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Economics 101
Solutions to Problem Set 3
Due Thursday, February 3, 2011
Soojin Kim
1. Nicholson and Snyder problem 4.2.
(a) Problem of a young connoisseur
Solution
She solves the following maximization problem.
max
w
F
,w
c
w
2
3
F
w
1
3
C
s.t.
40
w
F
+ 8
w
C
= 600
.
Since at optimum,
MRS
ij
=
p
i
p
j
,
we have
U
w
F
U
w
C
=
2
3
w

1
3
F
w
1
3
C
1
3
w
2
3
F
w

2
3
C
=
40
8
⇒
5 = 2
w
C
w
F
⇒
5
w
F
= 2
w
C
.
Thus, from the budget constraint, we get
w
*
C
= 25 and
w
*
F
= 10.
(b) Eﬀect of change in the price of Bordeaux on the demand for wine
Solution
Now, we have a diﬀerent budget constraint.
max
w
F
,w
C
w
2
3
F
w
1
3
C
s.t.
20
w
F
+ 8
w
C
= 600
.
Under the new price, the demand becomes (
w
**
F
,w
**
C
) = (20
,
25).
(c) Compare utilities in (a) and (b)
Solution
Note that
U
(
w
*
F
*
c
) = 10
2
3
25
1
3
≈
13
.
57
U
(
w
**
F
**
C
) = 20
2
3
25
1
3
≈
21
.
54
.
1
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View Full Document Utility is unambiguously higher in part (b), since the price of wine decreased, which
eﬀectively expands the budget set of the wine fancier.
In order to ﬁnd the monetary value of this utility increase, let us ask the following
question: what is the amount of income
I
that would leave the wine fancier indiﬀerent
between having the price of Bordeaux decline to $20 (and having budget of $600)
and having the price of Bordeaux at $40, i.e. what is
I
such that
V
(40
,
8
,I
) =
V
(20
,
8
,
600)?
Suppose the connoisseur had the budget of
I
to be spent on her wine cellar and the
price of Bordeaux were $40. Then, her optimal choice is (
I
60
,
I
24
) with the associated
utility of
(
I
60
)
2
3
(
I
24
)
1
3
.
Thus, the level of
I
that would leave the connoisseur indiﬀerent to having price of
Bordeaux devline to $20 is
I
that satisﬁes the following equation:
±
I
60
²
2
3
±
I
24
²
1
3
=
U
(
w
**
F
,w
**
C
)
≈
21
.
54
⇒
I
≈
952
.
25
.
Thus, if the connoisseur had the budget of $952
.
25 instead of $600, she would be
indiﬀerent between having the price of wines being (
p
F
,p
C
) = (40
,
8) and (
p
F
c
) =
(20
,
8)
.
2. Suppose Bruce enjoys opera, and while at the opera, champagne. Not surprisingly, better
seats at the opera cost more. For simplicity, suppose that all qualities of seats are available,
with
q
denoting the quality of a seat. His utility function is quasilinear, of the form
U
(
q,c
) = log
q
+
c,
where
c
is his consumption of champagne.
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This note was uploaded on 03/02/2011 for the course ECON 101 taught by Professor Dannicatambay during the Spring '08 term at UPenn.
 Spring '08
 DANNICATAMBAY
 Economics, Microeconomics

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