Glen,
Thanks for that. That makes your p(h) function very much more
interesting than what I had surmised. Depending on the value of h,
acceleration can be either positive or negative - as can be inferred
from your derivatives. So both cases get covered. Does Steve's position
also get included under the right conditions?
Grant
On 5/17/13 4:16 PM, glen e. p. ropella wrote:
Damn it Grant. Why do responses to you not go to the list by default? ;-)
Grant Holland wrote at 05/17/2013 02:41 PM:
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.
Not quite. The first one is a normal S curve. The second mode is
inverted. I don't know if I can add attachments. So, try this:
first mode:
https://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427eolc4anlkqf
second mode:
https://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427elo9c75852c
So, together, the bimodal function should look like a mesa.
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.
{ (e^(h+h_o))/(e^(h+h_o)+1)^2
d/dh = {
{ -(e^(h+h_o))/(e^(h+h_o)+1)^2
(The sign on h_o doesn't really matter, I suppose.) So, the curvature is
positive for the first mode and negative for the second. The 2nd
derivative will have the same sign as the 1st derivative, I think, which
means the convexity flips at h_o.
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?
Given the above, no. It goes through a high acceleration period near
h_o, but much less h << h_o and switches to mode 2 at h >> h_o.
Do these considerations reflect your thinking about technology growth?
Well, as I said before, I don't think it's accurate. But I do think my
"mesa" function might generally capture what people like Steve
_perceive_. I actually think that technology doesn't grow any faster or
slower on any variable. But I can see how one might _think_ it does.
E.g. with Geoff West's concept of more innovation in higher densities.
On 5/17/13 2:35 PM, glen e. p. ropella wrote:
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.
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