1

αβ
log
αβ
A
F
=
α
1

αβ
g
(
k
)
=
αβ
Ak
α
where
k
0*
=
g
(
k
)
is the policy function. Note that we can’t say
V
(
k
)
≡
W
(
k
0
)
yet, this is
the subject of next week’s lectures. For now take for granted that it holds in this case, and
I’ll use this example throughout to highlight some other facts.
1.
Iterating the value function:
Although in this case we can just solve out for the
value and policy functions pretty easily, usually we will be exploiting the Contrac
tion Mapping Theorem to get an approximation to the value function. Suppose we
use this technique using
v
0
≡
0 as an initial guess. The first iteration gives
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 Spring '08
 Staff
 Dynamic Programming, Recursion, Metric space, Bellman equation, policy function, complete metric space

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