Suppose a monopolist faces a market demand curve given by P = 50 - 0.5*Q. Marginal cost is initially equal to zero and constant.

a. (4) Calculate the profit maximizing price and quantity. Use the Lerner index to calculate the price elasticity of demand at this point. What is the amount of deadweight loss associated with this monopoly?

To solve this monopoly problem, we’ll follow these steps:

### 1. **Profit-Maximizing Price and Quantity**

The monopolist maximizes profit by setting Marginal Revenue (MR) equal to Marginal Cost (MC). Let’s first calculate MR.

The demand curve is given by:
\[
P = 50 - 0.5Q
\]

Total Revenue (TR) is given by:
\[
TR = P \times Q = (50 - 0.5Q) \times Q = 50Q - 0.5Q^2
\]

The Marginal Revenue (MR) is the derivative of Total Revenue with respect to \(Q\):
\[
MR = \frac{d(TR)}{dQ} = 50 - Q
\]

Given that Marginal Cost (MC) is zero (constant):
\[
MC = 0
\]

To maximize profit, set \(MR = MC\):
\[
50 - Q = 0
\]
Solving for \(Q\), we get:
\[
Q = 50
\]

Now, substitute \(Q = 50\) into the demand equation to find the profit-maximizing price:
\[
P = 50 - 0.5(50) = 50 - 25 = 25
\]

### **Result**:
- Profit-maximizing quantity: \( Q = 50 \)
- Profit-maximizing price: \( P = 25 \)

---

### 2. **Lerner Index and Price Elasticity of Demand**

The Lerner Index (L) is a measure of the monopoly’s market power and is defined as:
\[
L = \frac{P - MC}{P}
\]
Since \( MC = 0 \) and \( P = 25 \), the Lerner Index is:
\[
L = \frac{25 - 0}{25} = 1
\]

The Lerner Index is also related to the price elasticity of demand (\( \varepsilon \)) by the formula:
\[
L = \frac{1}{|\varepsilon|}
\]

So, setting \(L = 1\), we solve for \(\varepsilon\):
\[
1 = \frac{1}{|\varepsilon|}
\]
\[
|\varepsilon| = 1
\]

Thus, the price elasticity of demand at the profit-maximizing point is \( |\varepsilon| = 1 \).

---

### 3. **Deadweight Loss (DWL)**

Deadweight loss is the loss of economic efficiency due to the monopoly not producing at the socially efficient level, which would be where \( P = MC \).

To find the socially efficient output level, set \(P = MC = 0\) in the demand equation:
\[
0 = 50 - 0.5Q
\]
Solving for \(Q\), we get:
\[
Q = 100
\]

At the competitive equilibrium, the quantity would be \( Q = 100 \), and the price would be \( P = 0 \).

The deadweight loss is the triangular area between the monopoly output and the socially optimal output. The formula for deadweight loss is:
\[
DWL = \frac{1}{2} \times (Q_{\text{socially efficient}} - Q_{\text{monopoly}}) \times (P_{\text{monopoly}} - P_{\text{socially efficient}})
\]
Substitute the values:
\[
DWL = \frac{1}{2} \times (100 - 50) \times (25 - 0) = \frac{1}{2} \times 50 \times 25 = 625
\]

---

### **Final Answer Summary**:
- Profit-maximizing price: \(P = 25\)
- Profit-maximizing quantity: \(Q = 50\)
- Price elasticity of demand at this point: \( |\varepsilon| = 1 \)
- Deadweight loss: \(DWL = 625\)

Would you like more details or have any questions on this?

Here are five related questions:
1. How does a monopolist’s profit-maximizing behavior differ from a competitive firm’s?
2. What happens to the deadweight loss if the marginal cost is positive?
3. How would the results change if the demand curve was more elastic?
4. What is the relationship between the Lerner Index and the monopolist’s ability to mark up prices?
5. How does the elasticity of demand affect the monopoly’s pricing power?

**Tip**: The Lerner Index helps quantify a monopolist's power to set prices above marginal cost. A higher index indicates greater market power.

Suppose a monopolist faces a market demand curve given by P = 50 - 0.5*Q. Marginal cost is initially equal to zero and constant.

Calculate the profit maximizing price and quantity. Use the Lerner index to calculate the price elasticity of demand at this point. What is the amount of deadweight loss associated with this monopoly?

Explore how a monopolist maximizes profit given a demand curve P = 50 - 0.5*Q and zero marginal cost. Learn to calculate the profit-maximizing price, quantity, Lerner Index, price elasticity of demand, and deadweight loss with step-by-step solutions. Perfect for undergraduate economics students.

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