Concert Pricing: Elasticity, Profit Maximization, and a Block-Booking Offer
An independent concert venue has 1,200 seats and is planning one show. Fixed costs for the night-rent, insurance, and the artist's guarantee-total $18,000. Each attendee costs the venue $8 in ticketing, security, and staffing, and this marginal cost is roughly constant up to capacity. From past shows, the manager estimates ticket demand: at $48, attendance is 600; at $40, attendance is 800; at $32, attendance is 1,000. The manager currently prices at $32 to 'fill more seats' and points to the higher attendance as evidence of success. The board wants a profit-maximizing price, not a revenue-maximizing one, and asks whether the venue should ever leave seats empty. A local promoter offers to buy 200 seats at $25 each, but only if the venue does not discount to anyone else; the venue would still sell its other tickets at the chosen price. The promoter's audience is a separate group that would not attend at the regular price.
1. Compute the price elasticity of demand between $40 and $32 using the midpoint formula. Is demand elastic, unit elastic, or inelastic? What does the result imply about the manager's claim that higher attendance proves success?
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Use the midpoint formula between $40 and $32. Quantity changes from 800 to 1,000, so the average quantity is 900 and the percentage change is 200/900 = 22.2%. Price changes from $40 to $32, so the average price is $36 and the percentage change is -8/36 = -22.2%. Price elasticity of demand = 22.2% / (-22.2%) = -1.0, i.e., unit elastic. Total revenue is $40 x 800 = $32,000 and $32 x 1,000 = $32,000, unchanged, which is consistent with unit elasticity. Unit elasticity means the percentage change in quantity equals the percentage change in price, so cutting the price from $40 to $32 does not raise revenue. The manager's inference from higher attendance is wrong because revenue and profit are not attendance. To choose a profit-maximizing price, the venue must compare marginal revenue with the $8 marginal cost.
2. Assume the three demand points are linear. Derive the demand and marginal revenue curves, find the profit-maximizing price and quantity, and compare profit at that price with profit at the revenue-maximizing price and at $32.
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The three demand points are linear: price falls $8 for every 200 tickets, so the slope is -8/200 = -0.04. Using (800, 40), P = 72 - 0.04Q. Total revenue is P x Q = 72Q - 0.04Q^2, so marginal revenue is MR = 72 - 0.08Q. Profit is maximized where MR = MC: 72 - 0.08Q = 8, so 0.08Q = 64, Q = 800, and P = 72 - 0.04(800) = $40. Profit at Q = 800 is $32,000 - $8(800) - $18,000 = $7,600. At the revenue-maximizing point, MR = 0 gives Q = 900 and P = $36; revenue is $32,400, but profit is $32,400 - $7,200 - $18,000 = $7,200, which is less. At $32, profit is $32,000 - $8,000 - $18,000 = $6,000. The venue should leave seats empty: at $40 it sells 800 of 1,200 seats, and filling more seats would require a price cut that lowers profit because MR falls below MC.
3. Should the venue accept the promoter's offer of 200 seats at $25 each? Evaluate using incremental profit, capacity, and the 'no discount to anyone else' condition.
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Evaluate the promoter's offer on incremental profit. The venue would sell 200 extra seats at $25, earning $5,000 in added revenue. The added cost is 200 x $8 = $1,600, so incremental profit is $3,400. Capacity is 1,200, and current sales are 800, so the 200 seats do not displace regular customers. The condition that the venue not discount to anyone else constrains offering lower prices to other buyers, but it does not change the incremental calculation for this block. Accept if the promoter's group truly would not buy at $40 and if the block sale does not create capacity problems or undermine the $40 public price. If filling those seats required lowering the price for all 800 existing buyers, the calculation would be very different: revenue would fall by $15 x 800 = $12,000, far outweighing the gain. The key distinction is segmented pricing versus a uniform price cut.