Glen,
That's very good! And it captures the kind of hypolinearity that you I
think you have been suggesting.
Looks like to me that your p(h) function's sensitivity to human
population size is well-considered. If I understand your parameter
constants h_o and h_f correctly, then I believe the exponent of e in
both of your cases is a positive integer. I believe this means that your
p(h) is monotonically decreasing in both cases.
So, the next thing is to consider the acceleration of p(h) - its second
derivative. This means that we are interested in its convexity. I
suspect that it is always convex for positive h. If so, then its
acceleration is always positive. Of course, a more analytical approach
to taking these derivatives is called for.
So, assuming that the population h is always increasing with time -
probably a reasonable case, then p(t) is also convex. This implies, if I
am correct, that your production function is always accelerating. Is
this correct?
Do these considerations reflect your thinking about technology growth?
On 5/17/13 2:35 PM, glen e. p. ropella wrote:
Great idea!
I actually think an accurate approximation would involve an
impredicative hierarchical model. I don't think one can isolate
technology from the humans that create it.
But absent the time to put that together, I'll go with something like:
{ 1/(1+e^-(h-h_o)), h near h_o
p(h) = {
{ 1/(1+e^(h+h_f)), h >> h_o
where h is the population of humans and h_o is some
tech-accelerating-maximum population of humans. h_o becomes some sort
of "optimal clique size". h_f is some sort of failure size larger than h_o.
Grant Holland wrote at 05/17/2013 11:51 AM:
Glen's latest retort on this thread (see below) gave me this thought: It
would be interesting if you guys could offer an (admittedly
oversimplified) analytical model of your best guesses on what the
productivity function and the acceleration function (2nd derivative of
the production function) of "technology" over time would be. Such a
model, though simplistic, would force some careful thinking.
For example, if you believe that the production of technology over time
(p) is linear, or p = mt, then the acceleration would be 0. If you think
p is strict exponential, or p = e**t (as Steve might), then the
acceleration would be e**t. If you think it is cyclical (periodic) (say,
p = sin(t)), then the growth rate is cyclical, e.g. p = -sin(t). (Maybe
Glen thinks something like that.) Of course, the productivity function
is actually none of these but probably some analytic series, or whatever.
Anyway, this kind of thinking could at least be subjected to past
history and be a more quantifiable conversation promoter.
Just an idea.
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