3. A monopolist with total cost funtionc10+^(2 faces a market with two distinct
ID: 1166350 • Letter: 3
Question
3. A monopolist with total cost funtionc10+^(2 faces a market with two distinct consumer groups with the first group having a demand function of q1(P) 80-5P1 and the second having a demand function of q2(P)-40 2P2 a) Is this monopolist operating in the short run or the long run? Explain how you know b) Calculate the monopolist's profit-maximizing price and quantity sold if the monopolist must engage in uniform pricing. Find the monopolist's profits in this scenario. (Hint: use horizontal aggregation to form the total market demand curve) c) The monopolist can now engage in third degree price discrimination, setting a different price for each market segment. Calculate the monopolist's optimal prices charged to each market segment, the quantities consumed by each market segment, and the monopolist's profits. Would the monopolist rather engage in this price discrimination or would they rather seta uniform price?Explanation / Answer
3.(a) Q(P)= q1(P)+ q2(P)
Q(P)=120-7P
P(Q)=120/7 - Q/7
TR= P(Q)(Q)=120Q/7 – Q2/7
MR=120/7 – 2Q/7
Differentiate MR with respect to quantity we get
d MR/ d Q =(-) 2/7 = (-) 0.28 this is slope of MR curve
C(Q) = 10+Q2/2
MC= Q
Differentiate MC with respect to quantity we get
d MC/ d Q = 1 this is slope of MC curve
Since,
slope of MC curve> slope of MR curve
the monopolist is operating in short run.
(b)For uniform pricing we add demand curves q1(P) and q2(P)
q1(P)=80-5P1
q2(P)=40-2P2
Q(P)= q1(P)+ q2(P)
Q(P)=120-7P [P1=P2]
We express it in terms of quantity
P(Q)=120/7 - Q/7
Condition for profit maximization is given by: MR=MC
Profit = TR-TC
=P(Q).Q-C(Q)
=(120/7 –Q/7)Q– (10+Q2/2)
=120.Q /7 – Q2/7 – 10 –Q2/2
Now we differentiate profit with respect to quantity and equate it to 0
We get
120/7 – 2Q/7 – Q = 0
Q*=40/3
P*=120/7 – 40/21
P*=15.23
Profit = P(Q).Q-C(Q)
=(15.23)(40/3) – 10-(40/3)2/2
Profit =104.17
(c)Market 1
MR1=MC
q1(P)=80-5P1
P1=16 - q1/5
C(q1)=10+ q12/2
MC= q1
MR1=d TR1 / d q1
TR1=P1(q1). q1 = (16 - q1/5)(q1)
=16 q1 –q12/5
MR1 =16-2q1/5
MR1=MC
16-2q1/5= q1
q1=80/7 = 11.42
p1=16-16/7 = 96/7 = 13.71
Market 2
q2(P)=40-2P2
p2=20- q2/2
C(q2)=10+ q22/2
MC= q2
MR2= d TR2 / d q2
TR2=P2(q2). q2 =20 q2 - q22/2
MR2=20- q2
MR2=MC
20- q2= q2
10=q2
P2=15
Total profit = p1 q1 + p2 q2 –(10+ Q/2) [Q=q1+q2]
=(11.42)( 13.71) + (10)(15) – 10 – [11.42+10]/2
=166.02
Monopolist would rather engage in price discrimination because profit from price discrimination is more than that from uniform pricing.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.