The (inverse) demand for tickets to a concert by Paolo Conte at the local theater is given by P = 120 – 0.005Q. The theater has negligible (zero) fixed costs and can sell any number of tickets up to 25,000 at a constant marginal cost of MC = $5. How many tickets should the theater sell if it wants to maximize its profits?


The (inverse) demand for tickets to a concert by Paolo Conte at the local theater is given by P = 120 – 0.005Q. The theater has negligible (zero) fixed costs and can sell any number of tickets up to 25,000 at a constant marginal cost of MC = $5. How many tickets should the theater sell if it wants to maximize its profits?


11,500


12,000


24,000


25,000

To determine the profit-maximizing quantity of tickets, we need to follow a process involving marginal revenue (MR) and marginal cost (MC).

### Step 1: Write the Demand Function
The inverse demand function is given as:

\[
P = 120 - 0.005Q
\]

Here, $ P $ is the price and $ Q $ is the quantity of tickets.

### Step 2: Find the Total Revenue (TR) Function
Total revenue is calculated as price times quantity:

\[
TR = P \times Q = (120 - 0.005Q) \times Q = 120Q - 0.005Q^2
\]

### Step 3: Find the Marginal Revenue (MR) Function
Marginal revenue is the derivative of total revenue with respect to $ Q $:

\[
MR = \frac{d(TR)}{dQ} = 120 - 0.01Q
\]

### Step 4: Set Marginal Revenue Equal to Marginal Cost
To maximize profits, we set marginal revenue equal to marginal cost (MC). The marginal cost is constant at $5:

\[
MR = MC
\]

\[
120 - 0.01Q = 5
\]

### Step 5: Solve for $ Q $
Now, solve for $ Q $:

\[
120 - 5 = 0.01Q
\]

\[
115 = 0.01Q
\]

\[
Q = \frac{115}{0.01} = 11,500
\]

### Conclusion:
The theater should sell **11,500** tickets to maximize its profit.

Let me know if you want details or have any questions. Here are 5 related questions for further exploration:

1. How would the profit-maximizing quantity change if marginal costs were higher?
2. What happens to profit-maximizing quantity if demand becomes more elastic?
3. How does consumer surplus change at 11,500 tickets?
4. How would profits change if ticket prices were capped at $100?
5. What is the total revenue when 11,500 tickets are sold?

**Tip:** In economics, the marginal revenue is often lower than price due to the decreasing demand curve.

The (inverse) demand for tickets to a concert by Paolo Conte at the local theater is given by P = 120 – 0.005Q. The theater has negligible (zero) fixed costs and can sell any number of tickets up to 25,000 at a constant marginal cost of MC = $5. How many tickets should the theater sell if it wants to maximize its profits?

Find out how to determine the profit-maximizing quantity of tickets for a Paolo Conte concert using the demand function P = 120 - 0.005Q and constant marginal cost of $5. Explore the marginal revenue and profit maximization principles to solve this problem, suitable for undergraduate economics students.

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