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ECON 831  Mathematical Economics
ANSWERS MIDTERM
1. (Budget correspondence)
(see your notes)
2. (Metric)
The function
d
is a nonnegative real valued function and we show that it is a metric.
We have to check the 3 properties:
(i) We show that
∀
x,y
∈
R
n
, d
(
x,y
) = 0
⇔
x
=
y
.
d
(
x,y
) = 0
⇔
n
X
i
=1

x
i

y
i

= 0
⇔ 
x
i

y
i

= 0
∀
i
= 1
,
···
,n
since

x
i

y
i
 ≥
0
⇔
x
i

y
i
= 0
∀
i
= 1
,
···
,n
⇔
x
i
=
y
i
∀
i
= 1
,
···
,n
⇔
x
=
y
Note: you can also prove the equivalence by proving both "
⇒
" and "
⇐
".
(ii) We show that
∀
x,y
∈
R
n
, d
(
x,y
) =
d
(
y,x
)
.
d
(
y,x
) =
n
X
i
=1

y
i

x
i

=
n
X
i
=1

x
i

y
i

b/c for any real number
a
,

a

=
 
a

=
d
(
x,y
)
(iii) We show that
∀
x,y,z
∈
R
n
, d
(
x,y
)
≤
d
(
x,z
) +
d
(
z,y
)
.
d
(
x,y
)
≤
d
(
x,z
) +
d
(
z,y
)
⇔
n
X
i
=1

x
i

y
i
 ≤
n
X
i
=1

x
i

z
i

+
n
X
i
=1

z
i

y
i

1
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View Full Document n
X
i
=1

x
i

y
i

=
n
X
i
=1

(
x
i

z
i
) + (
z
i

y
i
)

≤
n
X
i
=1
(

x
i

z
i

+

z
i

y
i

)
b/c for any 2 real numbers
a,b,

a
+
b
 ≤ 
a

+

b

=
n
X
i
=1

x
i

z
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This note was uploaded on 02/19/2010 for the course ECON 831 taught by Professor Antoine during the Fall '09 term at Simon Fraser.
 Fall '09
 Antoine
 Economics

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