Suppose the market demand function is given by: Q=100-2P, where Q: total quantit

Question

Suppose the market demand function is given by: Q=100-2P, where Q:  total quantity, P: market price. And in this market there are two firms with MC=AV= $10. Find each of the following:   

a. Perfect competition price, quantity, and consumer surplus?   

b. Monopoly price, quantity, consumer surplus, profit, and welfare loss?    

c. Cournot  price, quantity, consumer surplus , each firm`s profit, and welfare  loss?   

d. Stackelberg price, quantity, consumer surplus, each firm`s profit, and welfare loss?  

f. Collusion quantity, profit  from collusion?

Explanation / Answer

Q=100-2P
MC=AV= $10
a)The competitive market equilibrium price should satisfy P=MC, so P=$10

Q=100-2(10)
=100-20
=80
The Consumer Surplus in perfect competition case is
= 0.5(100-10)*80
=3600

b)MR=MC
MC=10
Q=100-2P
P=(100-Q)/2
TR=(100-Q)/2*Q= (100Q-Q2)/2
MR=(100-2Q)/2
(100-2Q)/2=10
100-2Q=20
2Q=100+20
2Q=120
Q=60
P=20
CS=0.5(100-20)*60=2400
PS=(20-10)/60=0.1666

DEADWEIGHTLOSS=(3600+0)-(2400+0.166)=1199.834

c) TR=(100-Q)/2*Q= (100Q-Q2)/2
MR=(100-2Q)/2

Profit=(100Q-Q2)/2-10Q

=50Q-10Q-0.5Q2

=40Q-0.5Q2

Q=qa+qb
P=(100-Q)/2
=50-0.5(qa+qb)
TR(A)=Pqa
=(50-0.5qa-0.5qb)qa
=50qa-0.5qa2-0.5qaqb
MR(A)=50-qa-0.5qb=MC
50-qa-0.5qb=10
qa=40-0.5qb
This is the reaction function of A

Similarly for B
TR(B)=Pqb
=(50-0.5qa-0.5qb)qb
=50qb-0.5qaqb-0.5qb2
MR(B)=50-qb-0.5qa=MC
50-qb-0.5qa=10
qb=40-0.5qa
Substitutings B's reaction fucntion in reaction function of A we have
qa=40-0.5(40-0.5qa)
qa=40-20+0.25qa
qa-0.25qa=20
0.75qa=20
qa=20/0.75
qa=26.66
qb=40-0.5(26.66)=26.67

P=(100-Q)/2
=(100-53.34)/2
=23.33

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