Short notes on a simple global solution method
1
Consider the following recursive maximization problem:
V
(
a, e
) = max
c,a
0
(
u
(
c
) +
β
P
e
0
∈
{
e
1
,e
2
}
π
(
e
0

e
)
V
(
a
0
, e
0
)
)
subject to
c
≤
e
+ (1 +
r
)
a
−
a
0
a
0
≥ −
φ
The Euler equation (substituting in the budget constraint) is
u
0
(
e
+ (1 +
r
)
a
−
a
0
)
≥
β
X
e
0
∈
{
e
1
,e
2
}
π
(
e
0

e
)
V
a
(
a
0
, e
0
)
with equality if
a
0
>
0
.
The Envelope condition is
V
a
(
a, e
) =
u
0
(
e
+ (1 +
r
)
a
−
a
0
)(1 +
r
)
We are looking for a decision rule
a
∗
(
a, e
)
for savings (given values for
r
and
φ
)
.
1. Construct a grid on
a
=
{
a
1
, a
2
...a
n
}
.
Set
a
1
=
−
φ.
2. Guess a vector of values
ˆ
V
a
for all combinations for
a
and
e
on the grid:
ˆ
V
a
=
n
ˆ
V
a
(
a
1
, e
1
)
,
ˆ
V
a
(
a
2
, e
1
)
, ...,
ˆ
V
a
(
a
n
, e
1
)
,
ˆ
V
a
(
a
1
, e
2
)
, ...,
ˆ
V
a
(
a
n
, e
2
)
o
(e.g.
ˆ
V
a
= 1
)
3. Take the
fi
rst point on the grid
(
a
1
, e
1
)
.
4. Check whether
u
0
(
e
1
+ (1 +
r
)
a
1
+
φ
)
−
β
X
e
0
∈
{
e
1
,e
2
}
π
(
e
0

e
1
)
ˆ
V
a
(
a
1
, e
0
)
>
0
.
1
Jonathan Heathcote, May 13 2000, last updated April 2006.
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•
If it is then
a
∗
(
a
1
, e
1
) =
−
φ
•
If it is not then
a
∗
(
a
1
, e
1
)
>
−
φ.
Solve for
a
∗
(
a
1
, e
1
)
as follows:
1. Note that by concavity
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