Consider a Cournot duopoly with the inverse demand P = 260 - 2Q. Firms 1 and 2 c
ID: 1221938 • Letter: C
Question
Consider a Cournot duopoly with the inverse demand P = 260 - 2Q. Firms 1 and 2 compete by simultaneously choosing their quantities. Both firms have constant ma rgina 1 and average cost M C = AC = 20. a. Find each firm's best response function, b. Find the Cournot - Nash equilibrium quantities, profits and market pr ice. c. Plot the best response curves and illustrate the equilibrium point, d. Suppose, instead, that firm 1 chooses first and 2 follows. Find the Stackelberg equilibrium quantities, profits and market price.Explanation / Answer
Similarly by symmetry
Reaction function for firm 2
Q2 = 60 - Q1/2
Draw these 2 reaction equation on graph and show the point of intersection as equilibrium.
Soving the two reaction equation by putting 2 reaction equation into 1.
Q1 = 60 - [60 - Q1/2]2
Q1 = 60 - 30 + Q1/4
3/4Q1 = 30
Q1 = 30*4/3 = 40
Q2 = 60 - Q1/2 = 60 - 40/2 = 40
So, Q1 = Q2 = 40, is the nash equilibrium
Quantity Q = Q1 + Q2 = 40 + 40 = 80
Price P = 260 - 2*80 = 100
Profit of firm = peofit of firm 2 = 100*80 - 20*80 = 6400
. For Stackelberg Equilibrium
Firm 1 is the leader
As Firm 2 is follower, so Reaction function for firm 2, Q2 = 60 - Q1/2 (same as Cournot model)
Profit function for firm 1
Profit1 = PQ1 - AC1*Q1 = (260 - 2(Q1+Q2)*Q1 - 20Q1
Putting reaction function of firm 2 into this
Profit1 = PQ1 - AC1*Q1 = (260 - 2((60 - Q1/2) + Q1)*Q1 - 20Q1
Profit1 = (140 - Q1)*Q1 - 20Q1
dprofit/dQ1 = 140 - 2Q1
Putting dProfit/dQ1 = 0
Q1 = 140/2 = 70
Q2 = 120 -70/2 = 25
So, Q1 = 70 , Q2 = 25 is a stackelberg equilibrium.
Q Q1 + Q2 = 70 + 25 = 95
Price P = 260 - 2*95 = 70
Profit of firm 1 = 75*70 - 20*70 = 3850
Profit of firm 2 = 75*25 - 20*25 = 1250
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