Homework Assignment #7
18.4 Find the general expression (in terms of all the parameters) for the commodity bundle (
x
1
, x
2
) which
maximizes the CobbDouglas utility function
U
(
x
1
, x
2
) =
kx
a
1
x
1

a
2
on the budget set
p
1
x
1
+
p
2
x
2
=
I
.
Answer:
We assume
p
0
and
I >
0, so the problem has a nontrivial solution. We will also assume
that 0
< a <
1.
The constraint has derivative (
p
1
, p
2
)
6
=
0
, implying that the NDCQ condition is
satisfied.
The Lagrangian is
L
=
kx
a
1
x
1

a
2

λ
(
p
1
x
1
+
p
2
x
2

I
).
This yields firstorder conditions 0 =
akx
a

1
1
x
1

a
2

λp
1
and (1

a
)
kx
a
1
x

a
2

λp
2
. Collecting the
λ
terms on the lefthand side and dividing,
we obtain
a
1

a
x
2
x
1
=
p
2
p
1
.
Rearranging, we find
p
1
x
1
=
ap
2
x
2
/
(1

a
). We now use the budget constraint to find (1+
a/
(1

a
))
p
2
x
2
=
I
, so
x
2
= (1

a
)
I/p
2
and
x
1
=
aI/p
1
.
The secondorder conditions should also be checked, but Chapter 18 did not cover that. In this case we
know that CobbDouglas utility is concave (as shown in class), so we have a global maximum (Chapter
21).
18.9 Maximize
x
2
y
2
z
2
subject to
x
2
+
y
2
+
z
2
=
c
2
, where
c
is some fixed positive constant.
What is the
maximum value of the objective function on the constraint set? Show that for all
x
,
y
,
z
.
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 Fall '08
 STAFF
 Optimization, Utility, $1000, $80,000, thousand dollars, NDCQ

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