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Answer Key: Problem Set 3
February 8, 2005
Problem 1
Consider a market with inverse demand
P
=11
−
Q
,whe
re
Q
is the sum of
output produced by all
f
rms in the market. Firms have identical cost functions
c
(
q
)=2+
q
.
Question a
Suppose there is a single
f
rm in this market. What price and quantity will
the
f
rm choose? What will be the consumer surplus? What will be the
f
rm’s
pro
f
t?
ANSWER:
The
f
rm solves
max
q
{
(11
−
q
)
q
−
(2 +
q
)
}
It’s easy to see the f.o.c. is
11
−
2
q
−
1=0
which gives
q
m
=5
,p
m
=6
π
m
=(
1
0
−
5)5
−
2=23
CS
m
·
5
·
1
2
=12
.
5
Question b
Suppose there are
n
f
rms in the market that compete by simultaneously choos
ing quantities. What quantities,
q
c
(
n
)
, will they choose in equilibrium? How do
these quantities change with the number of
f
rms? What is the total quantity
Q
c
(
n
)
produced in equilibrium and how does it change with the number of
f
rms
in the market? What is the pro
f
t
π
c
(
n
)
and how does it change with
n
?Wha
t
is the consumer surplus and how does it change with
n
?
1
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View Full DocumentANSWER:
Firm
i
solves
max
q
i
{
(11
−
q
i
−
Q
−
i
)
q
i
−
(2 +
q
i
)
}
The f.o.c.(w.r.t.
q
i
)is
11
−
2
q
i
−
Q
−
i
−
1=0
Symmetry implies
Q
−
i
=(
n
−
1)
q
i
,which reduces the f.o.c. to
10
−
(
n
+1)
q
i
=
0
. (Note that if you make this substitution
before
deriving the FOC, you will
not
get the right soultion; remember that
f
rm
i
does not get to choose the
quantities of its competitors!) Therefore:
q
c
i
(
n
)=
10
n
+1
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 Spring '08
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