# Huang Wenzheng: the illusion of fertility in a hypothetical case

The illusion of fertility in a hypothetical case

When talking about population trends, we often hear that Chinas fertility rate is not low at all. There are not many only children in our childrens class. Many students have brothers and sisters, and some have several. So in this case, is fertility really high? Generally speaking, fertility rate refers to the average number of children per family. Lets discuss the illusion in fertility rate estimation through a hypothetical example.

Of the 14 children in this small class, only three are only children, less than 1 / 4 of the total; seven are from families with three or four children, accounting for half of the total. Seeing this situation, many peoples intuition is that the parents of these children have too many children, so the population will expand. Is this intuition reliable?

That is to say, according to this estimation method, each family has an average of 2.57 children. The intuitive feeling is that there are more children than parents, and the population will expand.

But in fact, this is just an illusion. This algorithm overestimates the fertility rate seriously. This is because the fertility rate reflects the average number of children in each family, so the basic sampling unit is the family, not the individual child. If the kindergarten class is regarded as a sample from the whole society, it is 14 children, not 14 families.

Whats the difference between family and child extraction? On the contrary, if the whole society randomly selects families, then no matter how many children a family has, the probability of being selected is the same. But when children are randomly selected, families with more children are more likely to be selected. Therefore, to estimate the fertility rate based on the number of childrens brothers and sisters, we need to consider the impact of the number of children in each family on the probability of family being selected.

Since an n-child family corresponds to n children, we can equally say that a child from an n-child family corresponds to 1 / N family. According to this relationship, each only child corresponds to one family, each child from two child family corresponds to one-half family, each child from three child family corresponds to one-third family, and each child from four child family corresponds to one-quarter family.

Therefore, 14 children in the kindergarten class correspond to 3 * (1 / 1) + 4 * (1 / 2) + 3 * (1 / 3) + 4 * (1 / 4) = 7 families. It can be calculated that there are 14 / 7 = 2 children in each family, far less than the previous calculation of 2.57 children.

When people make intuitive judgments, it is natural to default that a child represents a family and use arithmetic mean, so as to give multiple children a proportion of the number of children, but the real use should be harmonic mean. This is because for n-child families, n children can represent a family, and the equivalent is that a child can represent 1 / N family. In short, when judging the fertility rate according to the number of childrens brothers and sisters, peoples intuitive feeling is close to the arithmetic average, but actually what should be used is the harmonic average.

However, if the number of siblings is not equal, the harmonic average must be less than the arithmetic average. The greater the difference in the number of children between different families, the greater the arithmetic average is than the harmonic average, and the more serious the illusion of overestimation of fertility rate is.

Population trends in hypothetical cases

However, it should be noted that even if reconciliation is used, the fertility rate will still be overestimated, because those who are unmarried, infertile and have no children are not included in the denominator.

The figure below shows a possible situation of parents generation in the small class of this kindergarten. In the picture, seven couples have children, including one with four children, one with three, two with two and three with one. In other words, seven families have 14 children, with a fertility rate of 2. This is the same as the fertility rate calculated by harmonic average in our last section.

However, considering that there are 1 infertile couple in every 8 couples in China, the picture also includes 1 couple without children. In this case, the previous generation of eight couples had only 14 children, with a fertility rate of only 1.75. Moreover, because there are more boys than girls at birth, and not every girl can survive until the end of childbirth, there are more than 16 parents at birth. If the previous generation starts with 17 people and turns into 14 people by the childrens generation, this is equivalent to a 17.6% (i.e. 1-14 / 17) per generation and a 32.2% (i.e. 1 - (14 / 17) ^ 2) per two generations.

In demography, replacement level refers to the fertility rate needed to maintain the number of children equal to that of their parents. According to the sex ratio at birth and female survival rate, the replacement level of Chinas fertility rate is about 2.15, which means that each couple needs to have an average of 2.15 children to keep the number of children equal to that of their parents.

Compared with the replacement level of 2.15, the fertility rate of 1.75 in the figure means a reduction of 18.6% (i.e. 1-1.75 / 2.15) per generation and 33.7% (i.e. 1 - (1.75 / 2.15) ^ 2) per two generations. That is to say, even if half of the children come from families with three or four children, and less than one fourth of the only children, the population may shrink rather than expand.

