Pricing a 500-Seat Fest: Revenue Target or Profit Target?
A college fest committee must price tickets for a 500-seat auditorium. Last year, at Rs 200 a ticket, they sold 400 tickets. A survey indicates that for every Rs 10 reduction in price, 30 more tickets would be sold, and for every Rs 10 increase, 30 fewer. The hall cannot seat more than 500. The contractor charges a fixed Rs 20,000 plus Rs 30 per person admitted (security, seating, refreshments). The treasurer argues for the price that brings the highest total ticket revenue; the deputy argues for the price that leaves the largest surplus after the contractor is paid. Both agree the price must be a multiple of Rs 10, and that the hall must not be overbooked. Last year's event made a small loss, and the committee needs a defensible number to put before the principal at the next meeting.
1. Define a variable, build the revenue function, and find the price that maximises ticket revenue subject to the seating and pricing constraints.
Read the worked answer
Model. Let x = number of Rs 10 reductions in price (x may be negative for an increase). Price p = 200 - 10x, attendance n = 400 + 30x. Feasibility: 0 <= n <= 500 gives x <= 100/30 = 3.33, and p > 0 gives x < 20.
Revenue. R(x) = (200 - 10x)(400 + 30x) = 80,000 + 6,000x - 4,000x - 300x^2 = 80,000 + 2,000x - 300x^2.
Maximise. R'(x) = 2,000 - 600x = 0 gives x = 3.33; R'' = -600 < 0, so it is a maximum. The continuous maximum is at x = 3.33 (price Rs 166.67, attendance 500, R = Rs 83,333).
Respect the two integer/constraint rules.
- Price must be a multiple of Rs 10, so x must be an integer.
- x = 4 would give 520 tickets, which is over capacity - infeasible, however attractive the arithmetic looks (160 x 520 = Rs 83,200).
- x = 3 gives 490 tickets at Rs 170: R = 170 x 490 = Rs 83,300.
- x = 2 gives 460 x 180 = Rs 82,800; x = 1 gives Rs 81,700; x = 0 gives Rs 80,000.
Answer. Charge Rs 170, sell 490 tickets, take Rs 83,300 in revenue. The continuous optimum is unusable here, and the nearest integer on the other side (x = 4) is ruled out by capacity - both traps must be checked explicitly.
2. Now maximise the surplus after paying the contractor. Show that the answer differs from the revenue-maximising price, and state the final recommendation.
Read the worked answer
Surplus function. S(x) = Revenue - 20,000 - 30n = (80,000 + 2,000x - 300x^2) - 20,000 - 30(400 + 30x) = 60,000 + 2,000x - 300x^2 - 12,000 - 900x = 48,000 + 1,100x - 300x^2.
Maximise. S'(x) = 1,100 - 600x = 0 gives x = 1.83, S'' = -600 < 0 (maximum). Check the neighbouring integers:
- x = 1: p = 190, n = 430, R = 81,700; S = 81,700 - 20,000 - 12,900 = Rs 48,800.
- x = 2: p = 180, n = 460, R = 82,800; S = 82,800 - 20,000 - 13,800 = Rs 49,000.
- x = 3: p = 170, n = 490, R = 83,300; S = 83,300 - 20,000 - 14,700 = Rs 48,600.
Why the two answers differ. Revenue ignores the Rs 30 marginal cost of each admission. Every extra seat adds Rs 30 of cost, so the surplus function is penalised for attendance and its optimum slides towards fewer, dearer tickets (Rs 180, 460 seats) rather than the fullest possible hall. Because revenue and cost both depend on x, the stationarity condition changes and the optimum moves from x = 3.33 to x = 1.83.
Recommendation. Present Rs 180 with 460 tickets and a projected surplus of Rs 49,000 - it beats the revenue-maximising price by Rs 400 and is robust: the surplus changes by only Rs 400 across x = 1 to 3, so a modest survey error does not destroy the plan. Note the capacity constraint is not binding here, and flag that the linear demand model is untested outside x = 0 to 3.
3. A committee member proposes Rs 160 'to fill the hall'. Evaluate this proposal using the model, and state one assumption whose failure would overturn your recommendation.
Read the worked answer
Evaluation of Rs 160. This is x = 4, giving n = 400 + 120 = 520, which exceeds the 500-seat hall - so it cannot be implemented as stated. Capped at 500 seats, revenue would be 160 x 500 = Rs 80,000 and surplus 80,000 - 20,000 - 15,000 = Rs 45,000. That is Rs 4,000 worse than the Rs 180 plan, despite selling 40 more tickets: the extra bodies each cost Rs 30 and each ticket brings in Rs 30 less than at the higher price.
The general lesson. 'Filling the hall' is a revenue metric, not a surplus metric. When marginal cost per admission is positive, maximum attendance is almost never the best outcome; the Rs 180 plan earns Rs 49,000 on 460 seats, while a full 500-seat house at Rs 160 earns Rs 45,000.
Assumption that could overturn it. The linear demand law - each Rs 10 cut buys exactly 30 extra sales - is the fragile part. If demand saturates (say the last 100 seats are hard-core students who would pay Rs 250), or if the survey overstates price sensitivity, the optimum shifts upward. A second assumption is that the Rs 30 per-head cost is genuinely variable; if the contractor's Rs 30 is a flat charge regardless of attendance, the revenue optimum (Rs 170) becomes the surplus optimum too. The committee should therefore present Rs 180 as the base case and quote the Rs 170 figure as the alternative if the per-head charge turns out to be fixed.