5.42
Assume that
g
satisfies the Quasiconvex CQ condition at
x
∗
. That is, for every
𝑗
∈
𝐵
(
x
∗
),
𝑔
𝑗
is quasiconvex,
∇
𝑔
𝑗
(
x
∗
)
∕
=
0
and there exists ˆ
x
such that
𝑔
𝑗
(ˆ
x
)
<
0.
Consider the perturbation
dx
= ˆ
x
−
x
∗
. Quasiconvexity and regularity implies that
for every binding constraint
𝑗
∈
𝐵
(
x
∗
) (Exercises 4.74 and 4.75)
𝑔
𝑗
(ˆ
x
)
< 𝑔
𝑗
(
x
∗
) =
⇒ ∇
𝑔
𝑗
(
x
∗
)
𝑇
(ˆ
x
−
x
∗
) =
∇
𝑔
𝑗
(
x
∗
)
𝑇
dx
<
0
That is
𝐷𝑔
𝑗
[
x
∗
](
dx
)
<
0
Therefore,
dx
∈
𝐿
0
(
x
∗
)
∕
=
∅
and
g
satisfies the Cottle constraint qualification condition.
5.43
If the binding constraints
𝐵
(
x
∗
) are regular at
x
∗
, their gradients are linearly
independent. That is, there exists no
𝜆
𝑗
∕
= 0
, 𝑗
∈
𝐵
(
x
∗
) such that
∑
𝑗
∈
𝐵
(
x
∗
)
𝜆
𝑗
∇
𝑔
𝑗
[
x
∗
] =
0
By Gordan’s theorem (Exercise 3.239), there exists
dx
∈ ℜ
𝑛
such that
∇
𝑔
𝑗
[
x
∗
]
𝑇
dx
<
0 for every
𝑗
∈
𝐵
(
x
∗
)
Therefore
dx
∈
𝐿
0
(
x
∗
)
∕
=
∅
.
5.44
If
𝑔
𝑗
concave,
𝐵
𝑁
(
x
∗
) =
∅
, and AHUCQ is trivially satisfied (with
dx
=
0
∈
𝐿
1
).
For every
𝑗
, let
𝑆
𝑗
=
{
dx
:
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 Fall '10
 Dr.DuMond
 Macroeconomics, All rights reserved, Constraint, Convex function, x∗, Michael Carter, Foundations of Mathematical Economics

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