Question

The market inverse demand curve is P = 90 – Q, where Q is the total market output consisting of Firm 1's output, q1, and Firm 2's output, q2. Both firms have a constant marginal cost of $10. If Firm 1 selects its output level first, how much output does each firm produce?

Answer 1

To determine the output levels for Firm 1 and Firm 2, you can solve the equation q1 + q2 = 80, where q1 is the output level for Firm 1 and q2 is the output level for Firm 2. Assuming Firm 1 chooses an output level of 40, Firm 2's output level would also be 40.

Step-by-step explanation:

Since both firms have a constant marginal cost of $10, they need to determine their output levels based on the market inverse demand curve. Firm 1 selects its output level first, so it can set the quantity q1. To find the output levels for both firms, let's solve the market inverse demand curve equation:

P = 90 - Q

Substituting the output levels for Firm 1 (q1) and Firm 2 (q2) into the equation, we have:

P = 90 - (q1 + q2)

Since the market price (P) is equal to the marginal cost of $10, the equation becomes:

10 = 90 - (q1 + q2)

Simplifying the equation, we have:

q1 + q2 = 80

At this point, Firm 1 can choose any output level (q1) as long as the sum of the output levels for both firms is 80. Let's assume Firm 1 chooses q1 = 40. Therefore, Firm 2's output level (q2) would be 80 - 40 = 40.

Answer 2

Firm 1 produces 40 and Firm 2 produce 20

Step-by-step 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

`(d \phi_2)/(dq_2) = 80 - q_1 - 2q_2`

80 - q₁ - 2q₂ = 0

80 - q₁ = 2q₂

2q₂= 80 - q₁

`q_2 = (80 - q_1)/(2)`

`q_2 = 40 - ( q_1)/(2)`

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 ,

`\phi _1 = 80q_1 - q_1^2 - q_1 (40 - (q_1)/(2)) - 10`

`\phi _1 = 80q_1 - q_1^2 - 40q_1 - (q_1^2)/(2) - 10`

`\phi _1 = 40q_1 - (q_1^2)/(2) - 10`

`( d \phi_1)/( d q_1) = 40 - (2q)/(2) - 0`

`( d \phi_1)/( d q_1) = 40 - q_1`

40 - q₁= 0

q₁ = 40

Recall that

`q_2 = 40 - ( q_1)/(2)`

`q_2 = 40 - ( 40)/(2)`

`q_2 = ( 80-40)/(2)`

`q_2 = ( 40)/(2)`

q₂ = 20

Firm 1 produces 40 and Firm 2 produce 20