Question

To examine the​ trade-off between market efficiency and market power from a​ merger, consider a market with two firms that sell identical products. Firm 1 has a constant marginal cost of ​$​, and Firm 2 has a constant marginal cost of ​$2. The market demand is

Qp= 105-p.

Note that

dπ1/∂q1= [105-1(q1+q2)]- 1q1-1=0

dπ2/∂q2= [105-1(q1+q2)]- 1q1-2=0

The​ Cournot-Nash equilibrium occurs where q1 = 15.00 and q= 12.00. ​ Market output is 69. Futhermore , the equilibrium occurs at a price of $______--

Answer 1

`q_1 = 35` units

`q_2 = 34` units

Market output = 69

The equilibrium occurs at Price = $36

Explanation:

Given that:

`(\partial \pi_1)/(\partial q_1 ) = [ 105 -1 (q_1 + q_2) ] - 1q_1 -1 = 0`

`= 105 -q_1 -q_2 - q_1 -1 = 0`

`= 105 -2q_1 - q_2 -1 = 0`

`= 2q_1 +q_2 = 104 --- (1)`

`(\partial \pi_2)/(\partial q_2) = [ 105 -1 (q_1 + q_2) ] - 1q_2 -2 = 0`

`= 105 -q_1 - q_2- q_2 -2 = 0`

`= 105 -q_1 - 2q_2 -2 = 0`

`= q_1 + 2q_2 = 103 --- (2)`

By Solving (1) and (2):

Using elimination method: Multiply

`2q_1 +q_2 = 104` ---- (1)

`q_1 +2q_2 = 103` ---- (2)

(multiply equation (2) by 2 )

`2q_1 +q_2 = 104`

`2q_1 +4q_2 = 206`

`-3q_2 = -102`

`q_2 = (-102)/(-3)`

`q_2 = 34` units

From (1):

`2q_1 +q_2 = 104`

`2q_1 + 34 = 104`

`2q_1 = 104 - 34`

`2q_1 = 70`

`q_1 = (70)/(2)`

`q_1 = 35` units

Market output = (34 + 35 ) units

Market output = 69

The equilibrium occurs at Price = 105 - 69

The equilibrium occurs at Price = $36