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# The equation that changed the lives of a billion people

Song Jian is arguably the mathematician who has had the biggest impact on the twentieth century world. While other mathematicians have had their works applied to technology, health, agriculture and industry, and have greatly contributed to the fashioning of today’s world, Song Jian is unique in his direct impact on the lives of millions of individuals in a very short time. It was Song Jian who persuaded the Chinese government of the 1980s to implement a policy that would harshly influence their society for more than thirty years: the one-child policy.

## Historical context

During the 1960s and the 1970s an increasing concern regarding demographical growth developed. Anne and Paul Ehrlich’s alarmist book The Population Bomb, which predicted a worldwide famine in the 1970s and 1980s due to the growth of the world’s population, was published in 1968. That same year, the Club of Rome, a group of scientists, industrialists and public officials from about fifty countries, gathered for the first time to reflect on the issue of how to limit the exponential population growth.

Even if alien to these discussions in the Western World, communist leaders of the most populated place on Earth were nonetheless worried about the effects of demographic growth on their country’s well-being. During the 1970s, they had attempted to implement incentive policies such as the wanxishao — later marriages, spaced births and a minor number of children — and were eager for advice and perspective on the subject. It was in that context that the respected mathematician Song Jian intervened.

Born in 1931, Song Jian was a distinguished son of communist China: he had joined the National Revolutionary Army on the eve of the civil war that would end with Mao’s final success, he brilliantly completed his studies in Moscow, and came back to China to become one of the main experts in optimal control theory (the discipline on which missile guidance is based). Only military research was well considered in Mao’s China, and thus Song was always relatively well treated by the regime, in particular during the Cultural Revolution, while many intellectuals were being arrested or sent away to be “re-educated”. After Mao’s death, his successor Deng Xiaoping wished to rebalance Chinese research by encouraging researchers to pay attention to the social and economic issues China was facing. That is the reason that led Song Jian to take an interest in demographic growth.

## Developing the model

Being entitled to attend international conferences thanks to his privileged status, Song Jian discovered the works of researchers associated with the Circle of Rome in a 1978 congress in Helsinki. He understood that his discipline, optimal control theory, could be applied in an innovative way to predicting and, why not, controlling population growth.

Back in China, he organised a team of mathematicians and engineers that devised a population growth model. Like all models, this one aimed at representing reality in mathematical terms as a means to make predictions; in that case, the goal was to project an estimate of the evolution of Chinese population over the next century according to a birth rate imposed to the whole of society (see Box).

The model established by Song’s team falls within the context of the now classic Lotka-McKendrick equations. In the following box I try to describe a simplified version of the model that nonetheless includes all the essential elements.1 $$W(a,t)$$ represents the number of women aged $$a$$ in year $$t$$; for instance, $$W (20, 1980)$$ is the number of Chinese women aged 20 in 1980. We call $$M (a)$$ the share of women expected to die at age $$a$$. For instance, $$M (20) = 0.01$$ indicates that 1 percent of 20 year-old women will die within the year. From $$W (a, t)$$ we wish to predict next year’s population. Thus, we describe $$W (a + 1, t + 1)$$ (the number of women aged $$a + 1$$ in the next year) as: $$W (a + 1, t + 1) = W (a, t) – M (a) \times W (a, t)\qquad(1)$$ If we took, for instance, a = 20 and t = 1980, we would have $$W (21,1981) = W (20,1980) – M (20) \times W (20, 1980)$$ which simply indicates that the number of women aged 21 in 1981 equals the number of women aged 20 in 1980, minus those that died during the year. In order to complete the model we must somehow be able to describe $$W (0,t)$$, i.e. the number of women born during year $$t$$, whose their age is $$0$$. To do so, we can for instance introduce parameter $$g$$, which describes the share of women that give birth to a child during the year. We would thus have: $$W (0,t) = \frac{1}{2} \times g \times (W (1,t) + W (2,t) + …)\qquad(2)$$ where $$(W (1,t) + W (2,t) + …)$$ represents the aggregate of women within the population of year $$t$$.2 Factor $$1/2$$ simply indicates that there is a 50 percent chance of the child to be a girl. The model formed by equations (1) and (2) predicts the evolution of the population of women (and, by extension of the whole population) year after year. Even if a bit more precise, Song’s actual model rests on these principles. The main parameter Song thought politics could control, in the same way a missile is controlled, was birth rate ($$g$$), which, at least theoretically, could be chosen at will to adapt demographic evolution at one’s wishes.

But the model so far presented could not yet offer immediate predictions. Countless and tedious calculations would have then been necessary. In order to circumvent such difficulty, Song and his team developed a program and were able to use a bit of calculation time (only a few minutes according to one of the team’s member) from one of the few available computers in China, which were normally devoted exclusively to military use. The results presented by the machine were indisputable: given a population of almost a billion in 1979, it was absolutely necessary that Chinese women did not have more than one child throughout their lives in order to come back by 2080 to a population figure that was deemed reasonable (700 million people3) (see Figure).

