by complementary slackness. Moreover, the positivity of
du
implies
λ >
0. Complementary
slackness then implies
px
+
y
=
I
. The additional equation required is
px
+
y
=
I
, which gives
us three equations in three unknowns (
x
,
y
,
λ
).
c
) Are the secondorder sufﬁcient conditions satisﬁed?
Answer:
The Hessian of
L
is
d
2
u
, which is negative deﬁnite, so the secondorder conditions for
a maximum are satisﬁed.
d
) Consider the system from parts (a) or (b) (as appropriate) as implicitly deﬁning (
x
*
,y
*
,λ
*
) as
functions of (
p,I
). Is
x
*
(
p,I
) a continuously differentiable function? Why?
Answer:
The relevant system is
0 =
∂
u
∂
x

λp
0 =
∂
u
∂
y

λ
0 =
px
+
y

I
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentMATHEMATICAL ECONOMICS FINAL, DECEMBER 10, 2002
Page 2
If we call the right hand side
F
(
x,y,λ,p,I
), we can apply the implicit function theorem provided
d
(
x,y,λ
)
F
is invertible. We have
d
(
x,y,λ
)
F
=
∂
2
u
∂
x
2
∂
2
u
∂
x
∂
y

p
∂
2
u
∂
x
∂
y
∂
2
u
∂
y
2

1

p

1
0
.
The determinant is
2
p
∂
2
u
∂
x
∂
y

p
2
∂
2
u
∂
y
2

∂
2
u
∂
x
2
= (

1
,

p
)
d
2
u
±

1

p
²
.
Since
d
2
u
is negative deﬁnite, the determinant is negative, and so the matrix is invertible. The
implicit function theorem then shows (
x,y,λ
) is a
C
1
function of (
p,I
).
3. A consumer has the quasilinear utility function
u
(
x,y
) =
x
+
y
2
. The consumer consumes nonnegative
quantities of both goods, subject to the budget constraint:
px
+
y
≤
6. Find (
x
*
,y
*
) that maximizes
utility subject to the above constraints. Be sure to check the constraint qualiﬁcation and secondorder
conditions.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 STAFF
 Economics, Critical Point, Derivative, Optimization, Continuous function, Î»

Click to edit the document details