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Suppose demand is given by Q = 60 - P
Suppose demand is given by Q = 60 - P. Also, assume that there are no production costs.
a. Find the quantity and price chosen in the case of a four-firm oligopoly.
b. Now find the quantity and price chosen in the case of an n firm oligopoly.
c. What happens to total quantity and price as n gets arbitrarily large.
Expert Solution
a) In the case of 4 firms
q= 4Q
q=4(60-P)
q=240-4P
4P=240-q
P=60-0.25q
TR = Pq
TR=(60-0.25q)q
TR =60q-0.25q2
MR =60-0.5q
At equilibrium
MR = MC
60-0.5q = 0
q = 120
P =60-0.25q
P =60-0.25(120)
P =30
b) In the case of n firms
q =nQ
q=n(60-P)
q=60n-Pn
Pn=60n-q
P=60-q/n
TR=Pq
TR=(60-q/n)q
TR=60q-q2/n
MR=60-2q/n
At equilibrium
MR = MC
60-2q/n = 0
60=2q/n
60n = 2q
30n =q
P=60-30n/n
P =60-30
P =30
c) In the case of n+1 firms
q =(n+1)Q
q=(n+1)(60-P)
q=60n-Pn+60-P
Pn=60n-q+60-P
P=(60n/n-1)-(q/n-1)+(60/n-1)
TR=Pq
TR=((60n/n-1)-(q/n-1)+(60/n-1))q
TR=(60n/n-1)q-(q/n-1)q+(60/n-1)q
MR=(60n/n-1)-(2q/n-1)+(60/n-1)
At equilibrium
MR = MC
(60n/n-1)-(2q/n-1)+(60/n-1) = 0
(60n/n-1) +(60/n-1)= (2q/n-1)
q = 30n + 20
P =(60n/n-1)-(30n+20/n-1)+(60/n-1)
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