E
frqrplfv
401
1
Final Examination questions: Answers
1. The “Gorman form” means that the indirect utility function can be written as
a
+
bw
where
a
and
b
are functions of prices alone.
Quasilinear preferences do not necessarily
f
t the Gorman form. For example, if
u
(
x
1
,x
2
)=2
x
1
/
2
1
+
x
2
then
x
1
x
2
V
(
p
1
,p
2
,w
)
Range
(
p
2
/p
1
)
2
w/p
2
−
p
2
/p
1
p
2
/p
1
+
w/p
2
w
≥
p
2
2
/p
1
w/p
1
02
(
w/p
1
)
1
/
2
w<p
2
2
/p
1
The
f
rst row conforms to the Gorman form but the second row doesn’t. When
wealth is su
ﬃ
ciently low quasilinear preferences may
not
f
t the Gorman form.
If
preferences do
f
ttheGormanformandeveryonehasthesamepreferencesthen
redistributions of wealth will not a
f
ect aggregate demands. To see this in the form
of a twogood example write
v
j
(
p
1
2
j
)=
a
(
p
1
2
)+
b
(
p
1
2
)
w
j
,j
=
A,B.
Using Roy’s identity for good
1
x
A
1
(
p
1
2
A
x
B
1
(
p
1
2
B
−
2
∂
a
(
p
1
,p
2
)
∂
p
1
b
(
p
1
2
)
−
∂
b
(
p
1
,p
2
)
∂
p
1
b
(
p
1
2
)
(
w
A
+
w
B
)
,
from which the result is clear.
2. Using the de
f
nition of the pro
f
t function
π
(
p,w
1
2
Max
z
1
≥
0
,z
2
≥
0
p
(
z
1
+
z
2
)
1
/
2
−
w
1
z
1
−
w
2
z
2
This yields
y
(
1
2
)
≡
(
z
1
+
z
2
)
1
/
2
z
1
(
1
2
)
z
2
(
1
2
)
π
(
1
2
)
Range
p/
(2
w
1
)(
p/
(2
w
1
))
2
0
p
2
/
(4
w
1
)
w
1
≤
w
2
p/
(2
w
2
)0
(
p/
(2
w
2
))
2
p
2
/
(4
w
2
)
w
1
>w
2
Note that the standard properties hold here.
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401
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Output supply and input demnds are homogeneous of degree
0
in prices
The pro
f
t function is homogeneous of degree
1
in prices
And
∂
y
∂
p
≥
0
,
∂
z
1
∂
w
1
≤
0
,
∂
z
2
∂
w
2
≤
0
3. Write shortrun total cost as
C
S
(
q
)
and longrun total cost as
C
L
(
q
)
.
In
C
S
(
q
)
z
2
is
f
xed at
z
∗
2
and so
C
S
(
q
)
≥
C
L
(
q
)
and we know the two are equal at
q
∗
.
So
q
∗
is
a minimizer of
g
(
q
)
≡
C
S
(
q
)
−
C
L
(
q
)
.
Thus
g
±
(
q
∗
)=
C
±
S
(
q
∗
)
−
C
±
L
(
q
∗
)=0
g
±±
(
q
∗
C
±±
S
(
q
∗
)
−
C
±±
L
(
q
∗
)
>
0
.
So at
q
∗
the shortrun and longrun marginal costs are equal to each other and
the shortrun MC is steeper than the longrun MC. Given
p
±
>p
,
then, the
q
at
p
±
=
C
±
L
(
q
)
will exceed the
q
at
p
±
=
C
±
S
(
q
)
.
Se the picture on the last page.
4. (a) A pro
f
t function is the value function for a pricetaking competitive
f
rm,
that is,
π
(
p,w
1
,w
2
Max
z
1
,z
2
pf
(
z
1
2
)
−
w
1
z
1
−
w
2
z
2
,
where
w
i
are input prices,
p
is output price and
f
(
z
1
2
)
is the production function.
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 Fall '08
 Burbidge,John
 Economics, Utility, p1, π

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