Respuesta :
Answer: [tex]Q_{1}[/tex] = 24 - 0.5[tex]Q_{2}[/tex]
Explanation:
Given that,
There are two firms in the cournot market,
Market demand for their product ⇒ P = 204 - 4Q
Each firm's constant marginal cost (MC) = $12 per unit
Let [tex]Q_{1}[/tex] and [tex]Q_{2}[/tex] are the output produced by the firm 1 and firm 2.
P = 204 - 4Q
P = 204 - 4([tex]Q_{1}[/tex] + [tex]Q_{2}[/tex])
P = 204 - 4[tex]Q_{1}[/tex] - 4[tex]Q_{2}[/tex]
Total revenue for firm 1 = Price × Quantity([tex]Q_{1}[/tex])
= 204 [tex]Q_{1}[/tex] - 4[tex]Q_{1}^{2}[/tex] - 4[tex]Q_{1}[/tex] [tex]Q_{2}[/tex]
Marginal revenue for firm 1 = 204 - 8[tex]Q_{1}[/tex] - 4[tex]Q_{2}[/tex]
MR = MC
204 - 8[tex]Q_{1}[/tex] - 4[tex]Q_{2}[/tex] = 12
[tex]Q_{1}[/tex] = 24 - 0.5[tex]Q_{2}[/tex]
Above is the firm 1's reaction function.
The firm 1's reaction function is [tex]Q_1 = 24 - 0.5Q_2[/tex]
The given parameters are:
Market demand, P = 204 - 4Q
[tex]Q^i[/tex] represent the output produced by firm i, where i = 1,2
We have:
[tex]Q = Q_1 + Q_2[/tex]
So, we have:
[tex]P = 204 - 4(Q_1 + Q_2)[/tex]
Expand
[tex]P = 204 - 4Q_1 - 4Q_2[/tex]
Revenue is calculated as:
Revenue (R) = Price (P) * Quantity (Q)
For firm 1, we have:
[tex]R_1 = P_1 * Q_1[/tex]
This gives
[tex]R_1 = [204 - 4Q_1 - 4Q_2] * Q_1[/tex]
Expand
[tex]R_1 = 204Q_1 - 4Q_1^2 - 4Q_1Q_2[/tex]
Differentiate to determine the marginal revenue for firm 1
[tex]MR_1 = 204 - 8Q_1 - 4Q_2[/tex]
The marginal revenue and the marginal cost are equal ($12 per unit)
So, we have:
[tex]204 - 8Q_1 - 4Q_2 = 12[/tex]
Collect like terms
[tex]8Q_1 = 204 - 12 - 4Q_2[/tex]
[tex]8Q_1 = 192 - 4Q_2[/tex]
Divide through by 8
[tex]Q_1 = 24 - 0.5Q_2[/tex]
Hence, the firm 1's reaction function is [tex]Q_1 = 24 - 0.5Q_2[/tex]
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