Two-part pricing and bundling
Learning goals
After this page, you can:
- Explain a two-part tariff, and why with similar customers the best plan sells each unit at marginal cost and sets the fee equal to one customer’s consumer surplus.
- Compute the fee from a straight-line demand, Fee = (a − c)² ÷ (2b), and compare it with the best single price.
- Use a reservation-price table to compare separate prices, a pure bundle, and a mixed bundle, and explain when bundling pays and when it adds nothing.
The idea
A warehouse club makes you pay a yearly fee before you can buy anything. Inside, it sells at thin markups. A fast-food counter sells a burger, fries, and a drink one by one — and also as a value meal for less than the three together. Both are ways to collect more of what customers would pay than one price per item can.
Why one price leaves money behind. With one price per unit, the seller faces a trade-off. A high price earns a big markup but sells few units. A low price sells many units but earns little on each. Many customers would have paid more than the price for their first few units. That extra value is their consumer surplus, and one price cannot reach it.
Two-part tariff. A two-part tariff has two parts: an entry fee you pay to get in, and a per-unit price for each unit you buy. Gyms, warehouse clubs, phone plans, and theme parks with an entry ticket plus a price per ride all work this way.
How should the seller set the two parts? Take customers who are all alike. Each customer has the demand line P = a − b·q, and marginal cost is c.
- Price each unit at marginal cost: per-unit price = c. At that price the customer buys q = (a − c) ÷ b units. The seller makes no profit on the units themselves. That is fine.
- Set the fee equal to the customer’s consumer surplus at that price. That surplus is the triangle between the demand line and the price: ½ × (a − c) × q = (a − c)² ÷ (2b). The customer is just willing to pay it, because the units are worth that much to them above what they pay per unit.
Why put the per-unit price at cost and not above it? A lower per-unit price makes the triangle bigger, and the fee collects the whole triangle. Any markup per unit shrinks the triangle by more than the markup brings in. So the best plan makes the triangle as big as it can be, and then takes it as the fee.
How much better? On a straight demand line, the best single price is (a + c) ÷ 2, and the profit per customer is (a − c)² ÷ (4b). The two-part plan earns (a − c)² ÷ (2b). That is exactly twice as much. In real life customers are not all alike, so the fee cannot be set for each person. Firms then offer a short menu of plans, which is close to nonlinear pricing.
Bundling. A bundle sells two or more products together for one price. A pure bundle sells only the package. A mixed bundle offers the package and each item alone.
To compare them, use a reservation-price table: the most each customer type would pay for each good. Then use one rule everywhere: a customer buys if their value is at least the price.
- Separate prices: for each good, try each listed value as the price. Revenue = price × number of customers whose value is at least the price. Keep the best price. Do the same for the other good, and add.
- Pure bundle: add each customer’s two values first. Then do the same search on those sums.
- Mixed bundle: keep prices for the items alone and also offer the bundle. Each customer picks the option that leaves them the most value (value minus price), or nothing.
When bundling pays. Bundling helps when tastes are opposite: one customer loves good 1 and does not care about good 2, another is the reverse. Adding the values evens them out, so one bundle price can capture most of what everyone would pay. When the same customers value both goods highly, adding the values evens out nothing, and the bundle earns no more than separate prices. Mixed bundling goes one step further. The customers who want both buy the bundle. The customers who want only one item still buy it alone, at close to their full value, instead of being handed something they do not care about.
Your company. Look for a fee-plus-price plan (a membership, a subscription, a pass) and for any bundle or “plan” that packages several services. The deal designer below turns one of them into a single row for your Playbook P6 post.
Two-part tariff (member demand P = a − b·q, MC c): per-unit price = c · q = (a − c) ÷ b · Fee = ½ × (a − c) × q = (a − c)² ÷ (2b)
Best single price (a + c) ÷ 2, profit (a − c)² ÷ (4b) → the two-part plan earns twice as much. Bundles: a customer buys if value ≥ price; try each listed value as the price and keep the best.
