Lab 5: Designing a Menu
Textbook reading: Kreps, Chapter 8, especially “Nonlinear pricing” (page 182) and “Terms and Conditions and Other ‘Quality’ Distinctions” (page 181), plus the Class 9 page. This lab assumes you have had Classes 8 and 9. Summit’s Membership is Q25 at a whiteboard; Ledger’s Plan is Q27 at a whiteboard.
Problem of the Day
Groups of three at a whiteboard. The TA presents the question. Groups work for forty minutes; the last ten minutes are a room-wide comparison.
Summit’s Membership
Summit Climbing has 100 Regulars and 400 Casuals. A visit costs Summit $4. Each climber’s value of each visit in a month:
| Visit | Regular | Casual |
|---|---|---|
| 1st | $20 | $12 |
| 2nd | $16 | $8 |
| 3rd | $12 | $4 |
| 4th | $8 | |
| 5th | $4 |
What’s the profit-maximizing price for a per-visit pass?
The owner is thinking of adding a monthly fee on top of a per-visit pass. What’s the profit-maximizing menu:
- Monthly fee of $40 and a per-visit charge of $4
- Monthly fee of $12 and a per-visit charge of $4
- Monthly fee of $4 and a per-visit charge of $8
Intuitively, explain your choice in terms of how it influences regulars and casual members differently.
Ledger’s Plan
Ledger sells accounting software to 200 Firms and 800 Freelancers, one copy each. Pro costs Ledger $20 a copy to support, Basic $10. What each type would pay:
| Pro | Basic | |
|---|---|---|
| Firm | $500 | $200 |
| Freelancer | $150 | $120 |
Today Ledger sells Pro only, at $150, and everyone buys. It plans to add Basic and price each version at what its intended buyer would pay: Pro at $500 for Firms, Basic at $120 for Freelancers. A buyer takes the version with the higher net benefit, buys nothing if neither is worth its price, and when indifferent takes the more expensive version.
Will the plan work? If not, find the profit maximizing price menu.
Review
Write the condition every menu today had to satisfy in one sentence.
Extensions (optional)
A third type joins Summit: 200 Occasionals who value their first two visits at $8 and $4. Does your tariff still work? Which type’s value kept pins the fee now?
Ledger considers a third version, Plus, valued at $300 by Firms and $140 by Freelancers, costing $15 to support. Can a three-version menu beat your best two-version profit?
Summit’s cost per visit falls to $2. Which way do the best per-visit price and fee move?
Solutions
Published after the last section has met.
Summit’s Membership. Today, at $10, a Regular makes 3 visits (worth $20, $16, $12), pays $30, keeps $18; a Casual makes 1 visit, pays $10, keeps $2. Profit ($10 − $4) × (300 + 400) = $4,200. Per-visit pricing alone peaks at $12 (Regulars 3 visits, Casuals 1 at exactly their value, still coming): $8 × 700 = $5,600. Of the three menus: the $40 fee with $4 visits is a Regular’s value kept at that visit price (5 visits worth $60, pays $20); Casuals keep only $12, so they leave, and profit is 100 × $40 = $4,000, below today. The $12 fee with $4 visits keeps everyone: all 500 join, no margin on visits, $6,000. The $4 fee with $8 visits: Regulars make 4 visits (worth $56, pay $32, keep $24) and Casuals 2 (worth $20, pay $16, keep $4), so a $4 fee keeps everyone; fees 500 × $4 = $2,000 plus a $4 margin on 1,200 visits = $4,800, total $6,800, the best of the three. It beats the $12 fee because the fee falls $8 on all 500 members (−$4,000) while the margin collects $4 on 1,200 visits (+$4,800); Regulars make twice the visits, so the margin lands mostly on them. The sentence for the owner: the fee has to be one the lightest member you want will pay, and the per-visit price, not the fee, is how you charge the heavy users more.
Ledger’s Plan. Firms: Pro $500 − $500 = $0, Basic $200 − $120 = $80, so they take Basic. Freelancers: Pro negative, Basic $0, so they take Basic. Everyone buys Basic: 1,000 × $110 = $110,000, below the $130,000 Ledger earns today with Pro at $150 for all. The plan fails because the premium buyers prefer the cheap option. Price fix: hold Basic at $120 and cut Pro until a Firm prefers it, $500 − P ≥ $80, so P = $420; profit 200 × $400 + 800 × $110 = $168,000. The $80 cut per Firm is the Firm’s net benefit from Basic.
Review. Each type must prefer the option meant for it. When that fails, lower the premium price until it holds, or make the cheap option worse for the premium type.
Extensions. Occasionals at $8 a visit make 1 visit worth $8 and keep $0, so the $4 fee loses them; dropping the fee to $0 to keep them collapses the tariff to per-visit pricing ($4 × 1,400 = $5,600). Keeping F = $4 and losing them still earns $6,800, so the Casual’s value kept still pins the fee. Plus: a Firm prefers Pro to Plus only if the Pro price is at most $200 above the Plus price, so any Plus cheap enough for Freelancers ($140) pulls Firms off Pro unless Pro falls to $340, and 200 × $320 + 800 × $125 = $164,000 is below $184,000; Plus cannot help. Cost $2: at $4 a visit with F = $12 everyone joins, fees $6,000 plus $2 on 1,700 visits, $9,400; at $8 with F = $4, $2,000 + $6 × 1,200 = $9,200. The best per-visit price falls from $8 to $4 and the fee rises from $4 to $12: a cheaper visit means the gym wants members making more of them, and collects through the fee instead.