I think you've got it wrong for two reasons. 1) arithmetic growth - we can't deal with it; any growth eventually results in overpopulation, it might take a while if you start with a low population like 100 but we're not starting with a low population; we already have over 7 billion so your arithmetic growth would still be enormous. 2) you've rounded off fractions. I know you said you can't have fractions of children but if you increase your starting population to millions then you will not get that stable population. You are only getting that because you are rounding down the fractions in a small population. I hold to my original statement if you have mortality then an average NRR of 1 is necessary, if you have immortality then 0 is necessary. In both cases additional births can be permitted on any (accidental) deaths that have had no offspring.
I believe your rebuttals have 3 types of mistakes:
1-those pertaining to rounding
2-those pertaining to scaling
3-those pertaining to the distinction between exponential and arithmetic growth
I think the 1st 2 types are COMPARATIVELY simple matters of math and logic whereas the 3rd is much hairier, much more debatable, and involves not merely logic, but also physical facts about reality and assumptions about reasonable objectives and likely developments. For now I'll only deal with the 1st class - those pertaining to rounding. Later, perhaps, I may deal with the other 2, but this is becoming surprisingly time consuming.
You imply that rounding is some sort of logical mistake, a source of systemic error, or a cheat, and that without rounding the results would be fundamentally different.
First, let's dispose of the last item, the idea that without rounding the results would be fundamentally different. I'll do the same table again twice, with only rounding changed. First with no rounding, and then rounding up instead of down.
This is with NO rounding, which is actually LESS realistic, not more, a matter I'll return to, but it helps to illustrate the logic. For brevity, I'm going to omit some of this table, but you can verify the figures easily enough, either manually, algebraically, or with a spread sheet:
=======================================
Immortals, NRR = 0.9, no rounding, initial population 100
generation-0 100
population..................... 100
generation-1 90
population..................... 190
generation-2 81
population..................... 271
generation-3 72.9
population..................... 343.9
generation-4 65.61
population..................... 409.51
. . .
generation-26 6.4610818892
population..................... 941.850262997
generation-27 5.8149737003
population..................... 947.6652366973
generation-28 5.2334763303
population..................... 952.8987130275
generation-29 4.7101286972
population..................... 957.6088417248
generation-30 4.2391158275
population..................... 961.8479575523
. . .
generation-42 1.1972515183
population..................... 989.2247363357
generation-43 1.0775263664
population..................... 990.3022627021
generation-44 0.9697737298
population..................... 991.2720364319
. . .
generation-102 0.0021514733
population..................... 999.9806367402
generation-103 0.001936326
population..................... 999.9825730662
. . . to infinity asymptotically approaching:
generation-infinity 0
population..................... 1000
=======================================
You can see that these are headed asymptotically to a generation size of 0 and a final stable population size exactly 10 times the initial population. The first will be the same for any NRR less than 1, the second is specific to the particular value chosen, purely as an arbitrary example of values of NRR less than 1. Fundamentally, any value less than 1 will give you the same kind of results. If you don't use rounding the results will be similar to the preceding table. If you do use rounding, the results will be similar to the more realistic table I presented in my previous post. If you were aiming at a smaller ratio of final to initial population you'd use a smaller NRR. Obviously, the non-integer numbers for generation and total population size in the preceding table are without any physical meaning because saying you can't have fractional children wasn't just an excuse to quip, but an accurate statement about physical reality.
Now let's look at what happens when you round UP instead of down.
=======================================
Immortals, NRR = 0.9 population-immortal-rounding_up_start_at_100
generation-0 100
population.........100
generation-1 90
population.........190
generation-2 81
population.........271
. . .
generation-25 11
population.........997
generation-26 10
population.........1007
generation-27 9
population.........1016
generation-28 9
population.........1025
generation-29 9
population.........1034
. . . forever with generation size remaining 9, total population growing arithmetically.
=======================================
You can see that with rounding up, we reach the minimal value for generation size at generation-27, exactly the same as with rounding down. That's another hint that the tables WITH rounding are more realistic than the table without. At that point it reaches a floor, the exact value of which is a consequence of rounding up in conjunction with the arbitrary NRR used for illustrating the effects of NRRs less than 1. If this were a real population, this terminal sequence where generation size remains 9 forever, would, BECAUSE OF THE ROUNDING UP, correspond to a CHANGE in policy, NOT a constant policy, specifically a change of the NRR from 0.9 to 1.
That last sentence contains a hint as to what is going on here. I said at the beginning that the model for generating the tables was "oversimplified", but that "the gist is sound", meaning that none of the simplifications should affect the general nature of the result and pointed out some of the simplifications. There are many others I didn't point out, such as ignoring cloning and other non-traditional methods of reproduction, the effects of divorce and remarriage, the fact that generations don't come neatly divided into discrete waves like coordinated broadsides of cannons. These oversimplifications, and others, are for the purpose of illustrating the basic logic without horrendous complexity or a book length exposition.
The particular oversimplification relevant here is representing NRR as something that can have a constant value. When dealing with large numbers this is unimportant, but it does become important when dealing with the terminal portion of the sequences shown, where generation size becomes constant with rounding, either up or down, or, without rounding, falls between 0 and 1 and becomes physically meaningless.
