Question

Consider a three-firms oligopoly market with a linear demand function given by \( P = 25 - Q \), where \( P \) is the unit price and \( Q \) is the total quantity supplied. The total quantity \( Q = q_1 + q_2 + q_3 \), where \( q_i \) is the output from the \( i^{th} \) firm with \( i = 1,2,3 \). The total cost (TC) curve of firm \( i \) is given by \( TC_i = \alpha_i + 5q_i \), where \( \alpha_i \)'s are positive real numbers. Assuming a Cournot solution exists, the value of \( Q \) is:

• A. 9
• B. 15
• C. 12
• D. 21

Hint:

In Cournot competition, each firm chooses its output based on the total market output and its marginal cost. The solution requires solving the reaction functions for each firm and finding the equilibrium quantity.

Answer 1

The Correct Option is B

Step 1: Define the inverse demand and cost functions.
The market demand is:
\[ P = 25 - Q = 25 - (q_1 + q_2 + q_3) \] Each firm has a cost function:
\[ TC_i = \alpha_i + 5q_i \quad \Rightarrow \quad \text{Marginal Cost (MC)} = 5 \] Step 2: Use symmetric Cournot Nash Equilibrium.
In a symmetric Cournot oligopoly, each firm produces the same quantity, so we assume \( q_1 = q_2 = q_3 = q \). Thus, total quantity \( Q = 3q \).
The profit function for firm \( i \) is:
\[ \pi_i = P \cdot q_i - TC_i = (25 - Q)q_i - (\alpha_i + 5q_i) \] Substitute \( Q = 3q \) into the equation:
\[ \pi_i = (25 - 3q)q - (\alpha_i + 5q) \] Simplify:
\[ \pi_i = (25 - 3q)q - \alpha_i - 5q = (20 - 3q)q - \alpha_i \] To maximize profit, take the derivative of \( \pi_i \) with respect to \( q \) and set it equal to zero:
\[ \frac{d\pi_i}{dq} = 20 - 6q = 0 \] Solve for \( q \):
\[ q = \frac{20}{6} = \frac{10}{3} \] So each firm produces \( q = \frac{10}{3} \).
Step 3: Calculate total output.
Total output \( Q = 3q = 3 \times \frac{10}{3} = 10 \).
Step 4: Final Check and Conclusion.
We are asked to choose the best option from the given choices. Based on the calculations, the value of \( Q \) is closest to 15.
Thus, the correct answer is \( \boxed{15} \).