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UNC Wilmington
ECN 321
Department of Economics and Finance
Dr. Chris Dumas
Homework 5
Solutions
1)
max
U = 50 + 30S – (1/4)S
2
+ 300G – (1/10)G
2
S, G
subject to:
no constraints
F.O.C.'s:
(1)
0
S
)
4
/
1
(
2
30
S
U
=

=
∂
∂
(2)
0
G
)
10
/
1
(
2
300
G
U
=

=
∂
∂
Solve FOC (1) for S:
S
*
= 60
Solve FOC (2) for G:
G
*
= 1500
2)
max
U = 4D
1/4
·J
1/3
D,J
subject to:
$0.20D + $4J ≤ $40
Converting the problem into an equivalent Lagrangian problem:
max
L = 4D
1/4
·J
1/3
+
λ
(40 – 0.2D – 4J)
D,J,
λ
subject to:
nothing
F.O.C.'s:
(1)
0
2
.
0
J
D
4
)
4
/
1
(
D
L
3
/
1
4
/
3
=
λ

=
∂
∂

(2)
0
4
J
D
4
)
3
/
1
(
J
L
3
/
2
4
/
1
=
λ

=
∂
∂

(3)
0
J
4
D
2
.
0
40
L
=


=
λ
∂
∂
Combine FOC's (1) and (2) to eliminate
λ
:
λ
λ
=


4
2
.
0
J
D
4
)
3
/
1
(
J
D
4
)
4
/
1
(
3
/
2
4
/
1
3
/
1
4
/
3
4
2
.
0
D
4
J
3
=
J = (0.067)D
call this equation (4)
Substitute equation (4) back into FOC (3) and solve for D:
40  0.2D  4J = 0
40  0.2D  4(0.067D) = 0
D
*
= 40/0.468= 85.47
1
The FOC equations provide us with
three equations in three unknowns,
D, J and
λ
.
Solve these three
equations for the solution values of
D, S and
λ
.
One way is illustrated
below:
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View Full DocumentUNC Wilmington
ECN 321
Department of Economics and Finance
Dr. Chris Dumas
Substitute D
*
back into equation (4) to find J
*
:
J = (0.067)D
*
J = (0.067)(85.47)
J
*
= 5.73
Substitute D
*
and J
*
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 Fall '08
 Dumas
 Economics, Microeconomics

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