> If everyone has one or less children, the population does not grow exponentially.
False. An exponential function, and a growing exponential function, differ only in the value of the exponent.
Here's an equation that predicts future population based on the value of an exponential term:
The equation: pt - p0 e^rt
pt = population at time t
p0 = present population
r = rate of population growth, 1% = .01
t = time
A result at 40 years for a population rate of change of -1.7% per year:
e^(-.017 * 40) = .5066
This result says that, for a population growth rate of -1.7%, after 40 years the population will be about half its present size. It's an exponential result, it's even an exponential growth rate, but the growth rate is negative.
The red trace is an exponential rate of change for r = +1.7%, the green trace for r = -1.7%. Both are exponential results, and both reflect what a population would look like for given r values.
False. Put all your bullshit math aside because that's what it is, bullshit trying to move the goalpost. The statement I made is absolutely factually true. If everyone has one or less children, the population does not grow exponentially.
We're not talking about how to model population growth, or functions, we're talking about whether every couple having one child results in exponential growth and it simply does not. If the population decreases, that is not exponential growth. An exponential result != exponential growth. You are wrong and we are done.
And, everything else aside, all the described rates are exponential.
> We're not talking about how to model population growth ...
WHAT?? Here is what you said: "If everyone has one or less children, the population does not grow exponentially." How exactly is that NOT talking about how to model population growth?
Also, it's "one or fewer children," not "one or less children."
> An exponential result != exponential growth
False in a discussion about population and exponential growth rates, terms you chose.
It's sad that you can't admit that a population decrease is not exponential growth, but your ego clearly can't handle being wrong so troll somewhere else.
Quotation: "The formula for exponential growth of a variable x at the (positive or negative) growth rate r, as time t goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is xt = x0 (1 + r)^t"
Let me emphasize that for you, in case your eyes are giving out:
"THE FORMULA FOR EXPONENTIAL GROWTH OF A VARIABLE X AT THE (POSITIVE OR NEGATIVE) GROWTH RATE R ..."
Now circle the word you're most unfamiliar with, raise your hand, and the teacher will administer a sedative.
You seem unable to read English. Let me make it clear for you, you're trying to make a lay conversation about population into a math problem which it is not, in doing so you are changing the meaning of lay words into their math definitions and misunderstanding the points being made to you in the same way a scientist means something different by the word theory than a lay person does. Stop talking, stop doing math, and listen, pay attention to context, this is not a math class and as I'm the commenter you replied to, I'm the one who sets the context and I've said not a math conversation; this is a lay conversation, use the lay definition of words.
You've been told numerous times to stop thinking about math and read the words but you clearly don't grok or can't remove your math head and have normal conversation. You're still talking about modelling growth, no one else is or was. Turn off your math head and read the English words using their lay definitions, a decline in population is not exponential growth as a decline in population is not any kind of growth because it isn't growth. If you can't bring yourself to admit that, you are beyond hope.
Secondly, I'm a computer programmer, I understand the math, so stop trying to lecture me on math when you're the one not listening. I can't help it you don't understand the point being made to you because you lack the ability to be aware of the context of the conversation you're in. I know what I'm trying to communicate and I've made it as clear as I can numerous ways; if your brain is unable to work in multiple contexts, that's your failing in understanding how to communicate with others.
If you try and prove your point with equations one more time, or reference them one more time, then you've utterly failed to understand what is being communicated to you and we're done because I don't talk to brick walls who can't think.
Wait, what? Replying to a comment that is clearly about mathematics, with a comment that is clearly about mathematics, is trolling?
The OP's comment, including " ... changing the common definition of exponential by saying the exponent could be 1 or less than one" is (a) mathematical and (b) wrong and misleading. The "common definition of exponential" accommodates any valid argument to exp(x), and to claim otherwise requires a mathematical correction to avoid misleading people.
This is a very disappointing comment line. I know not everyone here is a CS major, but if you're even CS-knowledgable you should understand what an exponential process is. It is not, as many laypeople assume, something that is "very big".
That said, I was not intentionally trolling. I do think it's possible for a SS-style benefit to sustain itself provided that an exponentially growing population, whose exponent is greater than 1, is obligated to pay in.
A country with a stagnant or declining population will probably have trouble.
You may disagree, that's fine.
Nevertheless, any population whose growth (or lack of growth) is defined by the average number of children per member will behave exponentially.
No it isn't. If everyone has one or less children, the population does not grow exponentially.