Lab 1: The Market
Optional textbook reading: Varian, Chapter 1, “The Market.”
Q3: Who ends up living near campus?
A landlord owns 5 identical apartments next to campus. Eight students each want one, and each knows the most they’d be willing to pay per month rather than live far away: Ana $500, Ben $450, Cat $400, Dev $350, Eli $300, Fay $250, Gus $200, Hana $150. If the rent is equal to their reservation price, they are willing to rent the apartment. If the rent is over their reservation price, they are not.
The city council doesn’t trust the landlord to decide something as important as who gets to live near campus, so it is debating four schemes:
- The landlord auctions the apartments one at a time to the highest bidder, charging each winner a different rent.
- The landlord must post a single rent, take it or leave it, and rents to whoever accepts.
- Rent is capped at $150 and the five tenants are chosen by lottery.
- First come, first served at a rent of $150, and Gus and Hana happen to camp out overnight.
One thing the council cannot prevent: once tenants are in, students may quietly sublet to each other at any price they agree on, and they will whenever a deal makes both sides happier.
For each scheme, work out who is living in the apartments once all the mutually agreeable subletting has died down, what rents or sublet payments are being paid, and who ended up with the money. Then answer the council’s real question: does its choice of scheme matter? For what, and for whom?
Solutions
Reading the market. Order the reservation prices from highest to lowest: 500, 450, 400, 350, 300, 250, 200, 150. At a rent \(r\), the number of students who want an apartment is the number whose reservation price is at least \(r\). Five apartments are demanded exactly when \(250 < r \le 300\) (at \(r = 300\) Eli is just willing; at \(r = 250\) Fay joins and six students want five apartments). The five highest reservation prices sum to \(500 + 450 + 400 + 350 + 300 = 2000\); this is the total value created whenever the top five are the ones housed, and no assignment can create more.
Scheme 1: the auction. Each apartment goes to the highest remaining bidder, so the winners are Ana, Ben, Cat, Dev and Eli. If the landlord can extract each winner’s full willingness to pay, the rents are 500, 450, 400, 350 and 300, the landlord collects $2000, and the tenants keep no surplus at all. (This is Varian’s discriminating monopolist. In practice, an English auction stops when the runner-up drops out, so prices land near the next-highest remaining bid) No subletting occurs: every outsider values an apartment at $250 or less, every insider at $300 or more, so there is no price both sides would accept.
Scheme 2: a single posted rent. The landlord picks the rent that maximizes revenue given the demand schedule:
| Rent | Students willing | Apartments rented | Revenue |
|---|---|---|---|
| 500 | 1 | 1 | 500 |
| 450 | 2 | 2 | 900 |
| 400 | 3 | 3 | 1200 |
| 350 | 4 | 4 | 1400 |
| 300 | 5 | 5 | 1500 |
| 250 | 6 | 5 | 1250 |
The revenue-maximizing rent is $300, which is also a market-clearing rent, so nothing is withheld and there is no shortage. Tenants are again Ana through Eli. The landlord collects $1500; the tenants keep surplus of 200, 150, 100, 50 and 0 respectively, $500 in total. Again no sublet is mutually agreeable.
Scheme 3: rent cap and lottery. Five names are drawn at random and each pays $150, so the landlord collects $750 regardless of who is drawn. Now the subletting matters. Suppose Hana holds a lease and Ana does not. Hana values living there at $150 and pays $150, so she is indifferent about staying; Ana would pay up to $500. Any sublet price between $150 and $500 makes both better off, so a deal happens. The same logic applies to any leaseholder whose reservation price is below that of some student outside, and chains of sublets are possible (Eli takes Ana’s payment and then rents from a lower-value winner, and so on). Trading stops only when no outsider values an apartment more than any insider, which means the final residents are Ana, Ben, Cat, Dev and Eli, whoever the lottery picked. The money: $750 to the landlord, and the remaining $1250 of value split between the five lucky winners and the five final residents, in proportions that depend on who was drawn and on how each pair bargains over the sublet price. A low-value winner such as Hana can pocket up to $350 of pure lottery profit.
Scheme 4: queueing. Gus and Hana take two apartments at $150; the other three go to whoever was next in line. As in scheme 3, the landlord collects $750, and every leaseholder whose reservation price is below an outsider’s sublets. Gus sublets to some higher-value student at a price between $200 and that student’s reservation price; Hana does the same at a price of $150 or more. The five final residents are once more Ana through Eli. The queue has replaced luck with effort as the way to earn a sublet margin, and the night spent camping out is a real cost that the lottery did not impose.
Does the council’s choice matter? Not for who ends up living near campus, and not for the total value created. In every scheme the same five students are housed and the value is $2000, because subletting reallocates each apartment to whoever values it most until no mutually beneficial trade remains. The final allocation is Pareto efficient under all four schemes. What the scheme decides is the division of that $2000:
| Scheme | Landlord | Final residents | Others |
|---|---|---|---|
| 1. Auction (full price discrimination) | 2000 | 0 | nobody |
| 2. Posted rent of $300 | 1500 | 500 | nobody |
| 3. Rent cap and lottery | 750 | depends on bargaining | lottery winners, up to $1250 in total |
| 4. Rent cap and queue | 750 | depends on bargaining | early arrivals, up to $1250 in total |
Two points worth drawing out with students. First, the conclusion leans on the assumption that subletting is free and frictionless. If the council could actually stop it, schemes 3 and 4 would leave the apartments with lower-value tenants (Gus and Hana instead of two of the top five) and value would be lost. Second, the rent cap does not lower the price that the high-value students end up paying; it changes who receives that payment, from the landlord to whichever student got in first.
Check Your Understanding
The Market
Before the trading starts, make sure you can read the market itself: what a reservation price is, how many students want in at each rent, and which rents clear the market with all five apartments full. These questions work through schemes 1 and 2, the auction and the posted rent.
Allocation Schemes
Now let the subletting begin. These questions work through schemes 3 and 4, the rent cap with a lottery and the overnight queue, and then push on the lab’s big question: what the council’s choice can and cannot change, and why every scheme keeps landing on the same five names.
Now check yourself: