Macroeconomics B
Solution to problem set 1
The recursive problem can be written in the following form
W
(
a
t
, z
t
) = max
{
c
t
,a
t
+1
}
u
(
c
t
) +
βE
t
W
(
a
t
+1
, z
t
+1
)
(1)
s.t.
a
t
+1
= (1 +
r
)
a
t
+
y
t
−
c
t
,
a
t
given, lim
t
→∞
a
t
(1 +
r
)
t
≥
0
,
where
z
t
is the appropriate state variable (to be determined) characterizing the evolution
over time of the income process.
Replacing for
c
t
and maximizing with respect to
a
t
+1
we obtain the FOC
u
(
c
t
) =
β
E
t
∂W
(
a
t
+1
, z
t
+1
)
∂a
t
+1
.
(2)
Shifting the envelope condition
∂W
(
a
t
, z
t
)
∂a
t
=
β
(1 +
r
)
E
t
∂W
(
a
t
+1
, z
t
+1
)
∂a
t
+1
= (1 +
r
)
u
(
c
t
)
(3)
one period forward and using
β
(1 +
r
) = 1 we obtain the Euler equation
u
(
c
t
) =
E
t
u
(
c
t
+1
)
.
(4)
Given that
u
is linear we Euler equation can be rewritten as
c
t
=
E
t
c
t
+1
(5)
which we use in what follows.
1. Suppose labour income follows the stochastic process
y
t
= ¯
y
+
ε
t
−
δε
t
−
1
,
(6)
with
ε
t
white noise.
In choosing the state variables for the income process consider that
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 Spring '11
 DR.SEINFEILD
 Economics, Macroeconomics, Probability theory, Stochastic process, Markov chain, Random walk

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