So now what Montroll does is

He goes trough and looks at the value

of this two expectation values

He looks at the value, in particular,
of the average, alright?

this one here, one of the two
maximal entropy constraints

and the he also looks at the value

of this constraint right here

how that changes over time

So let's look at the average of
the Log of the price

This is the first that's constraint,
because remember

it's a gaussian distribution

N(x), where x is Log(P)

Logarithm of the price

So, this are the two quantities

that are being fixed if we wanna
work in price space

as opposed to Log of price space.

So the first thing he points out, is that
the average of the Log of the prices

is a slowly increasing function of time

except for a funny little 'blip' here, ok?

So this is something you would expect,

over time the average Log price,

and in fact, the average price as well,

is going to increase,

and is going to increase
solely because of inflation

a good that costed a dollar
ten years ago,

would generally cost
more than a dollar today.

Inflation is a multiplicative process

but what he draws your attention to

is this column here

this is the variance of
the Log price over time

and what you can see in

the variance of Log of the Price

is that over 75 years of
the Sears-Roebuck catalogue

and, it's important to say,
both world wars,

enormous social change,

so, here they are selling buggy whips,

and here they are selling cassette tapes

enormous economic change, so certainly 
the overall prices have gone up,

they go from 0.1 in Log price space to 4

in other words, prices increase by a
factor of 2,4,8... 16, by a factor of 16

the variance of the Log of the prices

stays almost constant

is roughly a factor of 2,

the deviation, in other words,

from the average price,
or the average Log price

the square deviation from
the average Log price

is constant at around 2, over 75 years

and Montroll sees
"this is worthy of explanation"

this here we already understand,

we understand why prices grow

but we don't understand why
their variance stays constant

why is it the case that
the Sears-Roebuck catalogue

presumably the stability in
the catalogue here,

and the catalogue here,
an entirely different group

why is it the case that that
they where able to,

or somehow ended up doing the following?

keeping this variance constant

and one of the things he
notes is the following

that lets say, in 1900

the Log price
had a certain distribution

and indeed a certain variance, sigma,

in 1975 if every single good
in the Sears Roebuck

was still in the catalogue in 1975,

and every single good inflated
at the same rate

then ok, yes, the mean would go up,

the mean would go up

but all this goods, all this columns here,

would all grow by the same amount

so if every price P

was multiplied by the same factor alpha

then of course every Log price

Log P

is simply added to Log alpha

so ok, you would spect the variance
to stay constant

in that very particular case, other wise
allowing for the natural drift of goods

and in particular, allowing for
different rates of inflation

allowing for alpha, in particular,
to be a random variable

just has it was previously on
our model of language growth

once alpha becomes a random variable

then generically speaking

what's gonna happen is

that the variance will rise

because some goods will multiplicatively

over time, just randomly accrue
lots of really large multiplications

they become expensive really quickly

sort of like, let's say,
college education, apparently

and other goods, deflate,
even though, become cheaper

so one of the things you see in
the Sears Roebuck catalogue

is that women dresses become cheaper,

because the materials to make them
become synthetic,

and then become better and better
at making synthetic dresses

so other goods, in fact, deflate quickly

another good that deflates quickly,

although is not presumably a
dominant feature of the

Sears Roebuck catalogue in the early years

is computers, so the cost of a device
with the same computing power

something 10 years ago, is minuscule

compared to the price it was then,
even in absolute terms

so there is some goods deflate enormously,

some goods inflate enormously,

in the organic larger scale
consumer culture

but

what we find is,

in the data, the deviations,

the square deviations from
the mean in Log space stay constant

that's against
the spectations that we have

and the problem is to explain it.