Question

Assume that you observe two firms operating in a Bertrand oligopoly. The inverse demand function for the market is P = 200 – 2Q and each firm has the same cost function of C(Q) = 20Q. What is the level of production for each firm, market price, and profit of each firm? What would happen if both firms merge to form a single monopoly with a cost function of C(Q) = 20Q?

Answer 1

(a) Bertrand Competition

MC = dTC(Q)/dQ = 20

P = 200 - 2Q

2Q = 200 - P

Q = 100 - 0.5P = 100 - 0.5P1 - 0.5P2

For Firm 1,

Total revenue (TR1) = P1 x Q = 100P1 - 0.5P1² - 0.5P1P2

Marginal revenue (MR1) = TR1/P1 = 100 - P1 - 0.5P2

Equating MR1 and MC,

100 - P1 - 0.5P2 = 20

P1 + 0.5P2 = 80.........(1) (Best response, firm 1)

For Firm 2,

Total revenue (TR2) = P2 x Q = 100P2 - 0.5P1P2 - 0.5P2²

Marginal revenue (MR2) = TR2/P2 = 100 - 0.5P1 - P2

Equating MR2 and MC,

100 - 0.5P1 - P2 = 20

0.5P1 + P2 = 80.........(2) (Best response, firm 2)

Bertrand equilibrium is obtained by solving (1) and (2). Multiplying (2) by 2,

P1 + 2P2 = 160.... ..(3)

P1 + 0.5P2 = 80.........(1)

(3) - (1) yields: 1.5P2 = 80

P2 = 53.33

P1 = 160 - 2P2 [From (3)] = 160 - (2 x 53.33) = 160 - 106.67 = 53.33

Q = 100 - 0.5 x (53.33 + 53.33) = 100 - 0.5 x 106.67 = 100 - 53.33 = 46.67

Profit, firm 1 = Q x (P1 - MC) = 46.67 x (53.33 - 20) = 46.67 x 33.33 = 1,555.4

Profit, firm 2 = Q x (P2 - MC) = 46.67 x (53.33 - 20) = 46.67 x 33.33 = 1,555.4

(b) Monopoly equilibrium is obtained by equating MR and MC.

P = 200 - 2Q

TR = P x Q = 200Q - 2Q2

MR = dTR/dQ = 200 - 4Q

Equating with MC,

200 - 4Q = 20

4Q = 180

Q = 45 (Quantity will fall by (46.67 - 45) = 1.67)

P = 200 - (2 x 45) = 200 - 90 = 110

Profit = Q x (P - MC) = 45 x (110 - 20) = 45 x 90 = 4,050 (Profit will rise by (4,050 - 1,555.4 - 1,555.4) = 939.2)