The impossible scapegoat for sex ratio imbalance in China: son preference
Among the theories and hypotheses that explain the overtly elevated male-to-female sex ratio at birth in China comes to fore the phenomenon of son preference. Usually sex ratio at birth reflects the birth bias towards a gender under the more natural status of a society, because sex ratio later in life tends to be impacted by myriads of social and economic disturbances such as war and poverty. The sex ratio at birth in China hovers over 110:100, or 1.1, according to the sixth and seventh census, an eminently higher level compared to most other countries in the world. The natural sex ratio at birth under laboratory condition should be 104:100.
When there are more boys born, people immediately look at the obvious elephant in the room — the boys, and trace that line of thought to the question of why more BOYS are born. This is the commonsensical and very comprehensible origin of the theory of son preference, an apparent explanation almost as simple and straightforward as tautology. There are academic ramifications of the son preference hypothesis and not all of them are wrong. 李树茁、韦艳、乔晓春, to name a few, are the prominent ones among Chinese demographers to point out it is the selective infanticides and abortions against female births that have inflated the sex ratio at birth in China. Unfortunately, infant death data, like any detailed death data, in China are highly classified despite many attempts by demographers such as 李若建 to obtain and analyze them. Sex-selection methods that allegedly led to more female fetuses abortions in China have received somewhat moderate level of attention and were studied by people like Therese Hesketh, 王朋岗、李树茁, although female infanticide studies are still taboo or simply missing due to data absence.
However, except for the few named scholars and studies above (a couple more people may have eclipsed my cursory reading but that’s mostly about it), most demographers, in China but also some are based in other countries, take for granted that son preference is the problem at core. For Chinese demographers, the misreading of the impact of son preference is understandable because son preference indeed causes unnatural sex ratio at birth if it is combined with other structural constraints such as a policy limit on the number of total children to be had. However, outside China’s one-child policy context, treating son preference itself as the root of the evil behind higher sex ratio reflects blatantly a lack of basic knowledge of probability and how numbers add up. My message is simple: son preference has to be combined with a fundamental context-level constraint to work against female births and son preference per se is not and cannot possibly be the problem.
The most common version of explaining how son preference distorts the natural sex ratio at birth goes like in this scenario: people under the spell of some toxic patriarchal culture want more sons for various valid reasons (economic, labor power, later life care, whatever), so they prefer to have at least one male birth in their parity. In imperial China, peasants often tried to have n number of kids until a son is born so at least someone could inherit the land and wealth under the same family name. Even sterile couples will adopt at least one male offspring, and that demand could have pushed up the “supply” of boys. But such son preference rarely caused problems until lately noticed in China.
Why? Because son preference by itself does not produce more sons than daughters. The math simply doesn’t solve. The attempt to have n number of child until a male is born exemplifies geometric distribution. Having each birth is a binomial trial with two outcomes: female or male. The probability of one outcome is just 0.5. In a single trial, the probability p for male is 0.5, 1-p for female is also 0.5. If the outcome is female with probability equal 1-p, one continues the trial until a male is born. In theory, the trial could go on forever, but for childbirth let’s say 10 trials set the maximum.
If the probability of having male in each trial is p, then the probability that the nth trial has the first male birth is: P(n)=p*(1-p)^n-1. Or, in another way, “the previous x trials all failed to produce a male until the x+1 trial” has a probability: P(x)=p*(1-p)^x.
The average number of trials (births) that one needs to have until a male is born is simply 1/p. This is the expected value of n, or the mean of the distribution. Because the expected value of n is the sum of the following infinite series of trials: E(n)=1*p+2*p*(1-p)+3*p*(1-p)² +4*p*(1-p)³+5*p*(1-p)⁴…+…+n*p*(1-p)^n-1=[1+(1-p)+(1-p)²+(1-p)³+(1-p)⁴…]. The expectation of a sum of random variables is the sum of their expectations, the above series just equals 1/p (trust me bro but proofs here).
If the probability of having a son is 0.5, then the average number of births until a son is 1/0.5=2. Since the last birth is a son, the first one has to be a daughter, so sex ratio at birth in this case is 1:1. Having a preference for son does not inflate sex ratio at birth.
In real life, the natural sex ratio at birth is 104:100, the p to have son is actually 0.51, slightly higher than equal chance. Then the average number of trials in geometric distribution is 1/.51=1.96. Because no one can have a fraction of a child, so that is just two kids. Or, the average number of failed trials is (1-.51)/.51=.96, on average people will have one daughter before they have one son. Still no inflated sex ratio at birth.
On the other hand, if we reverse the preference function and people will only stop when they have a female birth. The average number of trials is 1/.49=2.04, and average number of failed trial is (1-.49)/.49=1.04. Because no one can have a fraction of a child, and the needed expected failed trial exceeds 1, people on average must have two sons before having a daughter. That means, ironically but verily, daughter preference will inflate sex ratio at birth, not son preference.
Therefore, like I said, son preference by itself cannot possibly cause the much concerned sex ratio at birth in China today and it has borne with too much undeserved blame. However, demographers today still unthoughtfully pick on the easy one. Or they likely are simply diverting the attention for their safety in political correctness from China’s birth control policy and its ramifications on female welfare; or, they could be lying in their comfort zone and avoid committing to more nuanced and less betrodden hypotheses. For the latter, I actually have a theory that preference for daughter could be the cause of the inflated sex ratio. As we have seen, having births until a daughter results in an expected value of two sons per daughter, hiking sex ratio to the dramatic 2:1. Chinese people have a proverb “儿女双全” and many elderlies know that daughters take care of the parents. Also the tradition that men buy a real estate property for marriage and pay for bride price, which combined could easily eat up to half a million dollar, has started to persuade couples that having a daughter is a good idea for income. If Chinese people want both a daughter and a son, the fertility ideal will just push up sex ratio as we have showed. That being said, selective abortion and infanticide are still most likely the real causes imho. One child policy, or the child quota policy, literally sets an upper bound on the number of trials a couple could have in childbirth. Son preference only causes more boys to be born and stay alive when the birth trials are limited and any non-male births are terminated prenatally or even postnatally. In this context, son preference no longer describes the traditional Asian patriarchal desire “I must have a son”. Instead it has become a chilling Satanic secret desire “I would rather this daughter not born”.
The bottom line for the long story: according to simple probability math, having son preference doesn’t produce more male births at all. Actually, daughter preference will produce more sons. Son preference will elevate sex ratio at birth only through sex-selection means (e.g. abortion) which become necessary when a policy imposes a limit on the number of children.
