Alejandro Badel.
Macro II, Jonathan Heatchote. Homework 5 Answers.
1)
Recursive formulation of the individual°s maximization problem:
state:
(
a; w
)
where
a
represents beggining of period bond holdings and
w
is
the current realization of the wage process.
control:
(
n; a
0
)
where
n
are hours worked and
a
0
are end of period bond
holdings.
feasible set:
° (
a; w
) =
f
(
n; a
0
) :
wn
+ (1 +
r
)
a
°
a
0
±
0
; a
0
±
0
g
Bellman Equation:
V
(
a; w
)
=
max
a
0
2
°(
a;w
)
(
1
1
°
°
°
wn
+ (1 +
r
)
a
°
a
0
°
n
1+1
="
1 + 1
="
±
1
°
°
+
±E
[
V
(
a
0
; w
0
)
j
w
]
)
w
0
random draw from
²
(
w
0
j
w
)
2)
First order conditions
Intertemporal
u
c
±
±
(1 +
r
)
E
[
u
0
c
j
w
]
Intratemporal
wu
c
=
°
u
n
3)
From the Intratemporal FOC we have
w
=
n
1
="
ln
n
=
°
"
ln
+
"
ln
w
We can see that the elasticity of worked hours with respect to wages is given
by
":
4)
We could assume for simplicity that the wage distribution starts o/ at its
stationary value, this would make the planner°s problem invariant over time.
Mantaining the assumption that the initial distribution is uniform (50 50), we
can formulate the planner°s problem recursively as follows:
State:
p
current fraction of agents with high wages
(
w
h
)
.
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 '05
 would
 Optimization, Order theory, NH, wl nl

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