## 1. Motivation

In the standard model of house-price dynamics, there are two kinds of cities: supply constrained and supply unconstrained. In supply-constrained cities (e.g., New York, Boston, or San Francisco), it’s difficult and costly to build new houses because of geographic and regulatory hurdles. In supply-unconstrained cities (e.g., Las Vegas, Phoenix, or San Bernadino), these hurdles are much lower, and it’s much easier to build new houses. To get a sense of just how unconstrained places like Las Vegas are, take a look at this time-lapse video of Las Vegas from space. The number of houses balloons as housing demand in Las Vegas grows.

Now, suppose more people suddenly want to live in a particular city. If the city is supply constrained like New York, then people will have to outbid existing residents to move into that city, which will drive up the prices on existing houses in that city. More people want the same number of homes, so prices have to go up. By contrast, if the city is supply unconstrained like Las Vegas, then people who want to move into that city can just build new houses. In supply-unconstrained cities, the supply of housing adjusts to accommodate the additional demand. Why outbid an existing resident when you can just build the same house right next door? Thus, in the standard model, supply elasticity is a key determinant of house-price growth (Saiz, 2010).

But, during the mid 2000s things got weird. Supply-unconstrained cities like Las Vegas realized huge spikes in transaction volumes and house prices (Chinco and Mayer, 2016). What changed? Why did Las Vegas suddenly look like a supply-constrained city? Is there some economic mechanism that might make supply-unconstrained cities behave like supply-constrained cities when there is a lot of trading activity? This post outlines how the combination of trading volume by house flippers (i.e., people who buy and then quickly resell houses without living in them) and naïve beliefs can generate artificial supply constraints in housing markets with lots of trading volume.

## 2. Everyday Example

To see how this mechanism works, it’s helpful to start with an example from everyday life. I love bagels. Imagine you’re at a bagel shop and you want to buy an everything bagel. The shop has lots of different kinds of bagels displayed in bins that contain bagels each. So, at the start of each morning, there’s a bin of plain bagels, a bin of poppyseed bagels, a bin of everything bagels, and so on… There’s a line, and it takes each person several seconds to order. Each time someone orders a bagel, the clerk takes it from one of the bins. Whenever one of the bins runs out, a second clerk takes it to the back of the shop and refills it with more bagels, a task that takes minute to complete.

Without any sort of naïvety, supply and demand work exactly like you’d expect in this setup. If there is only everything bagel left, then you might be willing to pay more than the price listed on the menu for that last bagel. But, if there were lots of bagels left, then you would never do this sort of thing. You’d just wait your turn in line and pay the listed price on the menu when you got to the counter. If a bin happened to run out when you were at the front of the line, then you’d recognize that it’s going to be full in a minute and just wait until the second clerk got back.

Without any sort of naïvety, sales volume doesn’t have any affect on this equilibrium. To be sure, if there are lots of people in line and bagels are selling really quickly, then you’re more likely to find the everything-bagels bin empty when you get to the front of the line. It always takes the second clerk minute to fill an empty bin. So, if there is a big line and more people pass by the front of the line per minute, then bins are more likely to run dry and more people arrive at the register when the second clerk is back in the kitchen. But, if bagel buyers are fully rational, then they’ll realize that each bin will be replenished in a minute and just wait till the fresh bagels come out before buying.

Introducing naïve beliefs changes things. If you don’t recognize that empty bins will be replenished in a minute, then you might be willing to outbid the guy in front of you for the th everything bagel in a bin—or, at the very least try to talk him into a different order. And, when you get to the register during a busy time of morning, it’s going to look like the whole bagel shop is running low on supply since each individual bin is more likely to be in the process of being filled. If you could linger in the bagel shop for a while, then this naïvety wouldn’t matter. Any empty bins would get replenished while you were standing around making your decision. But, when there is a line out the door and you have to make a quick decision, your naïve beliefs make it look like there is an artificially low number of bagels available.

## 3. Simple Model

Investors play the role of the clerk that takes minute to replace a bin of bagels. They take houses off the market for a short period of time. If trading volume is low or home buyers are fully rational, then they shouldn’t affect equilibrium house prices too much. But, if trading volume is very high and home buyers don’t realize that investor homes will come back on the market in months to a year, then home buyers might get the impression that the supply of houses is getting low. I now outline a simple model to make these ideas more concrete.

How many different houses can a home buyer see on the market if he looked for months? Let denote the total number of houses, denote the number of owner-occupied houses, denote the number of investor-owned houses, and denote the number of houses that are currently for sale in month :

(1)

In any given month, home buyers can only visit houses that are currently for sale. Owner-occupied and investor-owned houses are off the market. A house might be owner occupied one month, for sale the next month, and owned by an investor several months later. I write the probability of transitioning from one state to another in matrix form:

(2)

Each entry in this matrix represents the probability that a house transitions from one state to another. For example, represents the probability a house goes from being owner occupied one month to for sale the next. And, represents the probability that a house is investor owned in month given that it was investor owned in month . The columns of this matrix sum to and since a house has to be for sale before it can pass from one owner to the next. The diagram below gives an alternative way of representing these transition probabilities that doesn’t use matrix notation.

The number of houses that are always owner occupied after months of looking is . So, when there aren’t any investors, the number of houses that a home buyer can choose from after looking for months is . The number of houses that are always investor owned after months is . So, when there are investors, the number of houses that a home buyer can choose from after looking for months is given by:

(3)

Thus, the supply constraint imposed by investors on the number of houses that a home buyer can view in months is given by:

(4)

This term is just the change in the observed housing supply after months due to the presence of investors, . If , then investors decrease the number of houses for sale over the course of months by . If there would have been houses for sale over the course of months without investors, there are only houses for sale with investors.

## 4. Plugging in Numbers

This model is nice because it’s easy to plug in numbers to see how investor holdings can affect the perceived housing supply for naïve home buyers. We can go back and forth between holding-period lengths and transition probabilities by using the negative binomial distribution. Investors typically hold onto their houses for months, implying that . Owners typically hold onto their houses for years, implying that . Owners and investors are equally likely to buy houses, . Suppose that the typical house stays on the market for year, implying that .

The figure above shows the fraction decrease in the housing supply perceived by naïve home buyers as a function of their search duration when of the housing stock is for sale in any given month (code). e.g., if a home buyer would have seen houses in months in the absence of investors, then he sees only houses in months when of houses are initially owned by investors. The dashed green line says that investor holdings can lead to a drop in the housing supply as perceived by naïve home buyers. As the number of months that a naïve home buyer searches drops, the impact of investor holdings rises sharply. When search durations are really short, like they were in Las Vegas during the mid 2000s, tiny amounts of investor ownership can have enormous impacts on the perceived housing supply.