To find the monopolist's profit-maximizing output and price, we need to set the marginal cost equal to the marginal revenue, as the profit-maximizing condition for a monopolist is where $\text{MC} = \text{MR}$.

Step 1: Set Up the Information

1. Marginal Cost (MC): The marginal cost is constant at $MC = 64$.
2. Marginal Revenue (MR): Given by $MR = 200 - 8Q$.
3. Demand Curve (P): The price function is $P = 200 - 4Q$.

Step 2: Set $\text{MC} = \text{MR}$

To maximize profit, we equate $\text{MC}$ and $\text{MR}$: $64 = 200 - 8Q$

Step 3: Solve for $Q$

Rearrange the equation to solve for $Q$: $8Q = 200 - 64$ $8Q = 136$ $Q = \frac{136}{8} = 17$

So, the profit-maximizing output is $Q = 17$.

Step 4: Find the Profit-Maximizing Price

To find the corresponding price, substitute $Q = 17$ into the demand equation: $P = 200 - 4Q$ $P = 200 - 4 \times 17$ $P = 200 - 68$ $P = 132$

Final Answer

The monopolist's profit-maximizing output and price are:

• Quantity ($Q$): 17
• Price ($P$): 132

Would you like further details or have any questions?

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Here are some related questions:

1. How would the profit-maximizing output change if the marginal cost increased?
2. What would happen to the monopolist's price if the demand curve shifted downward?
3. How does a monopolist’s price compare to a perfectly competitive firm’s price with the same cost and demand conditions?
4. Why does a monopolist set marginal cost equal to marginal revenue to maximize profits?
5. How would consumer surplus be affected by this monopolist's pricing?

Tip: In monopoly problems, understanding the relationship between marginal revenue and marginal cost is crucial for determining profit-maximizing outcomes.