Answer:
Firm 1 produces 40 and Firm 2 produce 20
Explanation:
Given that:
The market inverse demand curve is P = 90 – Q
Total TC = 10q + 10
and θ = q₁ + q₂
Both firms have a constant marginal cost
Then :
TC₁ = 10q₁ + 10
TC₂ = 10q₂ + 10
If Firm 1 selects its output level first, then it acts as a leader and firm 2 acts as a follower.
So, let have :
Ф₂ = pq₂ - TC₂
Ф₂ = (90 - θ)q₂ - 10q₂ - 10
Ф₂ = 90q₂ - (q₁ + q₂)q₂ - 10q₂ - 10
Ф₂ = 80q₂ - q₁q₂ - q₂² - 10
[tex]\dfrac{d \phi_2}{dq_2} = 80 - q_1 - 2q_2[/tex]
80 - q₁ - 2q₂ = 0
80 - q₁ = 2q₂
2q₂= 80 - q₁
[tex]q_2 = \dfrac{80 - q_1}{2}[/tex]
[tex]q_2 = 40 - \dfrac{ q_1}{2}[/tex]
Ф₁ = pq₁- TC₁
Ф₁ = ( 90 - θ )q₁ - 10q₁ - 10
Ф₁ = 90q₁ - (q₁ + q₂)q₁ - 10q₁ - 10
Ф₁ = 80q₁ - q₁² - q₁q₂ - 10
Replace the value of q₂ in the above equation ,
[tex]\phi _1 = 80q_1 - q_1^2 - q_1 (40 - \dfrac{q_1}{2}) - 10[/tex]
[tex]\phi _1 = 80q_1 - q_1^2 - 40q_1 - \dfrac{q_1^2}{2} - 10[/tex]
[tex]\phi _1 = 40q_1 - \dfrac{q_1^2}{2} - 10[/tex]
[tex]\dfrac{ d \phi_1}{ d q_1} = 40 - \dfrac{2q}{2} - 0[/tex]
[tex]\dfrac{ d \phi_1}{ d q_1} = 40 - q_1[/tex]
40 - q₁= 0
q₁ = 40
Recall that
[tex]q_2 = 40 - \dfrac{ q_1}{2}[/tex]
[tex]q_2 = 40 - \dfrac{ 40}{2}[/tex]
[tex]q_2 = \dfrac{ 80-40}{2}[/tex]
[tex]q_2 = \dfrac{ 40}{2}[/tex]
q₂ = 20
Firm 1 produces 40 and Firm 2 produce 20
