Class 9: Price Discrimination II
Textbook reading: Kreps, Chapter 8, from “Nonlinear pricing” (page 182) to the end, including sections 8.3, 8.4, and 8.5. Also read Problem 8.5 on pages 193 to 194, through the three consumers’ value tables; Q25 uses two of those consumers. Kreps says the Disneyland example needs R or Excel; in class we do a five-visitor version by hand.
Key Ideas
When the firm cannot tell customers apart
- Last class the firm could see or fence its groups; now it cannot
- It posts one menu for everyone and designs it so that customers sort themselves
- Three menus: nonlinear prices, bundles, versions
Nonlinear pricing and the two-part tariff
- Nonlinear pricing: the total paid for x units is not a price times x (quantity discounts, rising block rates)
- Two-part tariff: an entry fee F, then a per-unit price p (Costco, gyms, printers and ink)
- A consumer described by a table of values buys every unit whose incremental value is at least p, and pays the fee only if her net benefit covers it
Bundling
- Bundling: selling two goods together for less than the sum of their separate prices
- Example: a two-day park pass sold alongside single-day tickets
- Whether a bundle pays depends on how customers’ values for the two goods line up
Versioning
- Sell a high and a low version (hardcover and softcover, Pro and Basic) and let each customer choose
- The binding constraint: the high-value group must prefer its own version to the cheap one
Problems of the Day
Q25: The Fee Haters
Costco charges a membership fee and then sells at cost. A rival, Ralph’s, hates fees: free entry, one price per case. Suppose one shopper values 1, 2, 3, 4, and 5 cases of sparkling water per month at $30, $50, $60, $62, and $62.50 in total, and each case costs $4 to supply. Each store sets its terms to earn as much as it can from her.
What does Costco set its membership fee to be, and how much does Costco earn from her? What does Ralph’s set is price to be, and how much does Ralph’s earn from her?
Q26: The Two-Day Pass
A park charges $60 a day, two days running, and each visitor costs it $20 a day. The marketing intern proposes a two-day pass at $110. The CFO objects: “Anyone who comes both days already pays $120. You’re handing them $10.”
Five potential visitors value the two days at:
| Day 1 | Day 2 | |
|---|---|---|
| Ann | $90 | $55 |
| Ben | $45 | $95 |
| Cal | $70 | $70 |
| Dee | $58 | $58 |
| Eve | $30 | $30 |
Is the CFO right? Find the pass price that makes the park the most money. Then find the single-day price that makes the park the most money. Does the pass price earn the park higher profits?
Q27: The Binding That Fails
A publisher sells a textbook to 1,000 libraries (a hardcover is worth $150 to them, a softcover $90) and 5,000 students (hardcover $60, softcover $55). A hardcover costs $11 to make, a softcover $10. The sales team plans hardcover at $150, softcover at $55.
A designer proposes a softcover whose binding fails after fifty loans: libraries would value it at $40, students still $55, same $10 cost. The sales team calls it vandalism.
How much is the vandalism worth?
Finally: what needs to be true about a customer base to make versioning profitable?
Answers
Q25: Ralph’s. Her incremental values are $30, $20, $10, $2, $0.50, and she buys every case worth at least the price. At $30 she buys 1 case, profit $26; at $20 she buys 2, profit $32; at $10 she buys 3, profit $18. The best price for Ralph’s is $20, earning $32. Costco. Price at cost, $4: she buys 3 cases (the fourth is worth only $2), worth $60 to her, and pays $12, so she keeps $48. A fee of $48 takes all of it, and she still joins. Costco earns $48, half again more than Ralph’s. The fee collects what a per-unit price cannot: everything the earlier cases were worth above the last one.
Q26: Today. At $60, day 1 draws Ann and Cal, day 2 draws Ben and Cal: four visits at $40 over cost, $160. The pass at $110. Cal was paying $120 for two days and now pays $110: the CFO’s $10, lost. Ann came one day for a net benefit of $30; the pass gives her $145 − $110 = $35, so she takes it, and the park collects $50 more and spends $20 more: +$30. Ben likewise: +$30. Dee came neither day, but $116 clears $110: +$110 − $40 = +$70. Eve stays home. Net +$120, profit $280. The CFO is right about Cal and wrong about the pass: it is aimed at the people who were coming once or not at all.
Q27: The plan fails. Libraries get $150 − $150 = $0 from the hardcover and $90 − $55 = $35 from the softcover, so they buy the softcover. Students get nothing from the hardcover and $0 from the softcover, and buy the softcover. Everyone buys the $55 book: 6,000 × $45 = $270,000. The vandalism. With the failing binding a library’s softcover is worth $40, below its $55 price, so the softcover no longer tempts it and the hardcover returns to $150. Profit 1,000 × $139 + 5,000 × $45 = $364,000. The vandalism is worth $94,000 against the sales team’s plan. Making the cheap version worse for the premium buyer removes the temptation that was capping the premium price, and costs the cheap version’s intended buyer nothing.