Consider a Cournot duopoly with the inverse demand P = 260 - 2Q . Firms 1 and 2

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 l 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

P = 260- 2Q

Q = Q1 + Q2

Profit function for firm 1.

Profit1 = PQ1 - AC1*Q1 = (260 - 2(Q1+Q2)*Q1 - 20Q

dProfit1/dQ1 = 260 - 4Q1 - 2Q2 - 20

Putting dProfit/dQ1 = 0

Q1 = 60 - Q2/2 this is the reaction function for firm 1

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

If you don't understand anything then comment, I'ill revert back on the same.:)

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