Question

Q2. Consider a Bertrand game with differentiated products in which two firms simul- taneously choose prices. The marginal cost for each firm is zero and there are no fixed costs. The demand functions for each firm are:

Q1 = 80 − 2P1 + 2P2,

Q2 = 80 − 2P2 + 2P1.

where P1 is the price set by firm 1, P2 is the price set by firm 2, Q1 is the quantity demanded of firm 1’s product and Q2 is the quantity demanded of firm 2’s product.

a. What are the best response functions for each firm?

b. What is the Nash equilibrium of this game?

c. What is the equilibrium profit for each firm?

(Remember to show all working)

Answer 1

Q1 = 80 − 2P1 + 2P2

Q2 = 80 − 2P2 + 2P1

MC = 0

A)

Both the firms will maximise profit, given their price choices,

Profit for Firm 1 = [Hence profit equals Total revenue as there are no costs]

Profit for Firm 1 =

P1*(80-2P1+2P2)

P2*(80-2P2+2P1)

Differentiating both the profit function with respect to P1 and P2, we get,

Setting both them equal to zero, we get,

80-4P1+2P2 = 0, P1 = 80+2P2 = 4P1

P1 = 20-0.5P2 (This is the best response function of Firm 1)

Similarly for firm 2 we get,

80-4P2+2P1 = 0, P2 = 80+2P1 = 4P2

P2 = 20-0.5P1 ((This is the best response function of Firm 1)

B)

For Nash Equilibrium, we put the value of P2 in the best response function of Firm 1, Hence

P1 = 20-0.5P2

P1 = 20-0.5(20-0.5P1)

P1 = 20-10+0.25P1

0.25P1 = 10

P1* = 40

Similary for Firm 2 we get P2* = 40

C)

Put the value of P1 and P2 in the demand function we get,

Q1* = 80-2*40+2*40 = 80

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

Profit for Firm 1 =

= 40*80 = 3200

Profit for Firm 2 = 40*80 = 3200

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