Gala Ticket Pricing with a Capacity Cap
The student events society is planning its annual charity gala. The venue holds 360 people. Last year the society sold 300 standard tickets at 20 each and 80 VIP tickets at 50 each, for 380 attendees, which exceeded the venue's current licensed capacity; this year the cap is 360. Fixed costs are 7,200 for venue, insurance and licensing, and each attendee costs 6 in catering regardless of ticket type. If prices stay unchanged, the committee expects standard demand to fall by 20% and VIP demand to rise by 25% relative to last year's sales. A member proposes raising the VIP price to 60, predicting that VIP sales would then be 10% lower than under the no-price-change projection. The committee wants to maximise net revenue-ticket sales minus catering-while covering fixed costs. Unsold capacity cannot be resold, and ticket prices must be whole numbers.
1. Compute projected attendance and net profit under (i) prices unchanged and (ii) the VIP price raised to 60. Which option is better, and by how much?
Read the worked answer
No price change
- Standard demand: 300 × 0.80 = 240
- VIP demand: 80 × 1.25 = 100
- Total attendance: 340, which is within the 360 cap
- Ticket revenue: (240 × 20) + (100 × 50) = 4,800 + 5,000 = 9,800
- Catering: 340 × 6 = 2,040
- Contribution to fixed costs: 9,800 − 2,040 = 7,760
- Profit: 7,760 − 7,200 = 560
VIP price raised to 60
- Standard demand remains 240
- VIP demand: 100 × 0.90 = 90
- Total attendance: 330
- Ticket revenue: (240 × 20) + (90 × 60) = 4,800 + 5,400 = 10,200
- Catering: 330 × 6 = 1,980
- Contribution: 10,200 − 1,980 = 8,220
- Profit: 8,220 − 7,200 = 1,020
Conclusion: the price rise is better by 1,020 − 560 = 460 in net profit. The gain comes from higher VIP revenue (5,400 vs 5,000) and lower catering on 10 fewer attendees (1,980 vs 2,040), partly offset by 10 fewer VIP tickets.
2. A committee member argues: 'Under the proposal the average ticket price is 10,200 ÷ 330 ≈ 30.91, versus 9,800 ÷ 340 ≈ 28.82 now, a rise of about 2.09 per attendee. With 330 attendees that adds about 690, so the proposal is even better than you calculated.' Identify the error in that reasoning.
Read the worked answer
The error is comparing averages computed over different attendee totals and then multiplying the difference by only one of those totals.
- The actual revenue difference is 10,200 − 9,800 = 400, not 690.
- The average-price gap of about 2.09 is real, but it is an artefact of both a price change and a quantity/composition change: the proposal has 10 fewer attendees, and those missing attendees were standard-ticket holders paying 20. Dividing a smaller revenue total by a smaller attendance base raises the average even before any VIP price effect is isolated.
- Multiplying 2.09 by 330 double-counts the quantity change. The correct incremental revenue is the difference between the two totals, not the difference in unit averages times a selected quantity.
- The member also ignores catering. The profit comparison must subtract 6 per attendee: 330 × 6 = 1,980 versus 340 × 6 = 2,040, a saving of 60. That is why the profit gain is 460, not 400.
A sound comparison fixes a common quantity or, better, compares total contribution directly: revenue minus variable catering for each scenario.
3. Suppose the VIP price rise to 60 causes VIP sales to fall to 80 instead of 90. Does the proposal still beat the no-change plan? Find the smallest whole-number VIP attendance at which the proposal at least matches the no-change profit.
Read the worked answer
With standard attendance fixed at 240 and VIP price at 60, let v be VIP attendance.
- Ticket revenue: 4,800 + 60v
- Catering: 6(240 + v) = 1,440 + 6v
- Contribution: 4,800 + 60v − 1,440 − 6v = 3,360 + 54v
- Profit: 3,360 + 54v − 7,200 = 54v − 3,840
The no-change profit is 560, so set:
54v − 3,840 ≥ 560 54v ≥ 4,400 v ≥ 81.48
Because v must be a whole number, the smallest qualifying attendance is 82 VIP tickets. At v = 82, profit = (54 × 82) − 3,840 = 4,428 − 3,840 = 588, which beats 560 by 28. At v = 81, profit = 4,374 − 3,840 = 534, which is 26 below the no-change plan.
If VIP sales fall to 80, profit = 4,320 − 3,840 = 480, which is 80 worse than the no-change profit of 560. So at 80 the proposal fails.