Question

1. Consider the following asymmetric version of the Cournot duopoly model. Two firms compete by simultaneously choosing the quantities (q, and q2) they produce. Their products are homogeneous, and market demand is given by p- 260-2Q, where Q-q +q2. Firm 1 has a cost advantage; Firm 1 produces at zero cost, while Firm 2 produces at a constant average cost of 40. (The difference in costs is what makes this an asymmetric game.) a. Derive both firms' profit functions, as functions of only q, and q2. b. Compute the Nash equilibrium. c. Compute the corresponding market price and both firms' profits.

Answer 1

p = 260 - 2q1 - 2q2

MC1 = AC1 = 0

MC2 = AC2 = 40

(a)

For firm 1,

Total revenue (TR1) = p x q1 = 260q1 - 2q1² - 2q1q2

Total cost (TC1) = AC1 x q1 = 0

Profit (Z1) = TR1 - TC1 = 260q1 - 2q1² - 2q1q2

For firm 2,

Total revenue (TR2) = p x q2 = 260q2 - 2q1q2 - 2q2²

Total cost (TC2) = AC2 x q2 = 40q2

Profit (Z2) = TR2 - TC2 = 260q2 - 2q1q2 - 2q2² - 40q2 = 220q2 - 2q1q2 - 2q2²

(b)

For firm 1, profit is maximized when $\partial$ Z1/$\partial$q1 = 0

260 - 4q1 - 2q2 = 0

4q1 + 2q2 = 260

2q1 + q2 = 130.........(1)

For firm 2, profit is maximized when $\partial$ Z2/$\partial$q2 = 0

220 - 2q1 - 4q2 = 0

2q1 + 4q2 = 220.......(2)

Subtracting (1) from (2),

3q2 = 90

q2 = 30

q1 = (130 - q2) / 2 [From (1)] = (130 - 30) / 2 = 100 / 2 = 50

(c)

Q = 50 + 30 = 80

p = 260 - (2 x 80) = 260 - 160 = 100

Profit, firm 1 = q1 x (p - AC1) = 50 x (100 - 0) = 50 x 100 = 5,000

Profit, firm 2 = q2 x (p - AC2) = 30 x (100 - 40) = 30 x 60 = 1,800