Class 8: Price Discrimination I

Textbook reading: Kreps, Section 7.4, “Demand by Groups and in the Aggregate” (pages 166 to 169), then Chapter 8, section 8.1 and section 8.2 up to “Terms and Conditions and Other ‘Quality’ Distinctions” (pages 175 to 182). Read the chapter opening carefully: the two airline puzzles and the senior discount are solved in this class, and the rest are solved in Class 9. Where Kreps writes MR as a formula with ν(p) in it, ν is his symbol for elasticity (ours is $\varepsilon$), and the formula is the Class 7 formula MR = P × (1 + 1/$\varepsilon$) and nothing more.

Key Ideas

Two kinds of customers

A bakery has two types of customers: tourists and locals.

Each group has its own elasticity, so its own marginal revenue
- MR = P × (1 + 1/$\varepsilon$)
A single price sets the average MR equal to MC, so it is wrong for both groups at once
If the bakery can charge tourists a different price than locals, it can earn higher profits
Price discrimination: different prices to different customers for the same good

Fences

In order to charge different prices to different customers, build a fence:
- One group will be willing to climb over it to get lower prices
- The other group won’t bother
Time and effort (Coupons or a January sale)
Demography (senior ID or student ID)
Geography (a second storefront off the highway)

The Resale Problem

A fence has to hold: every scheme invites someone to buy low and resell high (next class).

Problems of the Day

Q22: Rosa’s Bakery

Rosa’s bakery on the coast charges everyone $2.50 a loaf; a loaf costs her $1.20 to make. Tourists buy 120 × (5 − p) loaves a day and locals buy 180 × (3 − p). Her nephew says raise prices, the tourists will pay. Her daughter says cut prices, the locals are leaving for the bakery on Via Roma.

Rosa wants to do both. First she needs a way to charge the two groups different prices for the same loaf. Think of three options for Rosa.

Then find the two prices. Guess a price for one group, find the loaves they buy, compute revenue, subtract cost, and adjust until profit stops rising. Repeat for the other group.

What should Rosa charge each group, and how much more will she earn each day?

Extension: A week later a local is buying forty loaves a morning and selling them on the square at $2.80. What does Rosa do?

Q23: Marco’s Unlabeled Numbers

Marco runs the bakery in the next town and wants Rosa’s trick. He has no demand functions, only a note from his accountant: “Elasticity of demand: −1.5 for one group, −5 for the other.” The accountant forgot to write which group is which. A loaf costs Marco $1.20 too.

Recall that elasticity is the percentage change in quantity demanded for a one percent change in price, and that MR = P × (1 + 1/$\varepsilon$).

Help Marco. Which number belongs to the tourists? What should he charge each group? And since Marco thinks in pictures, sketch what his two demand curves might look like on one graph, price on the vertical axis.

Extension: The accountant finds two more groups, at −3 and −2. Price them too. As a group’s demand becomes more inelastic, what happens to the profit-maximizing price, and what happens as $\varepsilon$ approaches −1?

Q24: Three Puzzles

You own a movie theater and charge everyone $10. You’re wondering whether a senior discount would raise your profits. What has to be true about seniors for the discount to pay off? And once you have a separate senior price, what happens to the price everyone else pays?
In October 2000 a round trip from San Francisco to Chicago leaving Sunday morning and returning Tuesday cost over $2,300. Leaving fifteen hours earlier, on Saturday, it cost under $450. Same seat, same plane home. Who buys each ticket, what keeps them apart, and which group plays the role of the seniors?
In 1991 first class from San Francisco to Tokyo to London and back was cheaper than business class from San Francisco to Tokyo and back. Why would anyone bound for Tokyo not buy the cheaper first-class ticket, and why would the airline want it that way?

