Suppose a monopolist faces a market demand curve given by P=50-.5*Q. Marginal cost is initially equal to zero and constant. 

Given the information provided:

1. **Market Demand Curve**: \( P = 50 - 0.5Q \), where \( P \) is the price and \( Q \) is the quantity demanded.
2. **Marginal Cost (MC)**: Initially equal to zero, i.e., \( MC = 0 \).

To analyze this monopoly, let's break down the problem:

### 1. **Total Revenue (TR)**:
Total revenue is the product of price and quantity:

\[
TR = P \times Q
\]

Substitute \( P = 50 - 0.5Q \) into the equation for \( TR \):

\[
TR = (50 - 0.5Q) \times Q = 50Q - 0.5Q^2
\]

### 2. **Marginal Revenue (MR)**:
Marginal revenue is the derivative of total revenue with respect to quantity:

\[
MR = \frac{d(TR)}{dQ} = \frac{d(50Q - 0.5Q^2)}{dQ} = 50 - Q
\]

### 3. **Profit Maximization**:
A monopolist maximizes profit by setting marginal revenue equal to marginal cost (\( MR = MC \)). Since \( MC = 0 \), the profit-maximizing condition is:

\[
50 - Q = 0
\]

Solve for \( Q \):

\[
Q = 50
\]

### 4. **Find the Price**:
Substitute \( Q = 50 \) back into the demand equation to find the price \( P \):

\[
P = 50 - 0.5(50) = 50 - 25 = 25
\]

### 5. **Conclusion**:
- The profit-maximizing quantity is \( Q = 50 \).
- The profit-maximizing price is \( P = 25 \).

#### Would you like to explore further details, such as profits, consumer surplus, or welfare analysis? 

Here are 5 related questions:
1. How would the outcome change if marginal cost were positive?
2. What is the monopolist's total revenue at the profit-maximizing quantity?
3. How does consumer surplus compare in a monopoly versus perfect competition?
4. How do we calculate the deadweight loss in a monopoly?
5. What happens to the monopolist's price and output if the demand curve shifts?

**Tip**: In monopoly settings, marginal revenue falls twice as fast as the demand curve because the monopolist must lower prices to sell additional units.

Suppose a monopolist faces a market demand curve given by P=50-.5*Q. Marginal cost is initially equal to zero and constant.

Explore the solution to a monopoly profit maximization problem where the monopolist faces a demand curve P = 50 - 0.5Q and marginal cost equals zero. The solution details include revenue, marginal revenue, and profit-maximizing quantity and price. Suitable for undergraduate economics students.

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