So one can look at a lot of different properties of cities,
for different datasets, and do, say, what was done here:
make a plot, observe a trend on a log-log plot, and estimate a beta
And if one does that, again, for different datasets and different quantities,
some interesting trends or patterns emerge.
So how do properties of cities depend on their population?
Well often, they are well described by a power law of this form,
so here, N is the population size, and Y is whatever we're interested in.
in the previous unit it was metabolic rate
now it might be GDP, or length of roads, or something.
And it turns out that, properties cluster into three different types.
So there's some quantities that scale superlinearly.
And that would mean a beta greater than 1,
and examples of those types of quantities are things like new patents, GDP, wages,
we've seen examples of those.
So these are all things that grow faster than linear.
So that's sort of good news, as we think about cities getting larger,
there's a sort of an increasing returns to scale.
They become more productive, more creative,
The number of patents more than doubles if you double the city size,
GDP more than doubles and so on.
But it's not all good news, some things that we tend to think of as bad,
like total electrical consumption, that also goes up faster than linear,
AIDS cases, crime, all of these things tend to scale, more or less,
with an exponent that is larger than one.
And it turns out, also, that many of these exponents tend to be maybe around 1.15.
There's a lot of spread around that,
they kind of tend to cluster roughly around that value.
So, these things tend to scale superlinearly.
Then, there are some things that are sublinear,
and those have a beta less than one,
and turn out often to be around .85 or 8,
and they vary but they sort of cluster around that value,
and things like that we've already seen, like road length,
right, so that was the dataset here, sure enough, beta about 0.85,
length of electrical cables, numbers of gas stations,
and so on. So these tend to grow less quickly than linear.
So, if you double the city size, you'd think,
"Oh, there are going to be twice as many gas stations"?
Turns out on average there aren't quite twice as many gas stations,
there's 2^(0.85) times as many gas stations,
it's a number less than 2.
And we'll talk a little more about how to think about this,
but the basic idea is, as cities grow, as population grows,
they tend to get denser. And so, the road length don't need to double,
the number of gas stations don't need to double,
the amount of gas that is used might double,
but the number of gas stations themselves wouldn't necessarily need to do that.
Okay, so there tend to be some properties that have an exponent of about 1.15,
there tend to be some that are around 0.85,
and then there's some that are linear,
and that's a beta of around 1,
and what are some examples here?
Well, total housing
and that's not really surprising,
if you double the population, you're going to need to double the housing stock
total employment,
so if you double the population in size,
you're going to double the number of jobs, on average,
those jobs might pay more, because wages go up faster than linearly,
but the total number of jobs themselves is approximately linear,
take a approximation.
Household water or electricity consumption
So if you double the size of the city, the amount of water that the city needs,
at least for its households, is going to be about double,
same thing for electricity. So, these are not exact statements,
exact laws in any sense, but there does seem to be a pretty fairly robust
statistical pattern, that a whole bunch of things associated maybe with
creativity, although I don't know if AIDS and crime, they're not really creative,
but, whole bunch of things associated with interactions,
tend to go up superlinearly. Some infrastructure sort of things,
tend to go up sublinearly. And then there are some quantities,
that are sort of inherently per-person, that tend to be linear.
So, if nothing else, looking at urban scaling,
in analogy to how we looked at metabolic scaling,
reveals some interesting and fairly robust statistical patterns.
And one can use these statistical patterns, to make statements about,
what might happen, say, as urbanization continues.
So, we know that more and more people, a larger and larger percentage of
the world's population, is living in cities,
and we can use these results to say, in a statistical sense, on average,
what some of the implications might be,
economically, in terms of crime, in terms of various sorts of resources.
So urban scaling, the starting point is these observations that there are some
scaling, power law like behaviour, that seems to hold true
for many different sets of data across the world,
and seems to be fairly constant in time as well.