Since at most two constraints can bind, constraint qualification is satisfied.
The Lagrangian is
L
=
x
+
y
2

λ
(
px
+
y

6) +
μx
+
νy
. The firstorder conditions are
0 = 1

λp
+
μ
0 = 2
y

λ
+
ν.
The first equation tells us that
λ
≥
1
/p >
0. Complementary slackness then implies
px
+
y
= 6. Now
there are two cases to consider.
If
x
= 0, then
y
= 6. By complementary slackness,
ν
= 0. The second firstorder condition implies
λ
= 12. Provided 12
p
≥
1,
μ
= 12
p

1
≥
0, and we have a critical point (0
,
6).
If
y
= 0, then
x
= 6
/p
. By complementary slackness,
μ
= 0. The first firstorder condition implies
λ
= 1
/p
. Substituting in the second equation, we find
ν
= 1
/p
. This is our second critical point, (6
/p,
0).
We now turn to the secondorder conditions. Since there are
n
= 2 variables and
m
= 2 constraints,
we have no secondorder conditions available.
What we can do is compare the values of utility:
u
(0
,
6) = 36 while
u
(6
/p,
0) = 6
/p
. If
p >
1
/
6, (0
,
6)
is the maximizer while if
p <
1
/
6, (6
/p,
0) is the maximizer. If
p
= 1
/
6, both are maximizers.
4. A consumer is endowed with
T >
0 units of a resource
r
that may either be consumed or sold at price
w >
0.
The resource cannot be bought on the market, only sold.
The consumer also consumes a
consumption good
c
which has price
p >
0. The budget constraint is
pc
+
wr
≤
wT
. The consumer is
also subject to the constraints
c
≥
0,
r
≥
0, and
r
≤
T
(the last constraint reflects the fact that the resource
cannot be purchased). The utility function is
u
(
c, r
) = ln
c
+2 ln
r
. Solve the consumer’s problem, paying
attention to constraint qualification and the secondorder conditions.
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MATHEMATICAL ECONOMICS FINAL, DECEMBER 10, 2002
Page 3
Answer:
The Lagrangian is
L
= ln
c
+ 2 ln
r

λ
(
pc
+
wr

wT
) +
μc
+
νr

ρ
(
r

T
). The firstorder
conditions are
0 =
1
c

λp
+
μ
0 =
2
r

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 Economics, Critical Point, Derivative, Optimization, Continuous function, λ

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