I have been pondering a comment in William Fischel’s generally excellent new book on zoning to the effect that:

…suburbanization and reduced urban density are worldwide phenomena. All but 16 of the 120 urban areas on every continent grew outward and reduced their overall population densities in the last decade of the previous millennium, even as almost all of them grew in total population.

This is an interesting claim, but one that I find very difficult to reconcile with the evidence on other “big picture” changes observed in cities over the last three decades.

In recent decades, agglomeration economies have gotten stronger and the structure of advanced urban economies has changed. This has in turn increased the premium that people place on proximity and centrality – a phenomenon well illustrated by Grimes and Liang (2007), who show how close proximity to the city centre shifted from being a “disamenity” to an “amenity” during the 1990s:

Relatedly, regulations limiting density are increasingly binding (and, in places like Auckland, San Francisco and New York, more costly than regulations limiting sprawl). This was nicely illustrated by three recent pieces of research that showed that (a) Auckland’s legacy planning regulations imposed twice as many costs on apartments as standalone houses, and that (b) the cost of legacy councils’ building height limits were higher than the cost of the city’s former urban growth boundary:

Type of regulation | Estimated cost of regulations | Source |

Rules affecting standalone houses | $32,500-$60,000 per dwelling | Grimes and Mitchell (2015) |

Rules affecting apartments | $65,000-$110,000 per dwelling | Grimes and Mitchell (2015) |

Metropolitan urban limit | $860 per Auckland household | NZIER (2014) |

Height limits | $933 per Auckland household | NZIER (2015) |

So colour me perplexed. On the one hand, I have a very respected and knowledgeable economist telling me that cities are getting less dense. On the other hand, I have a mass of evidence, often compiled by *other* respected and knowledgeable economists, that suggests that densities should be *increasing*, not decreasing.

As it turns out, **Fischel’s claim is what my mathematician friends describe as “true but trivial”. It’s not inaccurate, but it doesn’t tell you anything interesting about the world either.** The idea that urban densities are getting lower is in fact a statistical artefact – i.e. it’s a “fact” that arises from the way that Fischel has calculated population densities.

Calculating population densities is a slightly arcane subject. Last year, I wrote a short paper that explored alternative measures and explained why we should prefer a population-weighted density measure to a simple average density:

The most common approach to measuring population density is simply to divide the total population of a city by the total land area of the city. As shown above, this approach will tend to underestimate the density of cities with large expanses of lightly populated exurban land. However, this approach is commonly used for international comparisons due to the fact that it relatively straightforward to calculate…

The population-weighted density measure was introduced by Barnes (2001) to correct for the weaknesses of the simple average density measure. This measure was recently used by the US Census Bureau to produce consistent and meaningful data on American cities (Wilson et al, 2012). As the example above suggests, it more accurately reflects the density at which the average city resident is living (Eidlin, 2010).

Population-weighted density is estimated by calculating the density of all individual neighbourhoods within a city, assigning each neighbourhood a weight equal to its share of the city’s total population, and summing up the weighted density of all neighbourhoods. In other words, if a dense inner-city neighbourhood has ten times as many people as an outlying suburban neighbourhood, the inner-city area would be weighted ten times as heavily as the suburban area.

In other words, average density measures the density of the average hectare of land in the city, even if hardly anyone lives there, while population-weighted density measures the density of the neighbourhood that the average citizen lives in. Average density is not, therefore, a very meaningful number if you want insight about how people are living in a city.

Moreover, it turns out that **a city’s average density can be declining even though every part of the city is getting more dense!** This sounds counterintuitive, so I’ve put together a simple model showing how it can happen.

You can download the model here in case you’re interested in playing with it. For simplicity, I’ve assumed a linear city that extends in one direction from a centre. (This readily generalises to a city that extends in multiple directions.) Population densities are highest near the centre and decline exponentially with distance. (The distance decay parameter I’ve used loosely approximates observed outcomes in big NZ/Aus cities.)

I’ve tested the impact of a 20% increase in the urban population. In the model, intensification accounts for around 60% of growth, meaning that all existing developed areas get ~12% denser. The remaining 40% of growth occurs in low-density greenfield areas, which expand the city’s footprint by 33%.

Mathematically-minded readers will see where this is going. But for the visual learners out there, here’s what it looks like on a graph:

Because the urban footprint has expanded by 33% while the population has only grown by 20%, the city’s average density has dropped by 10%. But if you look at the graph, you can see that it would be ridiculous to say that the city is spreading out and de-densifying. In fact, every part of the city is getting more dense, and most growth is occurring within built up areas.

In other words, Fischel is wrong to say that cities are getting less dense. Horizontal growth is certainly happening – but so is vertical growth.

I would argue that the latter trend – towards proximity, density, and efficient use of land – is the more significant of the two. **We are currently seeing big changes in the function, structure, and use of cities. Recognising and responding to those changes is important, and policies that unwittingly stifle them will have large economic and social costs.**

That’s why it’s important to measure density (and other urban phenomena) accurately – bad measures often contribute to bad policy decisions.

How do *you* think cities are growing?

Nice Peter, averaging can indeed lead to curious and clearly misleading claims about the world, for example; the

averagehuman has one ovary and one testicle…!Actual humans; not so much.

Excellent post. Critical to understand this at mathematical, spatial and functional levels to get better policy.

Yes, population weighted density clearly seems the superior measure.

On a related note (although not addressing your question of whether cities are getting denser or not…) – ChartingTransport has done a really nice comparison and mapping of pop-weighted density for 30 European and Australian cities (just latest year, no change over time). Highlights just how dispersed Aus/ NZ cities are – e.g. just 10% of Londoners and Parisians live at typical Australian suburban densities. Plenty of room for liveable intensification!

http://chartingtransport.com/2015/11/26/comparing-the-densities-of-australian-and-european-cities/

Great post Peter.

So intensifying by building more homes within the existing city, and intensifying by converting empty farmland to residential suburbs means… a lower density city? Seems so absurd when you put it like that!