What is the equilibrium price in this case?

Question

From the Review Questions to Chapter 1: The Market in Hal Varian's Intermediate Microeconomics with Calculus, 9th ed.:

1. Suppose that there are 25 people who had a reservation price of $\\\$500$, and the 26th person had a reservation price of $\\\$200$. What would the demand curve look like?
2. In the above example, what would be the equilibrium price if there were 24 apartments to rent? What if there were 26 apartments to rent? What if there were 25 apartments to rent?

Please note that in this chapter, the supply of apartments is taken to be fixed.

I have drawn the demand curve, and attempted to solve the first two cases in the second question in the following way:

If there are 24 apartments, they go to the 24 people with the highest reservation price. This happens to be $\\\$500$, so that is the equilibrium price in this case.

When there are 26 apartments, we can imagine 25 landlords successfully leasing their property for $\\\$500$, and one landlord who is unable to net the $\\\$500$ lowering his price until he reaches $\\\$200$. This prompts a surge of potential tenants at his doorstep, but the apartment does not necessarily go to the tenant who had a reservation of price of $\\\$200$; it goes to whomever spends most time bargaining with the landlord, has the best connections, etc. Since this apartment generally goes to someone who had a reservation price of $\\\$500$, the new tenant vacates an apartment in her wake, which is then leased for $\\\$200$, and so on until everybody is living in a $\\\$200$ apartment.

The third case is confusing. The answer at the end of the book says that the equilibrium price would be "any price between $\\\$200$ and $\\\$500$" when there are 25 apartments. Why is this true? It seems to me that since the supply of apartments is fixed, the price of one apartment would depend only on the reservation price of tenants. If the equilibrium price is anything lower than $\\\$500$, landlords would have an incentive to arbitrarily increase the rent to the highest the market can absorb, i.e., $\\\$500$. What's keeping the equilibrium price below $\\\$500$?

Please also tell me if my reasoning about the other two cases is flawed, or there are other, better ways to reason about them.

Answer 1

I have quickly skimmed the chapter and there does not seem to be much about bargaining models so as the question says you have to solve this using supply and demand.

We know that an equilibrium price on the market will occur when supply intersect demand. Given that the supply is fixed at $24$ and $25$ apartments respectively it will be just straight vertical line in each case. Now when it comes to demand, if we assume each person wants only one single apartment the demand will be given in step-wise fashion, with two steps given by reservation prices ${\\\$}500$ and ${\\\$}200$ respectively. So this situation will look like shown on the tikz picture I made below (supply is blue, demand is red):

At point where supply is equal to $24$ the equilibrium price given by intersection of supply and demand ($S=D$) is clearly ${\\\$}500$. However, when supply is equal to $25$, supply and demand will intersect at a whole range of different prices $[200,500]$.

Consequently, in this case using this simple supply and demand model we cannot say anything more that price will be something between ${\\\$}200$ and ${\\\$}500$. There are more advanced ways how we could model this situation which might yield unique price instead of range (maybe some models with explicitly modeled bargaining process or some matching models), but here the author of the book gives you hints by asking how demand would look like etc., that you are supposed to use this solution concept (also the first chapter uses only simple supply and demand models so that is another hint).

Answer 2

With regard to situation with 25 apartments: The reservation prices of renters is not known in advance. The only price signals a landlord will get happen at the extremes. So if a landlord lists a price of 501, he finds that nobody will rent. So he rents below that. And if he rents at 200, he has excess demand, so he will rent above that. So the market finds a way to end up at say 201 to 499 inclusive (ignore partial dollars for now). If all the landlords rent at 300, there is no excess demand and no shortfall in demand, so everything is rented. No reason to do swaps. Yes, the landlords COULD have charged 499, but they don't know that since at 300 there was no excess demand.

With regard to situation with 26 apartments: Varian should have specified whether or not there was a 27th person here. If you have 26 people in the world and 26 apartments, they will rent out at 0<=rent<=200. Varian's answer which is $200 implies a 27th person in the mix.