Question

2 firms are engaged in Cournot competition; firm Z faces the cost curve CA (yZ) = 40yZ and firm W
faces the cost curve CW (yW) = 40yW. The inverse market demand curve is P(y) = 100 y, where y
represents the market level of output.

a) Define the Cournot game.

b) Explain why a firm has no incentive to deviate from the Cournot Nash equilibrium (holding their opponent’s strategy constant)

c) Find the Cournot Nash Equilibrium

d) Now suppose instead of playing their strategies at the same time, firm Z moves first and then firm W moves second (sequential game). Does firm Z earn higher profits in this game or the game in part c)?

Answer 1

a)

Cournot Game is the simultaneous move game. Under this game, each firm has a cost function and a inverse demand function. So, all firms face common aggregate demand. Every firm in the game chooses the output level based on what output its rival firm chooses. The best response of each firm gives the Nash Equilibirum.

b)

In Cournot Game, a firm has no incentive to deviate from the Nash Equilibrium position because at this equilibrium each firm plays their best response strategy. Deviation from such point would result in lower profits.

c)

p = 100y = 100(yZ + yW)

P = 100yZ + 100yW

MCW = 40 = MCZ

Profit of firm Z:

Z = (P - MC)*yZ = (100yZ + 100yW - 40)*yZ

Z = 100yZ^2 + (100yW - 40)yZ

FOC: dZ/ dyZ = 0

200yZ + (100yW - 40) = 0

200yZ = 40 - 100yW

yZ = 0.2 - 0.5yW ----Best Response of firm Z

Similarly, yW = 0.2 - 0.5yZ ----Best Response of Firm W

Solving both best responses,

yZ = 0.2 - 0.5(0.2 - 0.5yZ)

yZ = 0.2 - 0.1 + 0.25yZ

0.75yZ = 0.1

yZ* = 0.13

yW = 0.2 - (0.5*0.13)

yW* = 0.13

Cournot Equilibrium: (0.13, 0.13)

Aggregate Output, y = 0.13+0.13 = 0.26

P = 100*.26 = 26

Profits: Z = (26 - 40)*0.13 = -1.82 = W

d)

Stackelberg Game.

Max profit of firm Z wrt yW = 0.2 - 0.5yZ

Z = (P - MC)*yZ = (100yZ + 100yW - 40)*yZ

Z = [100yZ + 100(0.2 - 0.5yZ) - 40]*yZ

Z = [100yZ + 20 - 50yZ - 40]*yZ

Z = 50yZ^2 - 20yZ

FOC: dZ/ dyZ = 0

100yZ = 20

yZ = 0.2

yW = 0.2 - (0.5*0.2)

yW = 0.1

Stackelberg Equilibrium: (0.2, 0.1)

Aggregate Output, y = 0.2+0.1 = 0.3

P = 100*0.3 = 30

Profit of Firm Z = (30-40)*0.2

= - 2

Profit in Cournot = -1.82

Profit in Stackelberg = - 2

Profit in Cournot > Profit in Stackelberg

So, the firm Z earns more profit in game in part c)