University of California, Davis
ARE/ECN 200A Fall 2008
PROBLEM SET 7 ANSWER KEY
1. A Tale of Twelve Cities
As a prerequisite for application of either Theorem 1 or Theorem 2, all individuals must
have preference relations that can be represented by indirect utility functions in the Gorman
form,
v
i
(
p; w
i
) =
a
i
(
p
) +
b
i
(
p
)
w
i
.
City 1: No.
The two types of citizens have Gorman forms with
a
A
(
p
) = 0
; b
A
(
p
) =
1
3
p
1
±
1
=
3
2
3
p
2
±
2
=
3
;
a
B
(
p
) = 0
; b
B
(
p
) =
2
3
p
1
±
2
=
3
1
3
p
2
±
1
=
3
:
Since
b
A
(
p
)
6
=
b
B
(
p
)
;
Theorem 1 does not apply. We do have the Gorman form restriction for
Theorem 2 with
a
A
(
p
) =
a
B
(
p
) = 0
(indicating that both preference relations are homothetic).
However, the wealth domain is unrestricted so that Theorem 2 also does not apply. Therefore,
City 1 does not have a positive representative consumer (by either Theorem 1 or 2).
City 2: YES, by Theorem 2.
Beacuse
a
A
(
p
) =
a
B
(
p
) =
a
C
(
p
) = 0
and wealth vectors are restricted.
City 3: YES, by Theorem 1.
The two types of citizens have Gorman forms with
a
A
(
p
) = 0
; b
A
(
p
) =
1
3
p
1
±
1
=
3
2
3
p
2
±
2
=
3
;
a
F
(
p
) =
(
p
1
+ 2
p
2
)
1
3
p
1
±
1
=
3
2
3
p
2
±
2
=
3
; b
F
(
p
) =
1
3
p
1
±
1
=
3
2
3
p
2
±
2
=
3
:
Since
b
A
(
p
) =
b
F
(
p
)
;
Theorem 1 does apply this time. However, with
a
F
(
p
)
6
= 0
;
we do
not have the Gorman form restriction for Theorem 2. Therefore, City 3 does have a positive
representative consumer by Theorem 1, but not by Theorem 2.
City 4: YES, by Theorem 1.
Same reasoning as in City 3.
1
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View Full DocumentCity 5: No.
The two types of citizens have Gorman forms with
a
A
(
p
) = 0
; b
A
(
p
) =
1
3
p
1
±
1
=
3
2
3
p
2
±
2
=
3
;
a
G
(
p
) =
(
p
1
+ 2
p
2
)
2
3
p
1
±
2
=
3
1
3
p
2
±
1
=
3
; b
G
(
p
) =
2
3
p
1
±
2
=
3
1
3
p
2
±
1
=
3
:
Since
b
A
(
p
)
6
=
b
G
(
p
)
;
Theorem 1 does not apply, and with
a
G
(
p
)
6
= 0
;
Theorem 2 also does
not apply. (Theorem 2 would not apply anyway since the domain of wealth vectors is not
appropriately restricted.) Therefore, City 5 does not have a positive representative consumer
(by either Theorem 1 or 2).
City 6: No.
The two types of citizens have Gorman forms with
a
D
(
p
) = ln
p
1
p
2
±
1
; b
D
(
p
) =
1
p
1
;
a
E
(
p
) = ln
p
2
p
1
±
1
; b
E
(
p
) =
1
p
2
:
Since
b
D
(
p
)
6
=
b
E
(
p
)
;
Theorem 1 does not apply, and with
a
D
(
p
)
6
= 0
and
a
E
(
p
)
6
= 0
;
Theorem
2 also does not apply. (Theorem 2 would not apply anyway since the domain of wealth vectors
is not appropriately restricted.) Therefore, City 6 does not have a positive representative
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 Winter '08
 Silvestre,J
 Utility, Market Power, positive representative consumer

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