Student Number:
Name:
Fall 2007
QUIZ 5
Section B
Question 1 [5marks] Consider a duopoly. Firm A's cost function is C(Qa)= 10Qa. Firm B's cost
function is C(Qb)= 10Qb. They both face the demand relationship: P=100-Q
1) [3 marks] How much does each firm produce is Firm B acts as Stackelberg Leader and Firm A
acts as the follower?
Firm A, the follower’s profit maximizing condition given Qb (the output of firm B) is:
MR = MC implies 100 –Qa–Qb –Qa = 10
This gives: Qa = (100 –Qb – 10)/2 = 45 – (0.5) Qb
Keeping this in mind, Firm B has total revenue = (100 –Qb – [45- 0.5qb]) Qb
Thus MR = 100 –Qb– [45- 0.5 Qb] – qb/2 = 55–Qb
MR = MC implies 55 –Qb = 10 implies Qb = 45
This is the output for firm B when it is the leader. Firm A then produces Qa = 22.5
2) [2 marks] How much profit does each firm earn when they produce the quantities as part 1).
Firm A’s Profit
=P*Q-C(Q)=(100-Qtotal)*Qa-10*Qa=(100-45-22.5)*22.5-10*22.5=506.25
Firm B’s Profit
= P*Q-C(Q)=(100-Qtotal)*Qb-10*Qb=(100-45-22.5)*45-10*45=1012.5
Question 2 [5 marks] A homogeneous products duopoly faces a market demand function given by
P=500-10Q. Firm 1 has a constant marginal cost MC=50. Firm 2 has a constant marginal cost
MC=60.
1)
[3 marks] What are the Cournot equilibrium quantities and industry price?
Find firm 2 reaction function by setting MR=MC
(500-10Q1)-20Q2=60
Q2=22-0.5Q1
Find firm 1 reaction function by setting MR=MC
(500-10Q2)-20Q1=50
Q1=22.5-0.5Q2
Solving the two reaction functions, we get
Q1=15.33
Q2=14.33
The industry price is P=500-153.33-143.3=203.4
2)
[2 marks] Calculate the Cournot equilibrium profit for each firm.
Firm 1:
Profit= (P-MC)*Q=(203.4-50)*15.33=153.4*15.33=2351.62
Firm 2:
Profit= (P-MC)*Q=(203.4-60)*15.33=143.4*14.33=2054.92