Chinese Children Problem

I posted this problem about a year and half ago, but it didn't generate much interest. Yet I maintain that it's a nice problem, albeit too mathy for the taste of some. So here it is again (in a modified form):

*** The Chinese Children Problem ***

In an attempt to reduce its population, China institutes the following policy: all couples may have only one child, unless both partners are themselves only children, in which case they may have two.

Various academics are persuaded to lend their support to this (rather unpopular) policy. Among these is a mathematician who makes the following suggestion: 'The government should grant a second child to some of the couples who are allowed only one. In this generation, to 9% of them. In succeeding generations to some small number of them, as necessary.'

You are the mid-level bureaucrat who has been assigned the task of writing a pamphlet to promote this policy. Your boss demands that you highlight exactly how quickly the policy can be expected to decrease the population, but in going through the relevant memoranda, you can't find any such figure(*). Your social status is not high enough for you to contact so eminent a personage as a mathematician, so you have to figure it out for yourself.

(*) You did find the following quote, however. 'The (approximate) figure '9%' is based on a model in which: (1) all people in one generation marry, and have children of the succeeding generation (as many as they're allowed); (2) people choose their mates independently of how many children they would be allowed to have; and (3) 'only' and 'non-only' children are equally numerous in the present generation.'

Jim Ferry Center for Simulation

Where have you been? China has had a one child policy, without the exceptions you describe but with some others, since 1982 or so (although it hasn't been uniformly enforced).

I really don't see the point of your question. But anyway, in your model, if the current generation is 50% 'only' and 50% 'non-only', 25% of all couples would be 'only-only', and the average allowed number of children would be .75 + 2*.25 = 1.25 per couple. If 9% of the non-'only-only' couples were allowed a second child, that would make 1.3175 children per couple.

Of course, even with a strict one-child policy, the population does not start to decrease for quite a while, because (due to rapid expansion in the preceding generation) the current childbearing generation is larger than the generation that is dying off.

Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2

Oops, I mistated the problem. The line

should have been

Here is the corrected version:

*** The Chinese Children Problem ***

In an attempt to reduce its population, China institutes the following policy: all couples may have only one child, unless both partners are themselves only children, in which case they may have two.

Various academics are persuaded to lend their support to this (rather unpopular) policy. Among these is a mathematician who makes the following suggestion: 'The government should grant honorary 'only-child' status to some 'non-only' children. In this generation, to 9% of them. In succeeding generations to some small number of them, as necessary.'

You are the mid-level bureaucrat who has been assigned the task of writing a pamphlet to promote this policy. Your boss demands that you highlight exactly how quickly the policy can be expected to decrease the population, but in going through the relevant memoranda, you can't find any such figure(*). Your social status is not high enough for you to contact so eminent a personage as a mathematician, so you have to figure it out for yourself.

(*) You did find the following quote, however. 'The (approximate) figure '9%' is based on a model in which: (1) all people in one generation marry, and have children of the succeeding generation (as many as they're allowed); (2) people choose their mates independently of how many children they would be allowed to have; and (3) 'only' and 'non-only' children are equally numerous in the present generation.'

Jim Ferry Center for Simulation

I thought having one child something they promoted, but wasn't strict law. To the extent that it is strict, the 'rather unpopular' in the description wouldn't apply.

I heard that China was actually thinking about an exception of the form that the problem describes. Sort of an interesting incentive to have one child.

Oops, I mistated the problem (it's corrected in another post). Instead, let 9% of non-only children be honorary only-children.

So if the only/non-only ratio starts out at 50/50, the next generation has 62.5% the population of the current; whereas if it starts out at 54.5/45.5, the figure is 64.9%. So your point is well taken.

However: the next generation after that has 42.5% of the original population in the 50/50 case, but only 42.0% in the 54.5/45.5 case . . .