It may be said that many of the children in this case come from the same family, which is inconsistent with the reality. For example, four four children from the same family. If they come from different families, the use of harmonic averaging may not be right. But its a misunderstanding. We did not mention that these children come from the same or different families in the algorithm description in the previous section, because this does not affect the calculation conclusion at all.

As long as each record shows the number of brothers and sisters of the child, the arithmetic average reflects the average number of brothers and sisters of each child, including himself, rather than the average number of children per family. The two concepts look similar, but they are actually different. Whether the data comes from a sample or a complete population, we should use harmonic average instead of arithmetic average to estimate the average number of children in each family.

In order to make readers more familiar with this point, we assume that in every 100 children in the whole society, there are 35 only children, 20 from two childrens families, 20 from three childrens families, 15 from four childrens families, and 10 from five childrens families. What is the fertility rate of the childrens parents? The intuitionistic fertility rate is the arithmetic mean, that is (35 * 1 + 20 * 2 + 20 * 3 + 15 * 4 + 10 * 5) / 100 = 2.45, but the reasonable estimate should be the harmonic mean, that is, 100 / (35 * 1 + 20 / 2 + 20 / 3 + 15 / 4 + 10 / 5) = 1.742.

If one eighth of couples have no children, the actual fertility rate is only 1.524. In this case, although a large number of children come from families with four or five children, the fertility rate is still far below replacement level.

3. Illusion of Chinas fertility rate

As the previous example shows, even if two or even three or four children are common, the actual fertility rate may have been far below the replacement level. This can be confirmed in other countries. For example, visitors to Japan will find that families with two or even three children are common in Tokyo subway or in rural Kansai, while only children are rare. But Japan is actually one of the lowest fertility countries in the world. Japans fertility rate has been between 1.26 and 1.45 in the past 10 years, according to official data. No one questions the accuracy of Japanese statistics.

When Huang Wenzheng, one of the authors of this paper, went to Hopkins University in 1992 to study as a doctor, he argued with an American classmate of the same department about Chinas family planning policy. This student, sterling Hilton, comes from a large family that advocates multiple parenting. He has nine brothers and sisters and seven wives. The couples goal at the time was to have five or six children, but they ended up with four, including an adopted black child. Their nine brothers and sisters have 43 children.

In the early debates, Huang Wenzhengs position was to defend family planning. But when he taught at Harvard after graduation, he accidentally saw an article by Yi Fuxian listing the fertility rates of some countries in the world, and found that the fertility rate in East Asia was almost the lowest. This made him realize that the so-called view that Chinese people especially like to have children was totally taken for granted, and therefore began to doubt the direction of the whole population policy. Later, he found that all the opinions supporting fertility were specious. At first glance, it makes sense, but in-depth analysis, none of them can stand up. Therefore, after returning to China in 2010, Huang Wenzheng began to pay close attention to population issues. In 2012, he began to cooperate with Liang Jianzhang, another author of this paper, on population issues, calling for the abolition of birth restrictions and vigorously encouraging birth.

In the same year, Huang Wenzheng recalled the debate with his classmate sterling. He found the e-mail address of the other party through the Internet and sent an email to inform the other party that his early opinion was wrong. Now in China, he calls for the abolition of birth restriction. Sterling was very happy after receiving the email. She replied to a long letter with the following family photo attached. This is his fathers birthday, parents and all of their younger generation and spouse present and take photos. There are 57 people in the picture, 80 If the younger generation and spouse all arrive at the same time. Huang Wenzheng himself is the only child after 60, and he has only one daughter. His father is the same age as sterlings father. If his father had a birthday at that time, there would be only five people in the family, only one sixteenth of the total number of sterlings father.

Figure 2: parents and their younger generation of sterling Hilton, 23 others not present

According to the intuitive sense, the Americans around us have such a big family. Should the American population explode? But in fact, the fertility rate in the United States was only near the replacement level before, and even below the replacement level in recent years.