Actual counts of people, whether generation size, total population size, or females in the room, will ALWAYS be positive integers or 0. A model that represents them otherwise is faulty to the extent it does so. Introducing rounding is a step in the direction of realism, NOT a step away from it. Any actual practice, regardless of whether it is individual and voluntary or social and coerced, is going to have to be a decision for some particular woman to have some integral number of children. She can't have 1.47 children even if she and society both want her to. The only way you can have a constant NRR with only integers is to fudge it by picking a special value for initial population that is chosen to force it to come out that way.
So if we want the table to correspond to a possible reality in greater detail, we have to forgo an exactly constant NRR, which isn't something policy, whether personal and voluntary or social and coercive or some combination, can dictate any more than it can dictate the value of i. This comes at the expense of a more complex exposition, but we can do it.
The NRR now becomes a sort of limiting case. It is a bit like some concepts in economics such as "elasticity of demand". The underlying phenomenon is grainy. There are an integral number of discrete transactions that occur as discrete times. There are an integral number of people in generations and in total populations and they are born at discrete times. Representing transactions or people as non-count nouns that can be the subject of simple formulas and floating point arithmetic is a convenient fiction that can make underlying realities easier to understand, just as ignoring friction while talking about the principles governing dynamic interactions between physical bodies in a system where friction IS a factor, can still assist in the comprehension of the more complex reality.
Ultimately a policy, regardless of how compliance is brought about, that aims at a value of NRR, will have to settle for integral values of generation sizes, and will have to bring that about by assigning integral values of children to each couple. Those integral values mean that the ACTUAL NRR for any given generation can be a little under, a little over, or exactly on the target value, depending on the size of the parental generation, the nature of the policy, and the value of the target NRR.
The following table represents the results of a practice of each generation having a total number of children equal to the target NRR rounded down to an integer and also shows a decimal approximation of the ACTUAL NRR for that generation. There is a huge range of possible practices that could could allocate numbers of children to couples so that the generation size would be equal to this number. For the value of the target NRR used in this example, one example would be for most couples to have 2 children and a few to just have 1. This is a very common sense result. It is no different in principle from the table I presented in my earlier post. It just is an attempt to explicitly show that speaking of a constant NRR is merely a verbal shorthand because the physical reality being described doesn't come in non-integer values and therefore actual practice must deal with integers. Rounding to integers in any model of this is therefore essential to accurately represent reality.
=======================================
Immortals, rounding_down._start_at_100
show decimal approximation of actual NRR
Target NRR = 0.9
generation-0
number in generation: 100
total population: .............100
Actual value of NRR to produce this gen: N/A
generation-1
number in generation: 90
total population: .............190
Actual value of NRR to produce this gen: 0.9
generation-2
number in generation: 81
total population: .............271
Actual value of NRR to produce this gen: 0.9
generation-3
number in generation: 72
total population: .............343
Actual value of NRR to produce this gen: 0.8888888889
generation-4
number in generation: 64
total population: .............407
Actual value of NRR to produce this gen: 0.8888888889
generation-5
number in generation: 57
total population: .............464
Actual value of NRR to produce this gen: 0.890625
generation-6
number in generation: 51
total population: .............515
Actual value of NRR to produce this gen: 0.8947368421
generation-7
number in generation: 45
total population: .............560
Actual value of NRR to produce this gen: 0.8823529412
generation-8
number in generation: 40
total population: .............600
Actual value of NRR to produce this gen: 0.8888888889
generation-9
number in generation: 36
total population: .............636
Actual value of NRR to produce this gen: 0.9
generation-10
number in generation: 32
total population: .............668
Actual value of NRR to produce this gen: 0.8888888889
generation-11
number in generation: 28
total population: .............696
Actual value of NRR to produce this gen: 0.875
generation-12
number in generation: 25
total population: .............721
Actual value of NRR to produce this gen: 0.8928571429
generation-13
number in generation: 22
total population: .............743
Actual value of NRR to produce this gen: 0.88
generation-14
number in generation: 19
total population: .............762
Actual value of NRR to produce this gen: 0.8636363636
generation-15
number in generation: 17
total population: .............779
Actual value of NRR to produce this gen: 0.8947368421
generation-16
number in generation: 15
total population: .............794
Actual value of NRR to produce this gen: 0.8823529412
generation-17
number in generation: 13
total population: .............807
Actual value of NRR to produce this gen: 0.8666666667
generation-18
number in generation: 11
total population: .............818
Actual value of NRR to produce this gen: 0.8461538462
generation-19
number in generation: 9
total population: .............827
Actual value of NRR to produce this gen: 0.8181818182
generation-20
number in generation: 8
total population: .............835
Actual value of NRR to produce this gen: 0.8888888889
generation-21
number in generation: 7
total population: .............842
Actual value of NRR to produce this gen: 0.875
generation-22
number in generation: 6
total population: .............848
Actual value of NRR to produce this gen: 0.8571428571
generation-23
number in generation: 5
total population: .............853
Actual value of NRR to produce this gen: 0.8333333333
generation-24
number in generation: 4
total population: .............857
Actual value of NRR to produce this gen: 0.8
generation-25
number in generation: 3
total population: .............860
Actual value of NRR to produce this gen: 0.75
generation-26
number in generation: 2
total population: .............862
Actual value of NRR to produce this gen: 0.6666666667
generation-27
number in generation: 1
total population: .............863
Actual value of NRR to produce this gen: 0.5
generation-28
number in generation: 0
total population: .............863
Actual value of NRR to produce this gen: 0
=======================================
I think that should adequately address rounding. Next, I'll take up scaling in another post if I get around to it.
