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MIDTERM, GRADUATE MACRO, ECON 606
Jonathan Heathcote
March 2nd 2006
Consider the following economy. There are two sectors: an apple sector,
and an orange sector. Both sectors use land
F
and labor
n
to produce. The
amount of land in each sector,
F
o
and
F
a
is
f
xed:
F
o
=
F
a
=
F.
The production
technology is CobbDouglas:
Y
i
=
z
i
F
θ
i
n
1
−
θ
i
i
=
a, o
where
0
≤
θ
≤
1
.z
i
, which determines sectorspeci
f
c productivity, is given by
z
o
=7
0
+
(
T
−
70)
z
a
0
−
(
T
−
70)
where
T
∈
{
60
,
61
, ...,
79
,
80
}
is the average temperature in Fahrenheit in the
period, and evolves over time according to a
f
rstorder Markov process de
f
ned
by the transition probability matrix
Π
.
In
f
ntelylived consumers have identical preferences over a composite con
sumption good
C
and leisure
l.
Suppose, to start with, that labor markets are
segmented: half of the population can only work on apple farms, the other half
can only work on orange farms. Let superscripts denote the identity of workers
/ consumers: thus, for example,
c
o
a
denotes consumption of apples by workers
who work on orange farms. Preferences for workers of type
i
∈
{
a, o
}
are given
by
E
∞
P
t
=0
β
t
u
(
C
i
t
,l
i
t
)
where
u
(
C
i
i
)=
u
(
C
i
)+
v
(
l
i
)
C
i
=
μ
¡
c
i
a
¢
σ
−
1
σ
+
¡
c
i
o
¢
σ
−
1
σ
¶
σ
σ
=1
σ
≥
0
is the elasticity of substitution between apples and oranges.
Assume the standard inequality constraints must be respected:
c
i
a
,c
i
o
i
≥
0
,
l
i
≤
1
.
PART 1: Consider a planner who cares equally about apple workers and
orange workers.
1
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View Full Document1. Write down a recursive formulation of the planner’s problem. Take care
to choose appropriate state variable(s).
V
(
T
)=
max
n
o
,n
a
,c
a
a
,c
o
a
,c
a
o
,c
o
o
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1
2
∙
u
μ
³
(
c
o
a
)
σ
−
1
σ
+(
c
o
o
)
σ
−
1
σ
´
σ
σ
=1
¶
+
v
(1
−
n
o
)
¸
+
1
2
∙
u
μ
³
(
c
o
a
)
σ
−
1
σ
c
o
o
)
σ
−
1
σ
´
σ
σ
=1
¶
+
v
(1
−
n
0
)
¸
+
βE

T
[
V
(
T
0
)]
⎫
⎪
⎪
⎬
⎪
⎪
⎭
subject to
c
a
a
+
c
o
a
=
z
a
(
T
)
F
θ
(
n
a
)
(1
−
θ
)
c
a
o
+
c
o
o
=
z
o
(
T
)
F
θ
(
n
o
)
(1
−
θ
)
and the inequality constraints, where
z
o
(
T
)=7
0+
(
T
−
70)
z
a
(
T
0
−
(
T
−
70)
2. What are the planner’s
f
rst order conditions?
Assume none of the inequality constraints are binding. Then the FOCs
simplify to
u
c
o
a
=
u
c
a
a
u
c
o
o
=
u
c
a
o
(1
−
θ
)
1
2
u
c
o
a
z
a
(
T
)
F
θ
(
n
a
)
−
θ
=
v
0
(1
−
n
a
)
(1
−
θ
)
1
2
u
c
o
o
z
o
(
T
)
F
θ
(
n
o
)
−
θ
=
v
0
(1
−
n
o
)
3. Compare the optimal consumption bundle for apple farm workers to the
optimal consumption bundle for orange farm workers.
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