Question

What is the? homogeneous-good duopoly Cournot equilibrium if the market demand function is

Q=1,600?400?p,

and each? firm's marginal cost is

?$0.28 per? unit?

The? Cournot-Nash equilibrium occurs where

q 1=

and

q 2=

?(Enter numeric responses using real numbers rounded to two decimal? places.)

Answer 1

Q = 1,600 - 400p

400p = 1,600 - Q

p = 4 - 0.0025Q

p = 4 - 0.0025q1 - 0.005q2 [Since Q = q1 + q2]

MC1 = MC2 = 0.25

For firm 1,

Total revenue (TR1) = p x q1 = 4q1 - 0.0025q1² - 0.0025q1q2

MR1 = TR1/q1 = 4 - 0.005q1 - 0.0025q2

Equating MR1 and MC1,

4 - 0.005q1 - 0.0025q2 = 0.28

0.005q1 + 0.0025q2 = 3.72.......(1) [Best response, firm 1]

For firm 2,

Total revenue (TR2) = p x q2 = 4q2 - 0.0025q1q2 - 0.0025q2²

MR2 = TR2/q2 = 4 - 0.0025q1 - 0.005q2

Equating MR2 and MC2,

4 - 0.0025q1 - 0.005q2 = 0.28

0.0025q1 + 0.005q2 = 3.72.......(2) [Best response, firm 2]

Cournot equilibrium is obtained by solving (1) and (2).

0.005q1 + 0.0025q2 = 3.72.......(1)

(2) x 2 yields:

0.005q1 + 0.01q2 = 7.44...........(3)

(3) - (1) yields:

0.0075q2 = 3.72

q2 = 496.00

q1 = (7.44 - 0.005q1) / 0.01 [From (2)] = [7.44 - (0.005 x 496)] / 0.01 = (7.44 - 2.48) / 0.01 = 4.96 / 0.01 = 496.00