Question

Assume there are two firms in the bean sprouts industry and they play Cournot. The inverse market demand curve is given by p(y) = 100−2y_T, where y_T is the total output of all the firms. The total cost function for each firm in the industry is given by c(y) = 4y.

i . Find the marginal revenue and marginal cost equations for the firms.

ii. Determine the best response functions for each firm.

iii. Calculate the Nash equilibrium levels of output for each firm. What is the equilibrium price?

iv. Draw the best response functions in a graph and then show the Nash equilibrium from (ii).

v. Calculate the profits of the firms.

Answer 1

p = 100 - 2yT = 100 - 2y1 - 2y2 (since yT = y1 + y2)

(i)

For firm 1,

Total revenue (TR1) = p x y1 = 100y1 - 2y1² - 2y1y2

Marginal revenue (MR1) = TR1/y1 = 100 - 4y1 - 2y2

TC1 = 4y1, so MC1 = dc1/dy1 = 4

For firm 2,

Total revenue (TR2) = p x y2 = 100y2 - 2y1y2 - 2y2²

Marginal revenue (MR2) = TR2/y2 = 100 - 2y1 - 4y2

TC2 = 4y2, so MC2 = dc2/dy2 = 4

(ii)

Equating MR1 and MC1,

100 - 4y1 - 2y2 = 4

4y1 + 2y2 = 96

2y1 + y2 = 48..........(1) (Best response, firm 1)

Equating MR2 and MC2,

100 - 2y1 - 4y2 = 4

2y1 + 4y2 = 96........(2) (Best response, firm 2)

(iii)

Nash equilibrium is obtained by solving (1) and (2). Subtracting (1) from (2),

3y2 = 48

y2 = 16

y1 = (48 - y2) / 2 [From (1) = (48 - 16) / 2 = 32 / 2 = 16

p = 100 - (2 x 16) - (2 x 16) = 100 - 32 - 32 = 36

(iv)

From (1), when y1 = 0, y2 = 48 (Vertical intercept) and when y2 = 0, y1 = 48/2 = 24 (Horizontal intercept).

From (2), when y1 = 0, y2 = 96/4 = 24 (Vertical intercept) and when y2 = 0, y1 = 96/2 = 48 (Horizontal intercept).

In following graph, R1 and R2 are reaction functions of firm 1 and 2 respectively.

(v)

Profit, firm 1 = y1 x (p - MC1) = 16 x (36 - 4) = 16 x 32 = 512

Profit, firm 2 = y2 x (p - MC2) = 16 x (36 - 4) = 16 x 32 = 512