Ferry Concession: Capacity-Constrained Fares for Residents and Tourists
A private operator holds the ten-year concession for the only ferry route to a 9,000-resident island. Ticketing data give well-identified demand curves: residents Q_R = 1200 − 40P and tourists Q_T = 800 − 20P, with Q in passenger-crossings per day and P in euros. Each crossing costs €4 per passenger in fuel and crew, and the fixed daily cost of running the timetable is €2,000. Each vessel is certified for 700 passengers, and the pier can handle no more sailings, so 700 crossings per day is a hard capacity limit for the foreseeable future. The current single fare is €12, and every sailing departs full with a waiting list, which management reads as proof that the fare is far too low. Management proposes charging residents €20 and tourists €25 from next season and has asked you to test the proposal. A regulator must approve any fare schedule and has publicly signalled concern about price gouging of islanders.
1. Compute the profit-maximising single fare given the 700-passenger capacity, and the resulting daily profit. Explain why the fare a textbook monopolist would set is not available to this operator.
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Aggregate demand. For prices where both groups travel, Q = (1200 − 40P) + (800 − 20P) = 2000 − 60P, i.e. inverse demand P = 33.33 − Q/60 and MR = 33.33 − Q/30.
Unconstrained monopoly. MR = MC gives 33.33 − Q/30 = 4, so Q = 880 and P = €18.67. But 880 > 700, so this price cannot be honoured: the operator would have to ration, and rationing at a fare below market-clearing invites queues and resale.
Capacity-constrained fare. With capacity binding, the fare is set where demand equals 700: 2000 − 60P = 700, so P = €21.67 (residents 333.3, tourists 366.7).
Profit. 700 × (21.67 − 4) − 2,000 = €12,366.7 − €2,000 = ≈ €10,367 per day.
The point of the exercise is that scarcity, not the demand elasticity formula alone, sets the fare: the binding constraint shifts the operator up the demand curve, and the waiting list is the symptom of a price below the capacity-clearing level, not of a pricing error in the elasticity sense.
2. Test management's proposal (residents €20, tourists €25). Show that it is or is not optimal under the capacity limit, and quantify the profit difference against the single fare.
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Unconstrained third-degree pricing. Resident inverse demand P = 30 − Q/40, so MR_R = 30 − Q_R/20. Setting MR_R = 4 gives Q_R = 520, P_R = €17. Tourist inverse demand P = 40 − Q/20, so MR_T = 40 − Q_T/10. Setting MR_T = 4 gives Q_T = 360, P_T = €22. Total demand 880 again exceeds 700, so the capacity constraint binds.
Constrained optimum. Allocate the 700 seats so marginal revenue is equalised: 30 − Q_R/20 = 40 − (700 − Q_R)/10. This solves to Q_R = 400, Q_T = 300, hence P_R = 30 − 10 = €20 and P_T = 40 − 15 = €25, with common MR = €10 > MC = €4, confirming capacity binds.
Verdict. Management's numbers are exactly the constrained optimum. Profit = 400 × (20 − 4) + 300 × (25 − 4) − 2,000 = 6,400 + 6,300 − 2,000 = €10,700, which is €333 per day more than the single fare - a 3.2% gain, not the large windfall the proposal implies. The gain is small precisely because capacity, not willingness to pay, is the binding constraint.
3. The regulator can impose a single fare and will require the operator to carry up to capacity. Judge whether uniform pricing raises or lowers total surplus relative to the discriminatory scheme, and identify who gains and who loses.
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Under discrimination (P_R = 20, Q_R = 400; P_T = 25, Q_T = 300):
- Total willingness to pay = ∫₀⁴⁰⁰(30 − Q/40)dQ + ∫₀³⁰⁰(40 − Q/20)dQ = 10,000 + 9,750 = 19,750.
- Variable cost = 700 × 4 = 2,800, so surplus before fixed cost = 16,950.
- Consumer surplus: residents ½ × 400 × 10 = 2,000; tourists ½ × 300 × 15 = 2,250; total 4,250.
Under the uniform fare €21.67 (Q_R = 333.3, Q_T = 366.7):
- Willingness to pay = 8,611 + 11,306 = 19,917; surplus before fixed cost = 17,117, i.e. ≈ €167 per day higher.
- Consumer surplus: residents 1,389; tourists 3,361; total 4,750.
Judgement. With capacity fixed, discrimination misallocates seats: the marginal resident admitted has a willingness to pay of only €20, while tourists excluded at the margin would pay up to €25. Uniform pricing therefore raises total surplus slightly, despite lowering profit from €10,700 to €10,367. Distributionally, however, residents pay €21.67 instead of €20 and lose €611 of surplus, while tourists gain €1,111 - the regulator's uniform fare harms precisely the group it intends to protect.