VIPSolutionsExercise 1 A movie theater shows films for a community of 1 0 0 0 0 people shows films during weekends. Right now, the price per ticket is $ 1 7 . 5 0 . In the past, when they increased or decreased the price per ticket, they discovered that for every dollar ( or fraction ) that the price was increased or decreased, the attendance decreased or increased proportionally by 2 0 0 people. The theater owner pays the Film Distribution Company $ 1 0 ( incremental cost ) in royalties per person who views the film. Find a . The willingness - to - pay distribution of the movie goers. b . The average price they would be willing to pay. c . The price per ticket that maximizes revenue. d . Should the theater owner increase or decrease the price per ticket if she wants to maximize contribution? e . Find the consumer surplus if the tickets are priced at the price that maximizes contribution.
Question:
Exercise 1 A movie theater shows films for a community of 1 0 0 0 0 people shows films during weekends. Right now, the price per ticket is $ 1 7 . 5 0 . In the past, when they increased or decreased the price per ticket, they discovered that for every dollar ( or fraction ) that the price was increased or decreased, the attendance decreased or increased proportionally by 2 0 0 people. The theater owner pays the Film Distribution Company $ 1 0 ( incremental cost ) in royalties per person who views the film. Find a . The willingness - to - pay distribution of the movie goers. b . The average price they would be willing to pay. c . The price per ticket that maximizes revenue. d . Should the theater owner increase or decrease the price per ticket if she wants to maximize contribution? e . Find the consumer surplus if the tickets are priced at the price that maximizes contribution.
This is AI generated Response.
Asked by: Vikram Kumar
Created at: 2025-10-09 01:11:06
Asked by: Vikram Kumar
Created at: 2025-10-09 01:11:06
Thought Process:
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**Parsing demand scenario**
I need to break down this demand function problem. So, there’s a community of 10,000 people and a current ticket price of $17.50 for weekend shows. When they adjust prices, they found attendance changes by about 200 people for every dollar increase or decrease in price. This gives me a price elasticity slope of -200, suggesting a linear demand curve. The maximum attendance at a price of $0 would be 10,000, leading to a maximum ticket price of $50.**Analyzing demand function**
I’ve determined the inverse demand function is P = 50 - Q/200, which seems to align with the willingness-to-pay distribution from $0 to $50 among 10,000 moviegoers. However, I need to ensure I’m interpreting their observation correctly: attendance changes by 200 for every dollar in price. The setup implies a maximum attendance of 10,000 at a $0 price. So, I figure the linear demand function should be Q(P) = 10,000 - 200P for prices between $0 and $50. Now, for part a, I'm looking at the CDF of willingness to pay among these 10,000 people.**Evaluating willingness-to-pay distribution**
So, if demand is represented as Q(P) = number of people with willingness to pay (WTP) greater than or equal to P, it translates to Q(P) = 10,000 - 200P. This results in the probability that W is greater than or equal to P being Pr(W ≥ P) = (10,000 - 200P)/10,000 for P between $0 and $50. The PDF is constant at 0.02 per dollar, indicating a uniform distribution of WTP across that range.
For part b, the average price would typically be the mean of the distribution, which is $25, although there may be ambiguity in the phrasing.**Calculating optimal prices**
For part c, I need to find the ticket price that maximizes revenue. The revenue function is R(P) = P * Q(P) = P * (10,000 - 200P), which simplifies to 10,000P - 200P². When I take the derivative and set it to zero, I find the revenue-maximizing price is $25.
Moving to part d, for contribution, which is revenue minus the variable cost of $10, I establish the contribution function as C(P) = (P - 10) * Q(P). After maximizing it and finding the derivative, I determine that the price that maximizes contribution is $30.**Summarizing theater pricing analysis**
Given the current ticket price of $17.50, raising it to $30 makes sense because the revenue-maximizing price is $25, while the contribution-maximizing price is $30.
For part e, if tickets are priced at $30, the demand would be 4,000, leading to a consumer surplus of $40,000. The willingness-to-pay distribution is uniform between $0 and $50, with an average WTP of $25.
Summary points:
- WTP distribution: f(w)=1/50 for w∈[0,50]; F(w)=w/50.
- Average WTP: $25.
- Revenue-maximizing price: $25.
- Contribution-maximizing price: $30, suggesting an increase.
- Consumer surplus at $30: $40,000.
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Let Q(P) be weekend attendance when the ticket price is P, with at most 10,000 potential moviegoers.
