Question

Consider two identical firms (no. 1 and no. 2) that face a linear market demand curve. Each firm has a marginal cost of zero and the two firms together face demand:

P = 50 - 0.5Q, where Q = Q1 + Q2.

a. Find the Cournot equilibrium Q and P for each firm. Calculate the results rounded to the second digit after the decimal point

b. Find the equilibrium Q and P for each firm assuming that the firms collude and share the profit equally.

Answer 1

(a)

P = 50 - 0.5Q1 - 0.5Q2

For firm 1,

TR1 = P x Q1 = 50Q1 - 0.5Q12 - 0.5Q1Q2

MR1 = TR1/Q1 = 50 - Q1 - 0.5Q2

Setting MR1 = MC,

50 - Q1 - 0.5Q2 = 0

Q1 + 0.5Q2 = 50.............(1) [reaction function, firm 1]

For firm 2,

TR2 = P x Q2 = 50Q2 - 0.5Q1Q2 - 0.5Q22

MR2 = TR2/Q2 = 50 - 0.5Q1 - Q2

Setting MR2 = MC,

50 - 0.5Q1 - Q2 = 0

0.5Q1 + Q2 = 50.............(2) [reaction function, firm 2]

(2) x 2 yields:

Q1 + 2Q2 = 100...........(3)

Q1 + 0.5Q2 = 50.............(1)

(3) - (1) yields:

1.5Q2 = 50

Q2 = 33.33

Q1 = 100 - 2Q2 [from (3)] = 100 - (2 x 33.33) = 100 - 66.66 = 33.33

Q = 33.33 + 33.33 = 66.66

P = 50 - (0.5 x 66.66) = 50 - 33.33 = 16.67

(b)

With collusion, firms will act like a monopoly.

TR = P x Q = 50Q - 0.5Q2

MR = dTR/dQ = 50 - Q

Setting MR = MC,

50 - Q = 0

Q = 50

Q1 = Q/2 = 25

Q2 = Q/2 = 25

P = 50 - (0.5 x 50) = 50 - 25 = 25