Consider a Bertrand game with differentiated products in which two firms simulta

Question

Consider a Bertrand game with differentiated products in which two firms simultaneously choose prices. The marginal cost for each firm is zero and there are no fixed costs. The demand functions for each firm are: where P_1 is the price set by firm 1, P_2 is the price set by firm 2, Q_1 is the quantity demanded of firm l's product and Q_2 is the quantity demanded of firm 2's product. What are the best response functions for each firm? (4 marks) What is the Nash equilibrium of this game? (4 marks) What is the equilibrium profit for each firm? (2 marks) (Remember to show all working)

Explanation / Answer

Q1 = 80 - 2P1 + 2P2

MC = 0

Profit of firm1 S1 = P1Q = P1*(80 - 2P1 + 2P2)

differentiating wrt P1

dS1/dP1 = 80 - 4P1 + 2P2

Putting dS1/dP1 = 0

80 - 4P1 + 2P2 = 0

P1 = 20 - P2/2

P1 = 20 - P2/2 is the reaction function of Firm 1

Q2 = 80 - 2P2 + 2P1

MC = 0

Profit of firm2 S2 = P1Q = P2*(80 - 2P2 + 2P1)

differentiating wrt P2

dS2/dP2 = 80 - 4P2 + 2P1

Putting dS2/dP2 = 0

80 - 4P2 + 2P1 = 0

P2 = 20 - P1/2

P2 = 20 - P1/2 is the reaction function of Firm 2.

b.

  Solving the two reaction function to find the Nash equilibrium

  P1 = 20 - P2/2

   P2 = 20 - P1/2

   P1 = 20 - [20 - P1/2]/2

   P1 = 10 + P1/4

   P1 = 40/3

P2 = 20 - P1/2 = 20 - 40/6 = 40/3

  Q1 = 80 - 2*40/3 + 2*40/3 = 80

Q2 = 80 - 2*40/3 + 2*40/3 = 80

So Nash equilibrium is both firm produce Q = 80 at P = 40/3

Profit of firm 1 = P1*Q1 = 40/3*80 = 1,066.66

Profit of firm 2 = P2*Q2 = 40/3*80 = 1,066.66

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