If you pay any attention to online housing discourse you’ve heard repeatedly that we can eliminate homelessness by just building more houses. The most idiotic versions of this theory rely on the (putatively obvious) idea that if the demand for a good is fixed then the price is roughly inversely proportional to the supply. My personal feeling about all theories like this is that they are framing phenomena created by state violence as if they were the result of universal natural laws, but that doesn’t mean we can’t learn something from them.

In particular we can learn that this supply/price assumption implies that under apparently normal market conditions landlords with more than a few units can maximize their profit by intentionally keeping apartments vacant to artificially restrict supply. If it’s actually true that increasing housing supply functionally decreases housing costs then all else being equal it will lead to house-hoarding. This appears to be a contradiction in the theory.

According to the theory if the supply of apartments goes up then the price will come down. Consequently if the price is too high then the supply is too low. Increase the supply and you lower the price. In other words price is a decreasing function of supply. Since the function is decreasing it’s invertible, so we can read this as saying that if you raise the price you lower the supply.¹ This is our first glimpse of the problem. It turns out that raising the price can increase a landlord’s profit even as it makes some of his apartments too expensive for local renters. The market can’t tell the difference between unbuilt and intentionally unrented units.

Here’s an example with some made-up numbers. Suppose a landlord has 15 units, all occupied, rented for $1,000 per month. Suppose further that economic data in the area shows that for every $100 increase in the rent one of the apartments will remain vacant.² Here’s a table showing the landlord’s gross monthly income at various rents. I’m also assuming that the landlord’s monthly costs are fixed during the lease term, so that gross income is proportional to net income and therefore they have the same extrema.

So by keeping either two or three apartments vacant the landlord can increase gross receipts by $600, which is 4%. It’s not huge, but it’s not nothing, No landlord is going to voluntarily leave a 4% raise on the table just to have a few more apartments rented. They’re not in the landlord business because they like landlording, but to make a return on their investment. It’s literally their only purpose. If they could make their return by leaving all their apartments vacant they wouldn’t hesitate to do it.

Also, this calculation probably understates the landlord’s vacancy-induced gains, since vacant apartments don’t need maintenance. On this theory the landlord has an incentive to keep three rather than two apartments vacant even though the gross rent receipts are the same. Additionally landlords benefit from charging higher rents and incurring some vacancies because it elevates the appraised value of their property and thus allows them to liquidate newly materialized capital. This phenomenon is robust across relevant variable ranges.

To see this, suppose a landlord has $L$ units. Let $R$ be the highest rent for which all units will be rented. Assume that for every $$k$ increase in the rent the number of vacant units goes up by one. Now let $x$ be the number of vacant units, which is also the number of $$k$ increases in the rent. Then there are $L-x$ units rented at $R + kx$ dollars per month. Therefore the gross monthly income is

$$G(x) = (L-x)(R+kx)$$

$$=-kx^2+(Lk-R)x+RL$$

Hence $$G^{\prime}(x)=-2kx+Lk-R$$ This has a critical point when

$$x = \frac{Lk-R}{2k}$$

Note that $k > 0$, so $x > 0$ when $Lk-R > 0$. This means that $x>0$ when

$$L > \frac{R}{k}$$

Here’s one way to interpret this result. Since both $R$ and $k$ are positive their ratio is positive, so $L$ is positive. Hence no matter the values of $R$ and $k$ there is some number of units for which it will be more profitable to keep some of them vacant.

When $k$ is small compared to $R$ then $L$ will be large and landlords with fewer units will rent all of them. It makes some sense to link this situation to precarious conditions for renters. If $R$ is fixed then a low value of $k$ means that it takes only a small rent increase to price a tenant out. This interpretation is consistent with a large value of $L$. When times are tough only landlords with many units can profit by keeping some vacant.

But if $k$ is big compared to $R$ then $L$ can get very low. Clearly 2 is a hard minimum since there aren’t market conditions under which intentionally keeping a single unit vacant maximizes income.³ In the example above we saw that if $k$ is 10% of $R=1000$ then even keeping 3 of 15 units is more profitable than renting them all. In the model envisioned by this analysis the better the economy the greater the incentive for landlords to hoard some of their housing stock.

So does this all tell us something important about actually existing housing markets? I have absolutely no idea. I’m a pure mathematician, so I make up assumptions and reason from them until I get an interesting result. I don’t know anything about economics or housing markets even if I do know a lot about calculus. But whether or not this result says anything about the real world, it certainly does say something about arguments that it’s possible to end homelessness by doing nothing more than allowing unlimited numbers of market-rate apartments to be built.

Under fairly normal market conditions it follows from the same hypotheses this argument relies on that landlords will regularly be able to make more money by intentionally keeping some of their apartments vacant. That is, the hypothesis that increasing the supply of apartments will cause average rents to decrease implies that as the supply of apartments increases landlords become incentivized to intentionally keep units vacant so as to raise rents. Probably neither of these arguments ought to be taken very seriously, but one of them really seems to be for some reason.

1. The supply isn’t lowered necessarily by apartments becoming physically unavailable, but in the sense that as the price goes up the number of people able to rent from the given range of apartments shrinks, so some of the apartments become unrentable due to cost and are therefore not part of the supply.
2. Please don’t let the fact that I pulled these numbers from the air distract you. This is just an example to show how the arithmetic works and to show that there are reasonable seeming numbers that illustrate the point. The example does show that the phenomenon could happen, and the actual numbers involved can be found by research, which is certainly already being done by landlords if anything I say here makes sense.
3. There aren’t market conditions in the usual sense of the phrase, where we pretend that government action is a whole different thing. Clearly the government could make it more profitable to keep even the only unit the landlord owns vacant, but they don’t currently do this.