Why is the equilibrium price not anything between \$0 and \$200?

Question

Consider the second problem below from Varian’s Intermediate Microeconomics:

Suppose that there were 25 people who had a reservation price of \$500, and the 26th person had a reservation price of \$200.

In the above example, what would the equilibrium price be if there were 24 apartments to rent? What if there were 26 apartments to rent? What if there were 25 apartments to rent?

For 24 apartments, the equilibrium price is \$500, and for 25, it’s \$200–\$500. So far, the answers in the back agree. However, for 26 apartments, I think the the equilibrium price should be \$0–\$200, whereas the book just says \$200 flat. I draw the supply-demand curve like this, where the blue line represents supply and black is demand:

Here’s a screenshot of the answers for reference:

Is the textbook correct, and if so, what am I missing here?

Answer 1

Varian is indeed wrong here in that there is a critical omission: He should have specified whether there in a 27th person and what this person's reservation price is.

• If there is no 27th person, then the set of possible equilibrium prices is $[\$0,\$200]$, as you correctly point out. Explanation:

At each $p\in(\$0,\$200)$, we have $S(p)=26$ and $D(p)=26$, so that there is a possible equilibrium.

At $p=\$200$, we have $S(p)=26$ and $D(p)=\{25,26\}$, so that there is a possible equilibrium.

At $p=\$0$, we have $S(p)=\{0,1,\dots,26\}$ and $D(p)=26$, so that there is a possible equilibrium.

At any $p>\$200$, we have $S(p)=26$ and $D(p)\leq25$, so that there is no possible equilibrium.

• If there is a 27th person and her reservation price is $k<\$200$, then the set of possible equilibrium prices is $[k,\$200]$. Explanation:

At each $p\in(\$k,\$200)$, we have $S(p)=26$ and $D(p)=26$, so that there is a possible equilibrium.

At $p=\$k$, we have $S(p)=26$ and $D(p)=\{26,27\}$, so that there is a possible equilibrium.

At $0<p<k$, we have $S(p)=26$ and $D(p)=27$, so that there is no possible equilibrium.

At $p=\$0$, we have $S(p)=\{0,1,\dots,26\}$ and $D(p)=27$, so that there is a possible equilibrium.

At any $p>\$200$, we have $S(p)=26$ and $D(p)\leq25$, so that there is no possible equilibrium.

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Notes.

1. I assume throughout, as I believe Varian has, that each apartment-owner's reservation price or willingness-to-accept is \$0. I also assume that negative prices are impossible.

2. Following usual practice in economics, I've been a little sloppy about whether the elements of the codomains of the supply and demand functions $S$ and $D$ are numbers or sets of numbers. (Technically and most generally, they should be sets. But so as not to confuse students, we usually and more simply say they are numbers.)