1. (8) a.) (4) The United States Postal Service (USPS) currently charges $0.73 per stamp.  They allow stamps.com to sell a sheet of twenty stamps with personalized photos for $1.20 per stamp.  Stamps.com keeps the extra beyond the $0.73 it pays the USPS.  If stamps.com is acting as a profit maximizing monopolist, what is their Lerner Index and what is the price elasticity of demand for a customized stamp?

b.)   (4)  Suppose a movie theater determines that the elasticity of demand for movie tickets is -2.0 for senior citizens and –1.5 for adults under age 65, and the marginal cost is $5 per consumer.  Use the Lerner index to determine the price senior citizens should be charged and the price adults under age 65 should be charged in order to maximize profits

### Part (a): Lerner Index and Price Elasticity of Demand for Customized Stamps

The **Lerner Index** is a measure of a firm's market power and is calculated as:

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

Where:
- $ P $ is the price charged by the firm.
- $ MC $ is the marginal cost, which in this case is the amount stamps.com pays the USPS for each stamp ($0.73 per stamp).

For stamps.com:
- $ P = 1.20 $ (price of each stamp)
- $ MC = 0.73 $ (cost of each stamp from USPS)

Now, calculate the Lerner Index:

\[
L = \frac{1.20 - 0.73}{1.20} = \frac{0.47}{1.20} \approx 0.3917
\]

Thus, the **Lerner Index** for stamps.com is approximately **0.3917** or **39.17%**.

---

Next, the Lerner Index is related to the **price elasticity of demand** ($ E_d $) by the following formula:

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

Rearranging for $ E_d $:

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

Substituting $ L = 0.3917 $:

\[
E_d = -\frac{1}{0.3917} \approx -2.55
\]

So, the price elasticity of demand for customized stamps is approximately **-2.55**.

### Part (b): Pricing for Senior Citizens and Adults under Age 65

The movie theater wants to maximize profits for two customer groups: senior citizens and adults under age 65. We are given the price elasticity of demand for each group and the marginal cost ($ MC = 5 $).

The profit-maximizing price for each group is determined using the **inverse elasticity pricing rule**:

\[
P = \frac{MC}{1 + \frac{1}{E_d}}
\]

#### Senior Citizens:
For senior citizens, the price elasticity of demand ($ E_d $) is -2.0. Using the formula:

\[
P_{\text{senior}} = \frac{5}{1 + \frac{1}{-2}} = \frac{5}{1 - 0.5} = \frac{5}{0.5} = 10
\]

So, the price for **senior citizens** should be **$10**.

#### Adults under Age 65:
For adults under age 65, the price elasticity of demand ($ E_d $) is -1.5. Using the formula:

\[
P_{\text{adult}} = \frac{5}{1 + \frac{1}{-1.5}} = \frac{5}{1 - \frac{2}{3}} = \frac{5}{\frac{1}{3}} = 15
\]

So, the price for **adults under age 65** should be **$15**.

### Summary:
- The **Lerner Index** for customized stamps is approximately **0.3917**.
- The **price elasticity of demand** for customized stamps is approximately **-2.55**.
- The movie theater should charge **$10** for senior citizens and **$15** for adults under age 65 to maximize profits.

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Do you want more details or have any questions?

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Here are 5 related questions:
1. How would a change in marginal cost affect the Lerner Index and prices?
2. What factors influence price elasticity of demand in different markets?
3. How does market competition affect the Lerner Index for a firm?
4. Could price discrimination be applied to other consumer groups in this scenario?
5. What are some limitations of using the Lerner Index for monopoly pricing?

**Tip:** The higher the price elasticity (in absolute value), the more responsive consumers are to changes in price.

a.) The United States Postal Service (USPS) currently charges $0.73 per stamp. They allow stamps.com to sell a sheet of twenty stamps with personalized photos for $1.20 per stamp. Stamps.com keeps the extra beyond the $0.73 it pays the USPS. If stamps.com is acting as a profit-maximizing monopolist, what is their Lerner Index and what is the price elasticity of demand for a customized stamp?
b.) Suppose a movie theater determines that the elasticity of demand for movie tickets is -2.0 for senior citizens and –1.5 for adults under age 65, and the marginal cost is $5 per consumer. Use the Lerner index to determine the price senior citizens should be charged and the price adults under age 65 should be charged in order to maximize profits.

Understand how to calculate the Lerner Index and price elasticity of demand for customized stamps sold by stamps.com, and learn how a movie theater can determine profit-maximizing prices for different age groups. This problem covers key concepts like monopoly pricing, elasticity of demand, and uses real-world data, including USPS stamp prices and movie ticket pricing.

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