Answer to practice problem from lecture 2
1. Gordon will maximize his preferences
U
(
a; c
) =
a
1
3
c
2
3
subject to the budget
constraint
M
p
a
a
+
p
c
c
. His Lagrangian will be
L
=
a
1
3
c
2
3
+
(
M
±
p
a
a
±
p
c
c
)
@L
@a
:
1
3
a
2
3
c
2
3
±
a
= 0
(1)
@L
@c
:
2
3
a
1
3
c
1
3
±
c
= 0
(2)
@L
:
M
±
p
a
a
±
±
p
c
c
±
= 0
(3)
Gordon uses statements
(1)
and
(2)
to remove the Lagrange multiplier
occur where the budget constraint holds with equality and
1
3
a
2
3
c
2
3
2
3
a
1
3
c
1
3
=
c
±
2
a
±
=
p
a
p
c
c
±
=
2
p
a
p
c
a
±
He plugs this relationship into
(3)
M
=
p
a
a
±
+
p
c
c
±
=
p
a
a
±
+
p
c
2
p
a
p
c
a
±
±
=
3
p
a
a
±
a
±
(
p
a
; p
c
; M
)
=
M
3
p
a
c
±
(
p
a
; p
c
; M
)
=
2
p
a
p
c
M
3
p
a
±
=
2
M
3
p
c
Then, since the price of an album is $10 while the price of a concert ticket
is $30, while Gordon±s monthly income is $300, we know that he optimally
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 Summer '08
 cunningham
 Utility

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