Then again, for a0 = sqrt(sqrt(2)-1) ~= 0.6436, the initial ratio a0/(1-a0) yields a second generation with sqrt(2)-1 ~= 41.4% of the current generation, so the 9% figure doesn't seem so special.

But it is.

Maybe they could have a Youth-in-Asia policy too.

Jim Ferry Center for Simulation

What I don't see is something that says exactly what the 9% exception is intnded to accomplish. Maybe I'm missing something, but it seems 'obvious' to me that if the goal is a stable population then everyone should be allowed two children, and if the goal is to reduce population quickly then exceptions to the one person limit are counterproductive.

Yes, that's why the problem is interesting. One way to look at China's goals is that it wants to reduce the population in the n^th generation (for some n). Rather than referring to a solution that depends on n, the 9% figure corresponds to a limiting case (as n -> infinity).

Note that if every couple in one generation has only one child, then the next generation is only half as large. But the next generation after that is equally large (i.e., also half as large as the original generation). Thus, succeeding generations are, on average, 71% the size of the previous one. It's possible to do better.

Jim Ferry Center for Simulation

It is not clear to me just what you are asking, nor is it clear that you have provided enough data.

For example, if the members of the current generation are all only children, then they can all have two children and the next generation will have the same size. If the current generation is all from multi-child families, they all must have one child and the next generation will be half the size. So to answer the question that you *ask*, 'How quickly will this decrease the population', you have to know something about the existing distribution.

However, the phrasing of the problem makes me suspect that this isn't the question you are really asking. The business with the 9%, for example, suggests that you might really want to know what policy will produce either a constant population or a constant percentage of only children.

As constant population has a simple answer (two children per couple, plus a bit to compensate for early death and infertility), I would further guess that you are looking for a constant percentage of only children, in other words, what percentage of only children results (with reasonable assumptions about random mating) in the same percentage of only children in the next generation?

Once you have that percentage, you can then work out what the change in population size from generation to generation is. Similarly, you can find the fixpoint percentage for different policies (such as the 9% policy you mention) and work out the generational change in population under that policy.

Of course any such calculation would only be valid for a population that was at the fixpoint percentage of only children. A population that did not start out at that percentage might take many generations to reach it.

This all looks quite a lot like the models used in population biology for working out the expected frequency of a gene in a population, given the inheritance characteristics of the gene and the selective advantage of the various genotypes. For example, one can work out the frequency of HbS (sickle cell variant hemoglobin) that should be observed, given that the homozygote condition is lethal, and that heterozygotes have a selective advantage over homozygotes for HbA (normal hemoglobin) in the presence of malaria.

So do you mean that you want to find the value of that percentage that minimizes P(i+1)/P(i), P(i) being the size of the i^th generation?

Umm, so was that the puzzle? Trying to figure out what the mathematician's objective was? It sounds like the kind of 'answer' only a mathematician would come up with since a more practical person would instantly realize 1) the goal obviously can't be to maximize the population loss per generation forever. Sooner or later it'll be 'low enough' if it's decreasing and 2) you don't have to keep the same rules forever either.

And the assumption that people will always have the maximum number of children allowed, yet will ignore this fact when choosing a mate, is bizzare even for a mathematician.

Sorry about that. I didn't realize I'd posted the answer so soon.

There is one problem with both our solutions: there is no rigorous proof of optimality. I think keeping the number of only children at the fixed point (with the every-other-generation adjustments you suggest) is optimal (in the sense that the optimal solution for n generations approaches it as n -> infinity). But how do we know that it's better than using a sequence of every-other-generation adjustments that keep the solution in some periodic behavior?

Your solution was 100% correct for the problem that was asked, of course, even if it is unrealistic to expect a 'mid-level bureaucrat' to display that degree of mathematical sophistication. But the question of the proof remains.

Intuitively, it seems that letting the solution bounce around is bad in proportion to how big the bounces are, the worst solution being the stable one. Hmmm.

Jim Ferry Center for Simulation