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.