Intermediate Microeconomics, Winter 2008
Problem Set No 15 Solutions
Q1.
Suppose a monopolist faces a demand function
D
(
p
) = 100

Ap
and
suppose costs are
c
. If
A
= 5 and
c
= 2, ﬁnd the optimal quantity and the
optimal price that the monopolist will set. Find the optimal price and the
optimal quantity as a function of general
A
and
c
.
In order to write proﬁt as a function of quantity only, we need to ﬁnd the
inverse demand function faced by the monopolist. The demand function is
q
= 100

Ap
, so, solving for
p
, the inverse demand function is
p
(
q
) =
100
A

q
A
.
The monopolist’s problem is to choose
q
to maximize
π
(
q
).
max
q
π
(
q
)
max
q
q
*
p
(
q
)

C
(
q
)
max
q
q
*
(
100
A

q
A
)

cq
We can conﬁrm that proﬁt is a concave function of
q
, and therefore the maximum
is attained where:
π
0
(
q
) = 0
100
A

2
q
A

c
= 0
2
q
A
=
100
A

c
2
q
= 100

Ac
q
= 50

Ac
2
Plugging in
A
= 5 and
c
= 2, we get
q
= 45.
Q2.
Suppose the demand function for AA and UA for ﬂights between
Chicago and Detroit is
D
(
p
) = 100

Ap
. If AA and UA each choose to supply
a quantity of 30, what will be the price? If UA chooses a supply of
q
UA
=30,
what will be the price as a function of the quantity that AA supplies? Given the
choice
q
UA
= 30, and given constant marginal costs c = 1, what is the optimal
supply of AA? (You might want to choose
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 Winter '08
 Burbidge,John
 Microeconomics, Game Theory, Supply And Demand, demand function, strictly dominated strategies, QAA

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