Given: for each $1 increase in price, attendance falls by 200 people (and vice versa), so demand is linear with slope −200 people per dollar. With a market size of 10,000, we get:
Q(P) = 10,000 − 200 P, for 0 ≤ P ≤ 50
Inverse demand (willingness-to-pay of the marginal attendee at quantity q):
P(q) = 50 − q/200, for 0 ≤ q ≤ 10,000
The choke price (where Q = 0) is P = 50.
a) Willingness-to-pay (WTP) distribution
- Interpretation: Q(P) is the number of people whose WTP ≥ P. So Pr(WTP ≥ P) = Q(P)/10,000 = 1 − (200/10,000) P = 1 − P/50 for 0 ≤ P ≤ 50.
- Thus WTP is Uniform on [0, 50].
- PDF: f(w) = 1/50 for 0 ≤ w ≤ 50; CDF: F(w) = w/50 for 0 ≤ w ≤ 50.
- Equivalently, inverse demand: P(q) = 50 − q/200.
b) Average price they would be willing to pay
- The mean of Uniform[0, 50] is 25.
- So the average WTP is $25.
c) Price per ticket that maximizes revenue
- Revenue: R(P) = P · Q(P) = P(10,000 − 200P = 10,000P − 200P^2.
- dR/dP = 10,000 − 400P = 0 ⇒ P = 25.
- Revenue-maximizing price: $25, with attendance Q = 10,000 − 200·25 = 5,000.
d) Should the owner increase or decrease price to maximize contribution?
- Contribution = Revenue − Variable cost, with incremental royalty cost of $10 per attendee ⇒ marginal cost per attendee = $10.
- Contribution as a function of price: C(P) = (P − 10) Q(P) = (P − 10)(10,000 − 200P).
- dC/dP = (10,000 − 200P) + (P − 10)(−200) = 12,000 − 400P; set to zero ⇒ P = 30.
- Current price is $17.50; contribution-maximizing price is $30. Therefore, to maximize contribution, the owner should increase the price.
Additional numbers at P = 30:
- Attendance: Q = 10,000 − 200·30 = 4,000.
- Contribution: (30 − 10)·4,000 = $80,000.
e) Consumer surplus at the contribution-maximizing price
- At P = 30, the highest WTP (intercept) is 50, and the marginal WTP at Q = 4,000 equals 30. CS is the triangle between 50 and 30 over 4,000 attendees:
- CS = (1/2) × base × height = 0.5 × 4,000 × (50 − 30) = 0.5 × 4,000 × 20 = $40,000.
Summary
- Demand: Q(P) = 10,000 − 200P; inverse demand P(q) = 50 − q/200.
- WTP distribution: Uniform[0, 50]; average WTP = $25.
- Revenue-maximizing price: $25.
- Contribution-maximizing price (with $10 per-attendee royalty): $30; increase price from $17.50 to $30.
- Consumer surplus at P = $30: $40,000.
Question:
Consumer surplus at the contribution-maximizing price means
This is AI generated Response.
Asked by: Vikram Kumar
Created at: 2025-10-09 01:40:40
Asked by: Vikram Kumar
Created at: 2025-10-09 01:40:40
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# Consumer Surplus at the Contribution-Maximizing Price
## Definition of Consumer Surplus
Consumer surplus (CS) is the difference between what consumers are willing to pay for a good or service versus what they actually pay. It represents the benefit to consumers from participating in the market.
## Calculation of Consumer Surplus
At the contribution-maximizing price of $30, the consumer surplus can be calculated as follows:
1. **Identify the Highest Willingness to Pay**: The highest willingness to pay (intercept) from the inverse demand function is $50.
2. **Identify the Marginal Willingness to Pay**: For the quantity of 4,000 attendees, the marginal willingness to pay corresponds to the price of $30.
3. **Calculate the Consumer Surplus**:
- The consumer surplus is represented by the area of the triangle formed between the maximum willingness to pay ($50), the price ($30), and the quantity (4,000).
- Formula for the area of a triangle:
\[
CS = \frac{1}{2} \times \text{base} \times \text{height}
\]
- Here, the base is the quantity (4,000) and the height is the difference in willingness to pay ($50 - $30 = $20).
Plugging in the values:
\[
CS = \frac{1}{2} \times 4,000 \times (50 - 30) = \frac{1}{2} \times 4,000 \times 20 = 40,000
\]
## Conclusion
The consumer surplus at the contribution-maximizing price of $30 is **$40,000**. This amount represents the total benefit that consumers receive over what they pay for the tickets.
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