Suppose that a monopolist has a constant marginal cost curve. That is, for each unit of output that the monopolist produces, it costs an additional $64
. The monopolist's marginal revenue is MR=200−8Q
, where Q is the quantity produced. The demand curve is P=200−4Q
.
 
What is the monopolist's profit-maximizing output and price? Write the exact answer. Do not round.


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.

Suppose that a monopolist has a constant marginal cost curve. That is, for each unit of output that the monopolist produces, it costs an additional $64. The monopolist's marginal revenue is MR=200−8Q, where Q is the quantity produced. The demand curve is P=200−4Q. What is the monopolist's profit-maximizing output and price? Write the exact answer. Do not round.

Explore how to determine a monopolist's profit-maximizing output and price when given a constant marginal cost of $64 and a marginal revenue function MR=200−8Q. This problem illustrates the process of equating marginal revenue and marginal cost to identify optimal production levels in a monopoly market, with a step-by-step solution.

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