Cinema Pricing with Elastic Demand and Unused Capacity
A single-screen cinema in a college town seats 220. Its current uniform ticket price is $11. Average attendance is 140 on weekends and 80 on weekdays. The owners estimate the price elasticity of demand for weekend adults is -1.2, for weekday adults is -2.0, and for students is -2.5. They are considering raising the weekend price to $13 and offering a $7 student ticket on weekdays. Concessions earn an average profit of $4 per attendee. The cinema is not currently selling out on either weekend or weekdays, and the owners do not expect the two changes to affect each other's attendance. Student IDs can be checked at the door. A nearby multiplex charges $12 for all tickets and does not offer student discounts.
1. Using the elasticity estimates, predict the effect of the weekend price increase on ticket revenue and total profit, including concessions. Should the owners raise the price?
Read the worked answer
The weekend price rises from $11 to $13, an 18.2% increase. With elasticity -1.2, quantity falls by about 1.2 × 18.2% = 21.8%. New attendance is 140 × (1 − 0.218) ≈ 109.5. Ticket revenue before is 140 × $11 = $1,540; after it is 109.5 × $13 ≈ $1,423.50, a fall of about $116.50. Concession profit falls by about (140 − 109.5) × $4 = $122. Total profit falls by roughly $238.50. The demand is elastic enough that the price increase reduces ticket revenue, and the lost attendees also reduce concession profit. The owners should not raise the weekend price to $13 unless the price increase attracts a different, higher-spending audience, which the data do not show.
2. Evaluate the weekday student discount. Estimate the change in weekday attendance and total profit, and identify one condition under which the estimate would be too optimistic.
Read the worked answer
The student price falls from $11 to $7, a 36.4% decrease. With elasticity -2.5, quantity rises by about 2.5 × 36.4% = 90.9%. Weekday attendance rises from 80 to 80 × 1.909 ≈ 152.7, still below the 220-seat capacity. Ticket revenue before is 80 × $11 = $880; after it is 152.7 × $7 ≈ $1,068.90, a gain of about $189. Concession profit rises by about (152.7 − 80) × $4 = $291. Total profit rises by roughly $480. The estimate would be too optimistic if nonstudents can easily buy student tickets or if student demand is less elastic than estimated, because then the attendance gain would be smaller and full-price adult sales might be cannibalized.
3. Is this price discrimination? Discuss the welfare effects and one arbitrage risk.
Read the worked answer
Yes. Charging weekend adults, weekday adults, and students different prices for the same film is third-degree price discrimination: the cinema separates groups with different elasticities and checks student IDs to limit resale. Welfare effects are ambiguous but likely positive here because the student discount fills unused weekday capacity. Output rises, student consumer surplus rises, and the cinema earns more profit; if the discount does not displace many full-price adults, total surplus increases. The risk is arbitrage: students could buy $7 tickets and resell them to nonstudents, especially if ID checks are weak. That would erode the price separation. Another risk is that the discount trains weekday adults to seek student tickets, reducing revenue from the higher-elasticity adult group.