Festival Ticket Pricing with a Sponsor Constraint
At a school festival, the student council must set a single ticket price for a concert. The hall costs ¥180,000 to hire, and each audience member costs an additional ¥120 in programmes, cleaning and insurance. A survey of students suggests that at ¥500, 800 people would attend, and that for every ¥50 increase in price, attendance would fall by 40. The council may choose only prices that are multiples of ¥50, starting from ¥500. A sponsor will contribute only if at least 600 people attend. The council wants the largest possible surplus, meaning ticket income minus hire and per-person costs, while keeping the sponsor's condition. A member proposes charging ¥800, arguing that because the survey still predicts 560 attendees, the higher price must be better. The council has asked for a mathematical analysis before voting.
1. Let x be the number of ¥50 increases above ¥500. Write the surplus S(x) and find the price that maximises S if the sponsor's condition is ignored.
Read the worked answer
Price p = 500 + 50x; attendance n = 800 − 40x. Surplus is (p − 120)n − 180000.
S(x) = (380 + 50x)(800 − 40x) − 180000 = 304000 + 40000x − 15200x − 2000x² − 180000 = 124000 + 24800x − 2000x².
This is a downward parabola; the vertex is at x = −b/(2a) = 24800/(4000) = 6.2. So the unconstrained maximum is at p = 500 + 50(6.2) = ¥810, with n = 800 − 40(6.2) = 552 and S = ¥200,880. Because only multiples of ¥50 are allowed, the best integer x is 6 (p = ¥800), with S = ¥200,800 and n = 560; x = 7 gives ¥199,600. The unconstrained answer is therefore ¥810, or ¥800 if integer prices are required.
2. Apply the sponsor's requirement that attendance be at least 600. What price should the council set, and how much surplus is forgone compared with the best unconstrained integer price?
Read the worked answer
Requirement: 800 − 40x ≥ 600, so x ≤ 5. On the interval x ≤ 5, S(x) = 124000 + 24800x − 2000x² is increasing because its vertex is at x = 6.2, so the best feasible integer is x = 5. Price = ¥750, attendance = 600, S = 124000 + 124000 − 50000 = ¥198,000.
The best unconstrained integer price was ¥800 with S = ¥200,800, but it gives only 560 attendees and is infeasible. The forgone surplus is ¥2,800. Note that the continuous unconstrained maximum at ¥810 would give ¥200,880, so the cost of the constraint is about ¥2,880; the integer comparison is the relevant one for a vote.
3. A member argues that the sponsor's rule costs only ¥2,800, so the council should charge ¥800 and ignore the sponsor. Evaluate this argument.
Read the worked answer
The ¥2,800 is only the difference in ticket surplus between the feasible ¥750 plan and the infeasible ¥800 plan. It does not include the sponsor's contribution. If the sponsor gives more than ¥2,800, complying is financially better; if it gives less, the council still may have non-financial reasons to keep the agreement. Ignoring the rule also risks losing the sponsor entirely and damaging future relations. The correct comparison is: surplus at ¥750 plus sponsor contribution versus surplus at ¥800 with no sponsor contribution. Without the sponsor's amount, the claim that it costs only ¥2,800 is incomplete. Also, the survey is a model with uncertainty; a price near the 600-attendee boundary is risky because attendance could fall below 600 even at ¥750.