China childbirth stats analysed

I read this infographic today about China’s one-child policy. One of the things that jumped out at me is that China has more men than women in their population, and the infographic (at the bottom right) mentions that some feel the policy makes this worse, along with infanticide and preferential abortion. So, I wanted to know how big the effect of the policy is and I did some stats.

The infographic explains the process as follows:

• As it is traditionally enforced, couples are encouraged to have only one child
• If that child is a girl, parents would be allowed to have a second child to hopefully bear a son
• After the second child, the would be discouraged from having a second child.

Now, assuming that this is exactly enforced as written, there are only three outcomes of parents having children (let’s assume the chance of a birth resulting in a boy is x and ignore things like multiple births etc):

Now, we can calculate the expected number of children, assuming everyone wants children and follows this process, this is simply 1 child with probability x and 2 with probability (1-x), so expected value is 1x + 2(1-x)=2-x. x is typically near 0.5, and they say that the fertility is near 1.5. Success! But what about the expected ratio of boys to girls?

Using the table above, multiplying the “boys” column by the “probability” column and sum ming, we end up with 1x – (1-x)x=2x–x²=x(2-x) boys expected. For girls, it’s 1(1-x)x + 2(1-x)(1-x)=(2-x)(1-x). If x is 0.5, this ends up with 0.75 boys and 0.75 girls. That doesn’t explain what we observe in China… In terms of the ratio of boys to girls, we see that the (2-x) term is common to both expected values, so the ratio ends up being x/(1-x). But this is exactly the same ratio that we would expect if no birthing policy is in place, as every birth would produce x boys and (1-x) girls on average. So, either these assumptions don’t hold, or we need to look elsewhere for the discrepancy.