Try it
Design one deal for your company — a membership, a pass, or a bundle. The Lab does the same arithmetic for a regional airline.
- In Two-part tariff mode, raise the marginal cost from $50 to $100. What happens to the fee? Is the plan still twice the best flat price?
- Switch Start from to A typical customer. Keep p = $150, n = 12.5, |ε| = 1.5. Do you get the same fee as the demand line 250 − 8·q?
- Switch to Bundle mode. Change the Executive’s value of a checked bag from $0 to $45. Does the pure bundle still beat separate prices? Why not?
- Click Use my numbers, type your guess for n, then Export PNG and Copy as table. That row goes at the end of your P6 post.
Worked example
Part 1. The Shuttle Pass (two-part tariff). Lakeshore Air (a fictional airline) sells a yearly pass. A regular flyer’s demand is P = 250 − 8·q flights a year. Marginal cost is $50 per flight.
Step 1. Per-unit price at cost. Charge $50 a flight. She flies q = (250 − 50) ÷ 8 = 25 times.
Step 2. The fee. Her consumer surplus is ½ × 200 × 25 = $2,500. That is the yearly fee. Profit per member is $2,500 — the flights themselves just cover their cost. Revenue per member is $2,500 + 25 × $50 = $3,750.
Step 3. Compare with one price. The best single fare is (250 + 50) ÷ 2 = $150. She flies 12.5 times a year. Profit = (150 − 50) × 12.5 = $1,250 — half as much. Revenue = $150 × 12.5 = $1,875.
Part 2. Lounge pass (L) and checked bag (B). Marginal cost is $0 to keep it simple. Three traveler types, one of each:
| Traveler | Lounge pass | Checked bag | Both |
|---|---|---|---|
| Executive | $60 | $0 | $60 |
| Family | $40 | $45 | $85 |
| Skier | $10 | $55 | $65 |
Step 4. Separate prices. Lounge at $60: 1 buyer, $60. At $40: 2 buyers, $80. At $10: 3 buyers, $30. Bag at $55: 1 buyer, $55. At $45: 2 buyers, $90. Separate total $170.
Step 5. Pure bundle. Bundle values are 60, 85, and 65. At $85: 1 buyer, $85. At $65: 2 buyers, $130. At $60: 3 buyers, $180.
Step 6. Mixed bundle. Lounge alone $60, bag alone $55, both $85. The Executive buys the lounge ($60). The Family buys both ($85). The Skier buys the bag ($55). Total $200.
Why mixed wins. The Family values both and pays for both. The two travelers who want only one item still pay close to their full value for it, instead of getting a bag or a lounge they do not care about.
Check yourself
A regional airline’s commuter pass: each member’s demand is P = 200 − 10·q flights a year, and marginal cost is $40 per flight. The airline charges $40 per flight. What is the best yearly fee?
Two-part tariff: per-unit price = marginal cost, fee = one customer’s consumer surplus = (a − c)² ÷ (2b).
Why does a two-part tariff for similar customers set the per-unit price at marginal cost rather than above it?
Price each unit at cost to make the surplus as large as possible, then take it as the fee.
Three travelers value seat selection and priority boarding as follows (MC $0): A $25 and $5; B $15 and $15; C $5 and $25. What is the best revenue from selling the two only as a bundle?
For a pure bundle, add each customer’s values first, then try each bundle value as the price.
Traveler X values a checked bag at $40 and a seat upgrade at $30. Traveler Y values them at $20 and $10 (MC $0). Compared with the best separate prices, how much more does a pure bundle earn?
Bundling pays when tastes are opposite; when the same customer values both goods highly, it has nothing to even out.
An airline sells a lounge pass for $60, a checked bag for $55, and both together for $85. Why keep the separate prices instead of selling only the bundle?
Mixed bundling offers the bundle and the separate items, so each customer type picks the option that fits what they value.
Video
Video coming soon.