Step 1: Relationship between \(q_1\) and \(q_2\)

The problem specifies that Duopolist II has a \(\frac{1}{3}\) share of the market. This implies the total quantity produced in the market is shared between both firms in the ratio of 2:1, with Duopolist I producing more than Duopolist II.

Let the total quantity produced in the market be \(q_1 + q_2\), where:

• \(q_1\) is the quantity produced by Duopolist I.
• \(q_2\) is the quantity produced by Duopolist II.

Given that Duopolist II has a \(\frac{1}{3}\) share of the market, the remaining \(\frac{2}{3}\) of the market is produced by Duopolist I. Therefore, we can express the relationship between \(q_1\) and \(q_2\) as:

\(q_2 = \frac{1}{3} (q_1 + q_2)\)

Multiplying both sides by 3:

\(3q_2 = q_1 + q_2\)

Now, moving the terms involving \(q_2\) to one side:

\(2q_2 = q_1\)

This gives us the relationship between the output of Duopolist I and Duopolist II:

\(q_2 = \frac{q_1}{2}\)

Thus, Duopolist II’s output is half of Duopolist I’s output.

Step 2: Deriving Duopolist I’s Profit Function

Duopolist I’s profit \(\pi_1\) is given by the difference between its total revenue and its total cost. The revenue for Duopolist I is given by:

Revenue = \(p_1 \cdot q_1\)

Substitute the demand function \(p_1 = 200 - 4q_1 - 2q_2\) into the revenue equation:

Revenue = \((200 - 4q_1 - 2q_2) \cdot q_1\)

Now, substitute \(q_2 = \frac{q_1}{2}\):

Revenue = \((200 - 4q_1 - q_1) \cdot q_1 = (200 - 5q_1) \cdot q_1 = 200q_1 - 5q_1^2\)

The cost function for Duopolist I is given by:

\(C_1 = 5q_1^2\)

Thus, Duopolist I’s profit is:

\(\pi_1 = \text{Revenue} - C_1 = (200q_1 - 5q_1^2) - 5q_1^2 = 200q_1 - 10q_1^2\)

Step 3: Maximizing Profit

To find the optimal output for Duopolist I \(q_1^*\), we take the derivative of the profit function with respect to \(q_1\) and set it equal to zero:

\(\frac{d\pi_1}{dq_1} = 200 - 20q_1 = 0\)

Solving for \(q_1\):

\(20q_1 = 200 \quad \Rightarrow \quad q_1^* = \frac{200}{20} = 10\)

Thus, the optimal output for Duopolist I is \(q_1^* = 10\).

Step 4: Finding the Output of Duopolist II

From the relationship \(q_2 = \frac{q_1}{2}\), we can find Duopolist II’s output:

\(q_2^* = \frac{q_1^*}{2} = \frac{10}{2} = 5\)

Thus, the optimal output for Duopolist II is \(q_2^* = 5\).

Step 5: Determining the Price for Duopolist I

Substitute \(q_1^* = 10\) and \(q_2^* = 5\) into the demand function to find the price:

\(p_1 = 200 - 4q_1 - 2q_2 p_1^* = 200 - 4(10) - 2(5) = 200 - 40 - 10 = 150\)

Thus, the optimal price for Duopolist I is \(p_1^* = 150\).

Step 6: Calculating Duopolist I’s Profit

Finally, we calculate the profit for Duopolist I by substituting the values of \(q_1 = 10\) and \(p_1 = 150\) into the profit formula:

\(\pi_1 = p_1 \cdot q_1 - C_1\) 
\(\pi_1 = 150 \cdot 10 - 5(10)^2 = 1500 - 500 = 1000\)

Thus, the profit for Duopolist I is (\(\pi_1^* = 1000\)).

Conclusion:

Optimal Output of Duopolist I (\(q_1^*\)): 10 units

Optimal Output of Duopolist II (\(q_2^*\)): 5 units

Optimal Price for Duopolist I (\(p_1^*\)): 150

Profit of Duopolist I (\(\pi_1^*\)): 1000

In this scenario, Duopolist I produces 10 units and charges a price of 150, yielding a profit of 1000. Duopolist II, with a market share of (\(\frac{1}{3}\)), produces 5 units.