uppose a profit maximizing firm faces a demand curve with a price elasticity equal to 3​ (in absolute​ value).  Their marginal costs are constant at​ $30.  According to the Lerner​ Index, what is the profit maximizing price for this​ firm?

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Part 1

A.

​$20

B.

​$15

C.

​$60

D.

​$45

The Lerner Index measures a firm's market power and is defined by the formula:

\[
L = \frac{P - MC}{P}
\]

Where:
- $ P $ is the profit-maximizing price,
- $ MC $ is the marginal cost,
- $ L $ is the Lerner Index.

For a profit-maximizing firm, the Lerner Index is also related to the price elasticity of demand ($E_d$) by the formula:

\[
L = \frac{1}{|E_d|}
\]

Where $ |E_d| $ is the absolute value of the price elasticity of demand. Given that $ |E_d| = 3 $, we can solve for the Lerner Index:

\[
L = \frac{1}{3}
\]

Now, substituting this into the Lerner Index formula, we get:

\[
\frac{P - MC}{P} = \frac{1}{3}
\]

Solving for $ P $, where $ MC = 30 $:

\[
\frac{P - 30}{P} = \frac{1}{3}
\]

Multiplying both sides by $ P $:

\[
P - 30 = \frac{P}{3}
\]

Multiplying both sides by 3 to eliminate the fraction:

\[
3(P - 30) = P
\]

Expanding:

\[
3P - 90 = P
\]

Solving for $ P $:

\[
2P = 90
\]

\[
P = 45
\]

Thus, the profit-maximizing price is **$45**.

### Answer: 
D. $45

Do you want further details or have any questions?

Here are 5 related questions to explore:
1. How does the Lerner Index change with different price elasticities of demand?
2. What would happen to the price if the price elasticity were less than 3?
3. Can the Lerner Index be applied to firms in perfect competition?
4. How do firms use price elasticity of demand to inform pricing strategies?
5. What are the limitations of using the Lerner Index for real-world pricing?

**Tip**: Understanding elasticity helps businesses predict how changes in price will affect demand and, ultimately, revenue.

Suppose a profit maximizing firm faces a demand curve with a price elasticity equal to 3​ (in absolute​ value). Their marginal costs are constant at​ $30. According to the Lerner​ Index, what is the profit maximizing price for this​ firm?

A. ​$20
B. ​$15
C. ​$60
D. ​$45

Solve a profit-maximizing firm's price problem using the Lerner Index formula and a price elasticity of demand of 3. Learn how constant marginal costs of $30 affect the optimal price. Ideal for undergraduate economics students.

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