From: "Tim Rhodes" <proftim@speakeasy.org>
To: <memetics@mmu.ac.uk>
Subject: Re: Defection Rates and Classes
Date: Mon, 16 Aug 1999 08:00:15 -0700
Well, since I couldn't sleep much anyway...
Derek wrote:
>Tim:
>
>(( P * (1 - Ddown) ) * Rp) = the number of children from professional class
>parents that will themselves enter the professional class in the next
>generation (old money)
>(( W * (1 - Dup) ) * Rw) = the number of children from working class
parents
>that will also enter the working class in the next generation
>
>Derek:
>
>Yes, agreed.
>
>Tim:
>(( W * Dup ) * Rw ) = the number of children from working class parents who
>will enter the professional class in the next generation (new money, if you
>will)
>
>Derek:
>
>Not agreed. Doesn't this assume that the upwardly mobile workers enter the
>professional classes, but then for the first generation reproduce at worker
>rates? (hence the * Rw ????)
No, Derek, you're confusing the children with the parents.
>
>The number of workers entering the professional classes is just
>
>W * Dup
No, Dup & Ddown are percentages (in decimal terms) of their respective
populations. (W * Dup) will only give you _number of worker parents_ whose
children will be in the professional classes in the next generation. You
have not factored in *any* reproduction without an Rx.
>
>Tim:
>(( P * Ddown ) * Rp ) = the number of children from professional class
>parents who will enter the working class in the next generation.
>
>
>Derek:
>No, again it's just P * Ddown
No, again, you've confused a percentage of the population with a growth
rate.
They are not the same thing.
>
>I think I can see the difference we are having. I want reproduction to
>occur within classes, and then the adults move classes prior to the next
>round of reproduction (which they will carry out according to the
parameters
>of the class they have entered, not the one they were born in). Your
>equations seem (I think) to have reproduction and movement going on
together
>(????).
You didn't seem to understand me from the start. Let me go over this again
working from simple terms to more complex. When you don't agree, stop
reading and ask, because like any mathematics, everything from that point on
will be useless for you. I'm going to just concentrate on ONE generation to
make things even more simple.
Here goes:
W1 = workers in the first generation
P1 = professionals in the first generation
(W1 + P1) = total people in the first generation
So far so good, right? Okay,
Rw = reproduction rate of workers
Rp = reproduction rate of professionals
(W1 * Rw) = total number of children born to the first generation of workers
(P1 * Rp) = total number of children born to the first generation of
professionals
Okay, how are we doing so far? You still with me? Good.
Now, here's where it starts to get complicated. I'm asserting that some
percentage of the children of each class will end up in the other class by
the time they become adults. So...
Dup = % of workers children that grow up to be in the professional class
Ddown = % of professionals children that grow up to be in the working class
Okay? These terms are percents of the total number of children born of that
class from first generation parents, right?
Let's throw a quick example in to see if you're still with me:
W1 = 100
Rw = 2
Dup = 10%
So, given these numbers, how many first generation workers' children will
grow up to be professionals?
( If you said anything other than 20 you aren't with me anymore. )
(W1 * Rw) or (100 * 2) gives you the total number of workers' children,
((W1 * Rw) * Dup ) or ((100 * 2) * 0.10 ) gives you the number of workers'
children that leave the working class.
So (200 * 0.10 ) = 20, right?
So, are we good with this so far?
I hope so, `cause now I'm going to introduce the radical notion that those
workers' children that don't become professionals, become workers. Remember
we just showed a couple lines earlier that,
((W1 * Rw) * Dup) = the number of workers' children that grow up to become
professionals
and we know that,
(W1 * Rw) = total number of children born to the first generation of workers
So, how many workers' children grow up to become workers?
(W1 * Rw) - ((W1 * Rw) * Dup)
Or the total number workers children minus the ones that become
professionals.
Right? Or to simplify the expression: (W1 * Rw) * ( 1 - Dup)
See how I got that? Nothing complicated, just a little application of the
ol' Distributive Property of Real Numbers here.
So,
((W1 * Rw) * Dup) = the number of workers' children that grow up to become
professionals
((W1 * Rw) * ( 1 - Dup)) = the number of workers children that grow up to
become workers
and
(W1 * Rw) = ((W1 * Rw) * ( 1 - Dup)) + ((W1 * Rw) * Dup)
That is: the total # of children born to workers is equal to the number of
their children that grow up and become workers plus the number of their
children that grow up to become professionals.
Now -- if I haven't lost you yet -- I'm going to say the same thing holds
true for the professional class, but here we'll use "Ddown" to represent the
percentage of professonals' children that end up in the working class. So:
((P1 * Rp) * Ddown) = the number of professionals' children that grow up to
become workers
((P1 * Rp) * ( 1 - Ddown)) = the number of professionals children that grow
up to become professionals
so that,
(P1 * Rp) = ((P1 * Rp) * ( 1 - Ddown)) + ((P1 * Rp) * Ddown)
That is: the total # of children born to professionals is equal to the
number of their children that grow up and become professionals plus the
number of their children that grow up to become workers.
Now, we just have to figure out how many of each class we have in generation
two.
Let's look at those equations again:
((W1 * Rw) * Dup) = the number of workers' children that grow up to become
professionals
((W1 * Rw) * ( 1 - Dup)) = the number of workers children that grow up to
become workers
((P1 * Rp) * Ddown) = the number of professionals' children that grow up to
become workers
((P1 * Rp) * ( 1 - Ddown)) = the number of professionals children that grow
up to become professionals
So, for W2 (the # of workers in the second generation), we want all the
children from both classes that grow to be big strong workers. So that
would be:
W2 = ((W1 * Rw) * ( 1 - Dup)) + ((P1 * Rp) * Ddown)
Do you see where those equations came from? Workers' kids that become
workers plus professionals' kids that become workers. And similarly for P2
(the # of professionals in the second generation):
P2 = ((P1 * Rp) * ( 1 - Ddown)) + ((W1 * Rw) * Dup)
So that in the second generation the number in each class is:
W2 = ((W1 * Rw) * ( 1 - Dup)) + ((P1 * Rp) * Ddown)
P2 = ((P1 * Rp) * ( 1 - Ddown)) + ((W1 * Rw) * Dup)
Now if you made it this far, you should be able to see how,
W' = ((W * Rw) * ( 1 - Dup)) + ((P * Rp) * Ddown)
P' = ((P * Rp) * ( 1 - Ddown)) + ((W * Rw) * Dup)
will hold true for every generation.
And if you have that, you can go back to that little model and plug in the
equations and you'll realize that the ratio of workers-to-professionals in
the population will stabilize of it's own accord, and that where that ratio
stabilizes is solely determined by you're Dup & Ddown variables.
Hoping you've made it this far,
-Tim
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