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.