Monday, June 13, 2016

The urban environment as information equilibrium

Cameron Murray wrote a post a couple weeks ago that made me think about applying information equilibrium to urban economics. Cameron tells us "[t]he workhorse model of urban economics is the Alonso-Muth-Mills (AMM) model of the mono-centric city" and then goes on to look at some of its faults. Here's his picture:

Let's tackle this with the information equilibrium framework.

Let's set up the information equilibrium system (market) $h : R \rightleftarrows S$ where $h$ is building height (proxy for density), $R$ is distance (range) from the center and $S$ is the supply of housing (housing units). And let's assume $R$ varies slowly compared to $S$ -- i.e. transportation improvements and new modes (that fundamentally change the city's relationship with distance) happen slowly compared to adding housing supply. This puts us in the partial equilibrium regime with $R$ as a "constant information source" (see the paper). Height $h$ is the detector of information flowing from the distance from the center to the housing supply; in economics, we think of this as a "price".

The "market" above is shorthand for the information equilibrium condition (with information transfer index $k$)

$$h \equiv \frac{dR}{dS} = k \; \frac{R}{S}$$

which we can solve for slow changes in $R$ relative to $S$ (and then plug into the "price" $h$) to obtain (with free parameters $R_{0}$ and $S_{ref}$ are model parameters):

$$h = \frac{k R_{0}}{S_{ref}} \exp \left( -\frac{\Delta R}{k R_{0}}\right)$$

Here's a plot of this function:

You could of course substitute housing price $p$ or density $\rho$ for $h$ (or more rigorously, set up information equilibrium relationships $p \rightleftarrows h$ or $\rho \rightleftarrows h$ so that e.g. $p \sim h^{\alpha}$ so that $p \sim \exp \; -\alpha \Delta R$).

Now markets are not necessarily ideal and therefore information equilibrium does not hold exactly. In fact, it fails in a specific way. The observed height $h^{*} \leq h$ (because the housing supply $S$ can only at best receive all of the information from $R$, i.e. $I(R) \geq I(S)$, a condition called non-ideal information transfer), so what we'd see in practice is something like this:

Here's a logarithmic scale graph:

This is not too different from what is observed (assuming price is in information equilibrium with building height $p \rightleftarrows h$) from here [pdf]:

In short, information equilibrium provides a pretty good first order take as an urban economic model. You can see that height restrictions and other zoning ordinances or preserved green space end up impacting the observed height negatively -- i.e. non-ideal information transfer.

1. O/T: Jason, given access to the data, what is your feeling about the prospects for predicting the number of shooting related deaths and injuries in a country (say one with an intact government (i.e. not Somalia or Syria))? I'd think population and number of firearms might be a candidate set of minimal variables for a low order model. Also, I wouldn't be surprised to learn that a modeler was able to construct more accurate higher order models as well with more detailed data (how many firearm owning households and/or individuals, what kind of firearms, etc). Probably somebody out there has done such a thing.

1. Yes, I'd say that

N/P ~ n/P

is a good first order model where N is the number of deaths, n is the number of guns and P is the population.

2. I'm just wondering, so price gives us a kind of upper bound of perfection, but isn't the economically interesting question why different people are so found form the boundary? And what we can do to get them closer?

Thinking about this example, you have people essentially acting randomly within what I call the "site economic frontier". How can we say anything about the rate of new construction if the agent are random, and particularly the link between the frontier and this rate?

1. I'd agree that deviations from "ideal" would be the interesting question. However I think there are two things at play here (in my opinion, accepting the IE description):

1. We need to know what ideal is in order to know the deviation

Basically, we can't know how far we are below the site economic frontier unless we have a model of that frontier.

2. The deviation from ideal might not be "economics" per se

The ideal calculated from information equilibrium is a property of the available state space (i.e. locations of housing units), not the actions of agents. If agents fully explore that state space, then this ideal is the result.

I think (my opinion) that traditional mathematical "economics" of optimizing agents, incentives and expectations really only applies if we acheive the ideal. If we don't acheive the ideal, the reason probably is not within the purview of "economics" -- a mathematical description might not be available. Central Park in Manhattan continues to exist because were not rational economic agents -- and in that case social (people like the park) and historical (it's been there for awhile) explanations are probably more important. There's no equation that tells us why it hasn't been filled in with expensive flats.

In areas that don't have social or historical importance, I imagine the ideal model works better and the distance from the ideal (accounting for regulations) might be a good measure of the likelihood of building in that area.

2. were = we're