Given all the stuff being discussed on Gelman’s blog about the housing market, here’s some ideas I had about how one might formulate a continuous version of the model (rather than say an agent based version).

First, let’s talk about how many houses are being rented and how many people are renting them, as a function of price. Let’s posit an important component of the model, $$F(t,r)$$ which is the frequency distribution of houses renting for price r at time t. Now let’s also posit some variables $$N(t)$$ for the total number of houses, $$B(t,r)$$ the average number of Bedrooms per house renting for r, and $$C(t,r)$$ the average number of people per bedroom in houses renting for r (C is for crowding). Then we can describe the relevant population of the area as:

$\mathrm{Pop}(t) = \int_0^\infty N(t)F(t,r)C(t,r)B(t,r) dr$

Now F(t,r) can change in several ways. First of all there can be some turnover. When there’s turnover of a house currently renting for rent $$r_1$$ and it starts renting for $$r_2$$ then we have:

$\frac{d}{dt}(N(t) F(t,r_1)) = -\int N(t) F(t,r_1)To(t,r_1,r_2) dr_2 + \int N(t)F(t,r_2)To(t,r_2,r_1)dr_2 + \frac{dN(t,r_1)}{dt}$

An integro-differential equation, with a kernel $$To(t,r_1,r_2)$$ that describes the turnover transition of an existing apartment between different rents, and a function $$\frac{dN(t,r)}{dt}$$ that describes how fast we’re constructing housing and renting it out at price r, each changes in time.

Now, we’ll imagine that the number of units actually on the market for a day or whatever is negligible (we don’t have say 15 percent of all units empty ever for example). So we can talk about a turnover rate $$v(t,r)$$ for volume of transactions at time t, price r. This defines a kind of spot price distribution. There’s no one spot price, because there’s no “universal apartment”. Different locations and different qualities, and different sizes produce different market rental spot prices. But if $$v(t,r) > 0$$ then at time t, some volume is available at price r.

There is a relationship between the kernel $$To(t,r_1,r_2)$$ and the transaction volume $$v(t,r)$$, namely that the total number of apartments transitioning from anywhere into price r, plus the new sales, is the volume:

$v(t,r) = \int N(t) F(t,r^*) To(t,r^*,r)dr^* + \frac{dN(t,r)}{dt}$

and also, there is a relationship between $$N(t)$$ and $$\frac{dN(t,r)}{dt}$$ namely:

$\frac{dN(t)}{dt} = \int \frac{dN(t,r)}{dt} dr$

This more or less describes the dynamics of the supply, at least, if you posit specific forms for the $$F(0,r)$$ function, and some dynamics of the transition kernels and the building rate and soforth.

But, in order to get information about the transition kernels and the volume, etc, you need to have some information about demand, and for this you need to posit $$D(t,r)$$ a demand for an apartment (or a bedroom?) at a given rent r. And this describes the reservoir of people who are waiting in the wings to scoop up an apartment if one becomes available at a given price. We should probably split it out as a component from inside our region, and from outside our region. From inside our region will help us determine how often people are leaving their apartments.

It’s clear that the demand can be a function of the character of the population, as Phil says you can get more rich people demanding services and then increasing the demand and the lower end of r. So, there is feedback between statistics of the population, and the demand.

Now, I think we have a structure that we can start to build on, specific functional forms could be hypothesized and dynamics could be observed through solving integro-differential equations, perhaps by discretizing the system and doing sums. Probably we need a bit more than this, this isn’t quite a complete closed model, but it’s a good start for a structure if you wanted to go after this with a continuous statistical approach.