Question

Consider a market in which the market demand curve is given by P = 18 - X - Y, where X is Firm 1’s output, and Y is Firm 2’s output. Firm 1 has a marginal cost of 3, while Firm 2 has a marginal cost of 6. a) (2 marks) Find the Cournot equilibrium outputs in this market. How much profit does each firm make? b) (2 marks) Find the Stackelberg equilibrium in which Firm 1 acts as the leader. How much profit does each firm make?

Answer 1

a) Profit 1 = (P 1 - 3)*6 = 54

Profit 2 = (P 2 -6)*3 = 9

b) Profit 1 = (P 1 - 3)*6 = 36

Profit 2 = (P 2 -6)*3 = 9

Step-by-step explanation:

a)

Cournot equilibrium :

for firm 1 :

P 1 = 18 - X - Y

R 1 = (18-Y)X - X^2

MR 1 = (18-Y) - 2 X = MC = 3

X= (15-Y)/2 .........(1)

for firm 2 :

P 2 = 18 - X - Y

R 2 = (18-X)Y - Y^2

MR 2 = (18-X) -2 Y = MC = 6

Y= (12-X)/2 ...........(2)

using this value of Y in equation (1),

X = (15 -(12-X)/2)/2

X = (18+X)/4

3 X / 4 = 18 / 4

X = 6

Y = 3

P 1 = P 2 = 9

Profit 1 = (P 1 - 3)*6 = 54 (ANSWER)

Profit 2 = (P 2 -6)*3 = 9 ((ANSWER)

b)

for firm 1

X = 18-Y -P 1

R 1 = (18-Y)P 1 - P 1^2

MR 1 = (18-Y) - 2P 1 = 3

P 1 = (15-Y)/2

X = (18-Y - (15-Y)/2) = (21-Y)/2 ..........(3)

for firm 2,

Y = 18-X - P 2

R 2 = (18-X)P 2 -P 2^2

MR 2 =(18-X) -2P 2 = 6

P 2 = (12-X)/2

Y = (18-X - (12-X)/2) = (24 - X)/2

Using it in (3)

X = (21 - (24-X)/2))/2 = (18 +X)/4

X = 6, Y = 3 , P 1 = P 2 = 9

Profit 1 = (P 1 - 3)*6 = 36 (ANSWER)

Profit 2 = (P 2 -6)*3 = 9 ((ANSWER)