Question

An industry has 6 firms in Cournot competition. Each of the 6 firms has zero fixed costs, and a constant marginal cost equal to 20. The product is homogenous and the industry inverse demand function is given by 𝑃 = 230 − 𝑄, where 𝑃 is the market price and 𝑄 is the industry output (sum of outputs of the 6 firms). The market price under Cournot-Nash equilibrium is equal to _____ (in integer).

Answer 1

Correct Answer: 50

Cournot Model: Market Price Calculation

Step 1: Given Information 

The inverse demand function is:

\[ P = 230 - Q \]

where total industry output is:

\[ Q = \sum_{i=1}^{6} q_i \]

Each firm \( i \) maximizes its profit:

\[ \pi_i = P \cdot q_i - C(q_i) \]

Given that:

• \( P = 230 - Q \)
• \( C(q_i) = 20q_i \) (marginal cost is constant at 20)

Step 2: Profit Maximization for Firm \( i \)

Substituting \( P = 230 - Q \) into the profit equation:

\[ \pi_i = (230 - Q) q_i - 20 q_i \]

Simplify:

\[ \pi_i = (210 - Q) q_i \]

Each firm takes \( Q_{-i} = \sum_{j \neq i} q_j \) as given, so:

\[ Q = q_i + Q_{-i} \]

Substituting this into the profit equation:

\[ \pi_i = (210 - (q_i + Q_{-i})) q_i \]

Differentiate with respect to \( q_i \) and set to zero:

\[ \frac{\partial \pi_i}{\partial q_i} = 210 - 2q_i - Q_{-i} = 0 \]

Rearrange to get the reaction function:

\[ q_i = \frac{210 - Q_{-i}}{2} \]

Step 3: Symmetric Equilibrium

For \( n = 6 \) identical firms, symmetry implies \( q_i = q_j = q \) for all \( i, j \), and:

\[ Q = 6q \]

Since \( Q_{-i} = 5q \), substitute into the reaction function:

\[ q = \frac{210 - 5q}{2} \]

Solving for \( q \):

\[ 2q = 210 - 5q \]

\[ 7q = 210 \]

\[ q = 30 \]

Step 4: Total Industry Output and Market Price

The total industry output is:

\[ Q = 6q = 6 \times 30 = 180 \]

Substituting \( Q = 180 \) into the inverse demand function:

\[ P = 230 - Q = 230 - 180 = 50 \]

Final Answer:

The market price in equilibrium is 50.