## Application and consequences of the one-child policy

Then, Song managed to use his eloquence, his connections and his power of persuasion in order to bring the results of his works to the centres of power. Swollen with self-assurance thanks to an approach that was innovative for the time, difficult to grasp by the lay person, and also in line with the communist vision of an exact science that could predict the future of populations, Song succeeded in silencing the critiques coming from sociologists regarding the feasibility of such a policy. Led by self-confidence, convinced that his model stated the truth and that fertility could be controlled just as one could control a missile, Song saw his investigation reach the offices of the most important Chinese leaders.

Many witnesses5 have reported how Deng Xiaoping, Wang Zhen, Chen Yun and Hu Yaobang were impressed by the erudite equations (perhaps the reader discouraged by the boxed text in this paper will be able to imagine their dismay), and were shocked by the model’s conclusions. Sometimes exaggerating, Song succeeded in securing acceptance of the idea that the one-child policy was not the best but the only solution for China.

In June 1980, the Central Committee (the governing body of the Communist Party) proposed the radical measure through the publication of an open letter, the first version of which had been written by Song himself. On 18th September of the same year, the decision was officially enacted and the implementation of the one-child policy began.

Whereas during its first year it was an incentive measure, the implementation of the one-child policy rapidly became harsher. In particular, in 1983, 21 million people were forcefully sterilised and 14 million abortions were imposed.6 Given the problems it provoked in the countryside, the policy was softened: in rural areas it became legal to have a second child if certain conditions were met. Little by little, such conditions became less and less rigid until the one-child policy was completely abandoned in 2015.

Yet, even if it was abandoned, the one-child policy will leave a durable trace in Chinese society. First of all, due to the sex ratio at birth imbalance. Chinese society has traditionally privileged boys; thus, the one-child policy brought many families to try to avoid having a daughter through sex-selective abortion, which brought to a sex ratio at birth of 1.15 in favour of boys (instead of the usual 1.05).7 Besides, China has today millions of ‘black children’, as undeclared infants were called, most of which were girls who have had little or no access to education and healthcare.6 Finally, in backward rural areas without a public pension system, having large numbers of children have been a way to support the elderly; such a practice is nowadays endangered by the feeble number of children and the massive rural exodus.

## Conclusion

Even if the role of Song Jian must not be overstated (the Chinese birth-control policy had started a good ten years before), it is undeniable that his investigation ushered China to a radical treatment of its demographic problem. To succeed so self-assuredly in convincing the leaders, Song must have at first surely persuaded himself that his model was perfect and that it could predict the future of Chinese society for about a century without any possible doubt. Besides having underestimated the difficulties in imposing a fertility rate to a population, Song produced a model that appeared quite imprecise in its predictions.

As a modeller myself, this example perfectly illustrates for me the problems tied to the delicate art of modelling. A model always results from a context and its constraints: the question that it aims at answering, its desired precision, its simplicity, its solvability and its level of generality. The resulting model will never be able to represent reality in all of its complexity, but will instead be adapted to a limited framework. Personally, and with the benefit of 37 years of hindsight, I think Song’s model is shockingly naïve, especially to be used as the basis of a whole policy: it does not take into account issues tied to sex-ratio, to changes in women’s reproductive behaviours, or to changes in mortality patterns; issues all of which had an impact on the evolution of Chinese demography that was as determining as that of the one-child policy. To these we must add the difficulties of incorporating the social pain caused by a policy within a mathematical model, a perspective that could have been introduced if the objections from sociologists had been taken into account in an actual expertise synergy.

### Bibliography

About the model’s technical aspects:

• Bacaër, N. (2011). A short history of mathematical population dynamics (Springer Science & Business Media). Song, J., Kong, D., and Yu, J. (1988). Population system control. Mathematical and Computer Modelling 11, 11–16.

About Song Jian’s history:

• Greenhalgh, S. (2005). Missile Science, Population Science: The Origins of China’s One-Child Policy. The China Quarterly 253–276.

About the consequences of the one-child policy:

• Greenhalgh, S., and Winckler, E.A. (2005). Governing China’s Population: From Leninist to Neoliberal Biopolitics (Stanford University Press). (Chapter 8)

• Ebenstein, A. (2010). The “Missing Girls” of China and the Unintended Consequences of the One Child Policy. J. Human Resources 45, 87–115.

• Hesketh, T., Lu, L., and Xing, Z.W. (2005). The Effect of China’s One-Child Family Policy after 25 Years. New England Journal of Medicine 353, 1171–1176.

### Notes

1. See Bacaër (2011), Chapter 25, or Song et al. (1988).

2. Song’s original model introduced an aged-based weighting that took into account the fact that a woman in her twenties is more likely to have children than a woman in her forties. Besides, Song’s model operated continuously throughout time rather than on a year after year basis.

3. The reasons why Song and his team thought such a figure was optimal are not thoroughly known to us (Greenhalgh, 2005).

4. Based on Figure 1 of Greenhalgh (2005).

5. As reported in Greenhalgh (2005)

6. See Greenlagh and Winckler (2005), Chapter 8.

7. On sex-ratios see Ebenstein (2010) and Hesketh et al. (2005).