TA: Yankun Wang
Economics 617
Problem Set 10 Solution Key
1. [Using the Kuhn-Tucker Necessity Theorem]
Let
a, c
be arbitrary positive parameters, satisfying
a>c
.
De
f
ne the
following functions:
F
(
x
)=
a
−
xf
o
r
x
∈
[0
,a
]
G
(
x
R
x
0
F
(
z
)
dz for x
∈
[0
]
Consider the following constrained maximization problem:
Maximize xF
(
x
)
−
cx
+
y
subject to
y
≤
G
(
x
)
−
xF
(
x
)
x
≤
a
(
x, y
)
≥
0
⎫
⎪
⎪
⎬
⎪
⎪
⎭
(
P
)
(a) Use Weierstrass theorem to show that problem
(
P
)
has a solution.
Solution:
G
(
x
Z
x
0
F
(
z
)
dz
=
ax
−
x
2
2
,
and the problem becomes:
Maximize
−
x
2
+(
a
−
c
)
x
+
y
subject to
y
≤
x
2
2
x
≤
a
(
x, y
)
≥
0
It’s easy to verify that the constraint set is nonempty, closed and bounded;
the objective function is continuous as a polynomial. Thus Weierstrass the-
orem tells us problem
(
P
)
has a solution.
(b) Show that if
(ˆ
x,
ˆ
y
)
is any solution to problem
(
P
)
,
then
ˆ
x>
0
,
ˆ
y>
0
.
Proof: Suppose
ˆ
x
=0
,
by constraint
y
≤
x
2
2
,
we must have
ˆ
y
.
But
consider
(
x
∗
,y
∗
)=(
a
−
c
2
,
(
a
−
c
)
2
8
)
.
If we let
f
(
x, y
−
x
2
a
−
c
)
x
+
y,
f
(
x
∗
∗
)
>f
(0
,
0) = 0
,
contradicting the optimality of
x,
ˆ
y
)
.
Now suppose
ˆ
0
,
ˆ
y
.
Then let
(
x
∗
∗
³
ˆ
x,
ˆ
x
2
2
´
.
Again,
f
(
x
∗
∗
)
>
f
x,
ˆ
y
)
,
and contraction.
1