W3211 Spring 2009
Professor Vogel
Recitation for the Week of February 02, 2009
.
1.
(a)
MRS
=
MU
x
=MU
y
=
1
=x
1
= 1
=x
, It is diminishing with
x
but is independent of
y:
(b)
max
x;y;°
L
=
y
+ ln
x
+
°
(12
°
2
x
°
3
y
)
@
L
@x
=
1
x
°
2
°
= 0
@
L
@y
= 1
°
3
°
= 0
@
L
@°
= 12
°
2
x
°
3
y
= 0
By °rst two FOCs,
°
= 1
=
2
x
= 1
=
3
!
x
°
= 1
:
5
Substituting
x
°
= 1
:
5
into the last FOC,
y
°
= 3
2. In order to get elasticities we need demand functions of
x
and
y
max
x;y;°
L
=
y
+ ln
x
+
°
(
I
°
p
x
x
°
p
y
y
)
@
L
@x
=
1
x
°
p
x
°
= 0
@
L
@y
= 1
°
p
y
°
= 0
@
L
@°
=
I
°
p
x
x
°
p
y
y
= 0
By solving this maximization problem, we have
y
°
=
I
p
y
°
1
and
x
°
=
p
x
p
y
(a)
e
y;p
y
=
@y
@p
y
p
y
y
=
°
I
p
2
y
p
y
y
=
°
I
yp
y
e
y;I
=
@y
@I
I
y
=
1
p
y
I
y
=
I
yp
y
e
y;p
x
=
@y
@p
x
p
x
y
= 0
(b)
e
x;p
x
=
@x
@p
x
p
x
x
=
°
p
y
xp
x
e
x;I
=
@x
@I
I
x
= 0
e
x;p
y
=
@y
@p
x
p
x
y
=
p
y
xp
x
(c) The optimal consumption of
x
,
x
°
=
p
x
p
y
, is independent of income
I:
Ux
p
x
=
1
p
x
x
;
U
y
p
y
=
1
p
y
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2
3. By
MRS
=
0
:
4
x
°
0
:
6
y
0
:
6
0
:
6
x
0
:
4
y
°
0
:
4
=
1
2
=
p
x
p
y
and his budget constraint
10 =
x
+ 2
y
, the initial
optimal
(
x
°
; y
°
)
is
(4
:
3)
. After the price change of
x
,
p
x
= 2
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 Spring '08
 Govel
 Utility, Px Py, Px x Py

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