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.