Tuesday, 27 September 2016

A hyperbolic paraboloid interpretation of residential relocations in U.S. metropolitan regions

Hossein Estiri, Harvard Medical School, USA


Hyperbolic paraboloids (Figure 1) have always been the most fascinating geometric forms to me. They are simply quadratic surfaces from cross-section of a hyperboloid and a paraboloid. Not so simple? They are often called “saddles” and have been widely used in architecture – Sagrada Família nave roof by Antoni Gaudí, a well-known Spanish Catalan architect, is an illustrious example of the use of hyperbolic paraboloid forms in architecture.

Figure 1: A hyperbolic paraboloid from two different angles

Have you seen a demographic theory described with a hyperbolic paraboloid? That’s what we did in this article (Figure 2)! The Cohort Location Model (CLM) is a simple approximation of how we distribute across metropolitan areas (in the U.S., at least), based on age of the household head.

Figure 2. The Cohort Location Model – for more details read the article in Urban Studies

For years, demographers have tried to explain why we live where we live – from now on let’s abbreviate the phrase to: [WWL]2. We tried to simplify [WWL]2 by combining 2 assumptions about our housing consumption and land use patterns. First, we assumed that as we become older, our expectations from (or needs for) where we live increases – e.g., we need more home space, more neighborhood amenities, etc.). There is, by the way, scientific evidence for this assumption. For example, Jake, an imaginary college student in his early 20s, can live in a shared 4-bedroom unit that was built in the 60s and is not conveniently located with respect to healthy grocery stores. Many of us have lived like Jake. It is fun, for Jake and his cohorts though.

But 10 years later, when Jake is out of school, has a spouse, and has just welcomed a new addition to his family (a kid), things change. Now he needs more space, and access to a lot of neighborhood services, all of a sudden matter to him. We found that age of 35 is about that time. Now this about 35-year-old former college student is seeking to relocate – probably to his first owned home in a decent neighborhood for his growing family.

The CLM says that the 35-year-old Jake most likely will end up going to the suburbs. This residential relocation pattern, according to our assumption number 2, is partially due to the land use patterns in the U.S. (i.e., the widespread suburbanization).

Those nice neighborhoods close to the city center(s) are way above Jake’s budget. This is probably the story of many young adults in the American metropolis these days. Our model follows Jake each 10 years until he is above 84. We think at some point Jake will either come back closer to the city center(s) – either will be able to afford to live in one of those nice central city neighborhoods or reduce his expectations to live closer to some important services – or will decide to go further out to get more natural amenities.

The pattern CLM presents will probably be different in other countries. But, I bet it will still be a hyperbolic paraboloid, if you do a linear approximation. We have made this research reproducible. Plug in your data into our code on GitHub (https://github.com/hestiri/hhLocation) and let’s see what kind of hyperbolic paraboloid you will get.

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