Chinese society has a very contradictory response to the comparison of fertility rates between China and foreign countries. No matter in the domestic streets, scenic spots or even rural areas, many people will be surprised to see Chinese families with many children, and they will take it for granted to see foreign parents with two or three children in developed countries. Nevertheless, when it comes to fertility, many people are adamant that Chinas fertility rate is higher than that of developed countries. For example, the 2000 census shows that Chinas fertility rate is at a very low level. Some scholars question how Chinas fertility rate can be lower than that of Europe?

In fact, even in the domestic media can occasionally see reports about the European family. For example, on October 20, 2019, the mirror reported that a British couple excitedly announced that their 22nd child would be born next April. Despite the prevalence of multi child families in the UK, the fertility rate in the UK is now less than 2. Most of the comments on Chinas online reports of such foreign multi child families are positive, envied and blessed. In contrast, comments on reports of families with multiple children in China are more negative, and even some readers think that Chinas fertility rate is still too high.

Behind this assumption, in addition to the previously mentioned illusion of intuitively overestimating the fertility rate, there may also be a long-term one child policy, which makes many people default that there are only one or two children in each family as normal. Against this kind of subconscious background, it is easy to cause a huge psychological impact to witness the so-called super birth family of three or four children.

This kind of impact is especially strong for those who live in big cities. They generally have only one child, but they see a steady influx of young people from other places. When they learn that many young people from the countryside have several brothers and sisters, their natural response is that after so many years of family planning, how can there be so many super births? From this, they will come to the wrong conclusion: there are too many rural residents.

But even if there are several brothers and sisters in rural young people, our previous example also shows that this does not mean that the fertility rate was high in that year. Moreover, only children from rural areas are not rare, which in turn shows that the fertility rate in rural areas of China was already far below the replacement level. The data of the past two decades have proved this point again and again.

Behind the sharp decline in rural fertility. It is a large number of rural people who come to work and live in the city, facing greater pressure of life, and divorced from the previous atmosphere of family and friends, and the concepts of family succession, raising children and preventing the elderly are weakened. At the same time, the level of social and economic development in rural areas is gradually improving, and the reality and opportunity cost of raising children are increasing. Numerous surveys over the past decade or two have shown that the willingness to have children in rural China is even lower than that in Japan and South Korea. The era of a lot of rural life has long passed.

IV. Chinas fertility and population trends

This judgment is not surprising, because under the long-term birth restriction, Chinese cities have regarded giving birth to a child as the normal default choice, while the rural areas are looking to the city. This is an unprecedented phenomenon in human history. South Korean society does not take one child as the default choice like China. The survey data also shows that the ideal number of children in South Korean families exceeds 2, which is higher than that in rural China. But according to the latest data, the fertility rate in South Korea has fallen below 1.0. Is there any reason to think that Chinas natural fertility rate will be significantly higher than that of South Korea in the future?

This rate of decline may exceed many peoples expectations, because population changes are exponential under constant fertility. For exponential changes, people usually overestimate the short-term trend and underestimate the long-term trend.

When it comes to index changes, many people are familiar with this story. Put one grain of rice in the first lattice and two meters in the second lattice of the chessboard, and then the number of grains in each lattice will double, until the 30th lattice, there will be 1.07 billion grains of rice. To turn this story around, that is to say, if each generation is halved, after 30 generations, it will be less than 1000 years, and the population of 1 billion will disappear completely. Of course, this is a simple extrapolation based on the current trend. If, after the full liberalization, we vigorously encourage childbearing, especially reduce the burden of raising families, so that young people are willing to have children, it is not entirely impossible to significantly increase the fertility rate in the future.

What we hope to use this article to illustrate is that when people feel that children from three or four child families are very common, the fertility rate may not be at an alternative level. When people feel that there are a lot of two child families, but often can see only one child, the fertility rate has been far below the replacement level, and the birth population is rapidly shrinking. When people feel that the only child is very common, the fertility rate is already at a very low level, and there is no doubt that the birth population is shrinking rapidly.

In fact, only when there are generally three or four children around, can the deficit caused by those families who have one or no children be made up, and the nation can continue. If you understand this articles explanation of the illusion of overestimation of fertility, you should realize how far away Chinese society is now from the state of fertility that can maintain the normal reproduction of the nation.

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