Answers

Q22: The fence. Any three of: a loyalty punch card (needs repeat visits), coupons in the local paper (locals clip them, tourists never see them), an end-of-day discount (locals know when to come, tourists have moved on), a resident ID discount, preorder pricing (locals plan ahead), a second counter off the main square, a higher price for the gift-wrapped loaf tourists want. Each asks for time, knowledge, or a credential that locals have and tourists lack. The search. Today, at $2.50, tourists buy 300 loaves and locals 90, so 390 loaves at $1.30 over cost: $507 a day. Tourists alone: $3.00 gives 240 loaves and $432.00; $3.10 gives 228 and $433.20; $3.20 gives 216 and $432.00; so $3.10. Locals alone: $2.00 gives 180 loaves and $144.00; $2.10 gives 162 and $145.80; $2.20 gives 144 and $144.00; so $2.10. Total $579.00, which is $72 a day more than now, about 14%. The nephew and the daughter were both right. Extension. The reseller buys at $2.10 and sells at $2.80, under Rosa’s $3.10, so tourists buy from him and Rosa’s tourist sales fall. Her options: cap purchases per card, make the loyalty card personal and show it at the till, or shrink the gap between the prices until reselling is not worth the morning. The lesson for next class: a fence that can be climbed with a shopping bag does not hold, and resale caps how far apart two prices can sit.
Q23: Which is which. Tourists are −1.5. They cannot escape a high price: one loaf, one afternoon, no idea where the other bakery is. Locals are −5: a 1% price rise loses 5% of their business, because they know Via Roma. The prices. Setting MR = MC with MR = P(1 + 1/$\varepsilon$) gives $P = \dfrac{\text{MC}}{1 + 1/\varepsilon} = \text{MC} \times \dfrac{\varepsilon}{\varepsilon + 1}$. Tourists: 1.20 × (−1.5)/(−0.5) = $3.60. Locals: 1.20 × (−5)/(−4) = $1.50. The sketch. Two lines through a common point, the tourists’ steep and the locals’ flat: a price rise moves the locals a long way along the quantity axis and the tourists hardly at all. Flatter means more elastic, at a common point. (Marco’s prices are not Rosa’s because his customers are not hers; at Rosa’s own prices her tourists have elasticity −1.63 and her locals −2.33, and the same formula gives $3.10 and $2.10.) Extension. −3 gives 1.20 × 3/2 = $1.80 and −2 gives 1.20 × 2 = $2.40. Lining them up: −5, $1.50; −3, $1.80; −2, $2.40; −1.5, $3.60. The more inelastic the group, the higher its price. As $\varepsilon$ approaches −1 the denominator 1 + 1/$\varepsilon$ approaches zero and the price climbs without bound; past −1, into inelastic territory, the formula turns negative, which is the Class 7 result that a profit-maximizer never prices where demand is inelastic.
Q24:
- 1. Seniors must have the more elastic demand: more able to escape a $10 ticket by waiting for the matinee, the discount night, or the streaming release, and more likely to notice $10 on a fixed budget. If so, the single price was a compromise, too high for seniors and too low for everyone else. Once seniors have their own price, the price for everyone else rises: it was being held down to keep seniors in the seats, and now it need not be. A senior discount is a price increase for everyone who is not a senior.
- 1. The Sunday-to-Tuesday ticket is a business trip: the employer pays, the dates are set by the meeting, and nobody gives up their weekend to save the company money, so demand is inelastic. The Saturday ticket is a leisure trip: the traveler pays, could drive or stay home, so demand is elastic. The fence is the Saturday-night stay, which costs the leisure traveler nothing and the business traveler a weekend. Leisure travelers play the seniors; business travelers are everyone else, paying $1,850 more for the same seat.
- 1. A Tokyo-bound traveler cannot use the cheap ticket without flying on to London and back, and airlines forbid throwing away segments, so the itinerary itself is the fence. The airline wants it that way because the two passengers differ in elasticity: the business traveler whose destination is Tokyo has few substitutes and pays the high business-class fare; the passenger going on to London has many ways to get there, so the airline must price low to win them, even in first class. The cheaper first-class ticket is a low price aimed at an elastic group, fenced off from the inelastic one by a routing the inelastic group does not want.