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Economics 617
Problem Set 7 Solution Key
1. [Convex Sets]
(a) Define the set:
Δ
n
∈
R
n
:
∑
i
1
n
i
1
This set is called in the
unit simplex
in
R
n
.
Show that
Δ
n
is a convex subset of
R
n
.
Solution:
Let
′
,
′′
∈
Δ
n
, and
∈
0, 1
. We need to prove:
′
1
−
′′
∈
Δ
n
. It’s
obvious that
′
1
−
′′
∈
R
n
.
∑
i
1
n
i
′
1
−
i
′′
∑
i
1
n
i
′
1
−
∑
i
1
n
i
′′
1
−
1,
where we have used the facts that
∑
i
1
n
i
′
1
,
∑
i
1
n
i
′′
1.
Therefore,
′
1
−
′′
∈
Δ
n
. By definition,
Δ
n
is convex.
(b) Let
a
∈
R
n
.
Define the set:
T
x
∈
R
n
:
ax
≥
1
Is
T
a convex set in
R
n
?
Explain.
Remark: for
n
1,
T
x
∈
R
:
x
≥
1/
a
,
T
is convex by definition. For
n
2,
T
is a closed half space, which is also convex.
Solution: Let
x
′
,
x
′′
∈
T
, and
∈
0, 1
. It’s easy to verify that
x
′
1
−
x
′′
∈
R
n
.
a
x
′
1
−
x
′′
ax
′
1
−
ax
′′
≥
1
−
1,
where the inequality follows because
ax
′
≥
1,
ax
′′
≥
1
and
∈
0, 1
.
2. [Concave Functions]
(a) Let
A
be a convex subset of
R
m
,
and
f
a realvalued function on
A
.
Show that
f
is concave on
A
if and only if the set:
V
x
,
∈
A
R
:
f
x
≥
is a convex subset of
R
m
1
.
Solution:
(i) the "only if" part:
Suppose
f
is concave on
A
.
Let
x
′
,
′
and
x
′′
,
′′
be elements of
V
, and
∈
0, 1
. We know
f
x
′
≥
′
and
f
x
′′
≥
′′
. Mutliply the first equation by
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This note was uploaded on 08/29/2009 for the course ECON 617 taught by Professor Staff during the Fall '08 term at Cornell University (Engineering School).
 Fall '08
 STAFF
 Economics

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