Peak-load pricing
Learning goals
After this page, you can:
- Explain peak-load pricing: when demand presses against a fixed capacity, the price rises until the seats are just filled; off-peak, the usual MR = MC rule sets the price.
- Find the peak and off-peak prices for a fixed capacity from two straight-line demands, and say whether capacity binds.
- Tell peak-load pricing apart from group price discrimination, and explain the fairness constraint: what a price cap does, and which designs reduce backlash.
The idea
The Wednesday before Thanksgiving, a flight to Florida can cost three times what the same flight costs the next Tuesday. Same plane. Same crew. Same fuel. On Wednesday every seat is wanted. On Tuesday half of them fly empty.
Capacity is the most a firm can serve at one time: the seats on a plane, the rooms in a hotel, the lanes on a toll road. In the short run it is fixed. Demand is not. It is high at some times (the peak) and low at others (off-peak).
Peak-load pricing means charging more when demand presses against capacity and less when it does not. The price follows the time and the scarce seat. It does not follow the person.
Off-peak: the usual rule. When there are more seats than people who want them, capacity is not scarce. The firm sets MR = MC, just as on the price-discrimination page. With demand P = a − b·Q, the best quantity is Q = (a − c) ÷ (2b) and the best price is (a + c) ÷ 2. Some seats may fly empty. That is fine. Cutting the price to fill them would lose more on the seats already sold than it gains on the new ones.
Peak: capacity binds. Now suppose the MR = MC rule asks for more seats than the plane has. The firm cannot sell seats it does not have. So it sells every seat, K, and charges the price that just fills them: P = a − b·K. We say capacity binds. The price now does a job: it rations the seats. It decides who flies — the travelers who value the seat most.
A quick check tells you that capacity binds. Find MR at capacity: a − 2b·K. If it is above marginal cost, one more seat would be worth more than it costs. The gap is the value of one more seat. That gap is also the signal for the long run: if it stays large, it may pay to add capacity.
You see the same pattern everywhere capacity is fixed. Hotel rooms cost more on a big game weekend. Movie tickets cost less at a weekday matinee. Electricity can cost more on a hot summer afternoon, when every air conditioner is running. In each case the busy time is when one more unit of capacity is worth the most.
Peak-load pricing is not price discrimination. In group price discrimination, different people pay different prices at the same moment. In peak-load pricing, everyone who flies on Friday evening pays the same fare, business or leisure. The price changes with the time because the seat is scarcer at that time. Surge pricing on ride apps is the real-time version: the price rises when cars are scarce, for every rider in that area.
The fairness constraint. Customers often dislike high peak prices, even when the logic is sound. Suppose a rule caps the peak price below the price that fills the plane. Then more people want seats than there are seats. That gap is a shortage. The seats still get rationed — but by speed, luck, or “first come, first served” instead of by price. The firm also earns less.
Firms reduce backlash without giving up peak pricing:
- Publish the schedule. Toll roads post their peak hours. Theaters post matinee prices. People can plan.
- Limit how far prices rise. A cap on the surge multiple keeps the worst cases away.
- Avoid emergencies. Big price jumps during a storm or a crisis look like gouging, and many places restrict them.
The goal is a price that still rations scarce capacity, under rules that customers can see in advance.
Your company. Does it charge more at busy times — holidays, rush hour, weekends, launch days? That belongs on your Playbook P6 inventory as a peak or surge row. Export the picture from the tool below and paste it next to that row.
Each period: P = a − b·Q, capacity K, marginal cost c · best quantity without a limit Qu = (a − c) ÷ (2b)
If Qu ≤ K: Q = Qu, P = (a + c) ÷ 2 (capacity not binding). If Qu > K: Q = K, P = a − b·K (capacity binds; MR at K = a − 2b·K is above c). Price cap P̄ below the peak price: seats wanted = (a − P̄) ÷ b, shortage = seats wanted − K.
Try it
Two panels: Friday evening (peak) and Tuesday midday (off-peak). The black dashed line is the plane’s capacity.
- Read both prices. On which day does capacity bind? How big is the red bracket — what is one more Friday seat worth above its cost?
- Lower capacity to 80 seats. What happens to the Friday price? To the Tuesday price? Why does only one of them move?
- Turn on Price cap on the peak at $250. How many travelers are turned away? How much profit is lost?
- Turn off the cap and turn on One price for both periods. What happens on Tuesday?
Worked example
Problem. Lakeshore Air (a fictional airline) flies the same 100-seat plane on Friday evening and Tuesday midday. Marginal cost is $40 per passenger. Friday demand is P = 500 − 2·Q. Tuesday demand is P = 200 − 1·Q. What fare should it charge each day?
Step 1. Friday without a seat limit. MR = 500 − 4·Q = 40 gives Q = (500 − 40) ÷ 4 = 115 seats. The plane holds only 100. So capacity binds.
Step 2. Friday fare. Sell 100 seats at 500 − 2 × 100 = $300. Check: MR at 100 seats is 500 − 400 = $100, well above $40. One more seat would be worth $60 more than it costs, but there is none. Profit = (300 − 40) × 100 = $26,000.
Step 3. Tuesday. MR = 200 − 2·Q = 40 gives Q = (200 − 40) ÷ 2 = 80 seats, at 200 − 80 = $120. Twenty seats fly empty. That is fine: capacity is not scarce, so the price only has to satisfy MR = MC. Profit = (120 − 40) × 80 = $6,400.
Step 4. Total. $26,000 + $6,400 = $32,400.
Step 5. One price all week. The best single fare is $300. At that price nobody flies on Tuesday. Profit = $26,000. Two prices earn $6,400 more and fly 80 more passengers.
Step 6. The fairness constraint. Suppose a rule caps Friday at $250. Then (500 − 250) ÷ 2 = 125 travelers want the 100 seats. Twenty-five are turned away by “first come, first served” instead of by price. Friday profit = (250 − 40) × 100 = $21,000, which is $5,000 less.
Why it is not price discrimination. On Friday every traveler pays $300, business or leisure. The price follows the scarce seat, not the person.
Check yourself
Which of these is peak-load pricing rather than price discrimination?
Peak-load pricing follows the time and the scarce capacity, for everyone; price discrimination follows the customer.
A 120-seat Friday flight faces demand P = 560 − 2·Q (seats, dollars per seat). Marginal cost is $50. What fare maximizes profit?
When capacity binds, the price is the one that just fills the seats: P = a − b·K.
The same plane flies on Tuesday, when demand is P = 180 − 1·Q. Capacity is still 120 seats and marginal cost $50. What fare maximizes profit on Tuesday?
Off-peak, capacity is not scarce, so the usual MR = MC rule sets the price, and some seats stay empty.
On the 120-seat Friday flight with demand P = 560 − 2·Q, a rule caps the fare at $260. How many travelers want a seat at $260 and cannot get one?
A cap below the market-clearing price creates a shortage: seats are rationed by luck or speed instead of by price.
A ride app wants the benefits of surge pricing with less customer backlash. Which design does that best?
Peak pricing rations scarce capacity; clear rules and limits keep it from looking unfair.
Video
Video coming soon.