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Solutions to Final Exam
EC720.01  Math for Economists
Peter Ireland
Boston College, Department of Economics
Fall 2010
Due Tuesday, December 14, 2010 at 11:00am
1. Growth and Pollution, Part I
The social planner’s problem is
max
z
(
Akz
)
1
−
σ
−
1
1
−
σ
−
°
B
γ
±
(
Akz
β
)
γ
subject to 1
≥
z.
a. The Lagrangian for the social planner’s problem can be written as
L
(
z,λ
)=
(
Akz
)
1
−
σ
−
1
1
−
σ
−
°
B
γ
±
(
Akz
β
)
γ
+
λ
(1
−
z
)
.
b. According to the KuhnTucker theorem, if
z
∗
is the value of
z
that solves the social
planner’s problem, then there exists an associated value
λ
∗
of
λ
such that, together,
z
∗
and
λ
∗
satisfy the ±rstorder condition
L
1
(
z
∗
,λ
∗
)=(
Ak
)
1
−
σ
(
z
∗
)
−
σ
−
βB
(
Ak
)
γ
(
z
∗
)
βγ
−
1
−
λ
∗
=0
,
the constraint
L
2
(
z
∗
,λ
∗
)=1
−
z
∗
≥
0
,
the nonnegativity condition
λ
∗
≥
0
,
and the complementary slackness condition
λ
∗
(1
−
z
∗
)=0
.
c. Let’s look ±rst for a solution with a nonbinding constraint, that is, with 1
>z
∗
.B
y
the complementary slackness condition, this solution must have
λ
∗
= 0. Hence, the
±rstorder condition implies that
z
∗
=
°
1
βB
±
1
βγ
+
σ
−
1
°
1
Ak
±
γ
+
σ
−
1
βγ
+
σ
−
1
For this solution to apply, it must be that
z
∗
<
1o
r
,inl
igh
to
ftheexp
re
s
s
ionju
s
t
derived, that
Ak >
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View Full DocumentIf, on the other hand, the constraint is binding, then
z
∗
=1andtheFrstordercond
it
ion
implies that
λ
∗
=(
Ak
)
1
−
σ
−
βB
(
Ak
)
γ
.
±or this solution to apply, it must be that
λ
∗
≥
0o
r
,inl
igh
to
ftheexp
re
s
s
ionju
s
t
derived, that
°
1
βB
±
1
γ
+
σ
−
1
≥
Ak.
Putting these results together, the optimal choice for
z
∗
is given by
z
∗
=
1i
f
²
1
βB
³
1
γ
+
σ
−
1
≥
Ak
²
1
βB
³
1
βγ
+
σ
−
1
´
1
Ak
µ
γ
+
σ
−
1
βγ
+
σ
−
1
if
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 Fall '09
 IRELAND
 Economics

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