Question

In \(1978,\) in an effort to reduce population growth, China instituted a policy that allows only one child per family. One unintended consequence has been that, because of a cultural bias toward sons, China now has many more young boys than girls. To solve this problem, some people have suggested replacing the one-child policy with a one-son policy: A family may have children until a boy is born. Suppose that the one-son policy were implemented and that natural birth rates remained the same (half boys and half girls). Using geometric series, compare the total number of children under the two policies.

Short answer

Answer: Under the one-son policy, each family will have, on average, one more child compared to the one-child policy.

Step-by-step solution

One-Child Policy

Under the one-child policy, each family can have only one child; therefore, the total number of children for N families is equal to N.

One-Son Policy Model

We need to create a geometric series to model the one-son policy. If a family has a boy on the first try (which occurs with a probability of 1/2), they won't have more children. If they have a girl (which occurs with a probability of 1/2), they try again to have a boy. The probability of having a boy on the second try after having a girl is (1/2)*(1/2) = 1/4. If they have two girls in the first two tries, the probability of having a boy in the third try is (1/2)^3 = 1/8. We can represent this as a geometric series, where each term represents the expected number of children in a single family: Geometric Series: \(\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\cdots\)

Find Expected Number of Children per Family under One-Son Policy

To find the expected number of children in a single family under the one-son policy, we need to find the sum of the given geometric series. Let S be the sum of the series: \(S = \frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\cdots\) Multiply both sides by 1/2: \(\frac{1}{2}S = \frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\cdots\) Now, subtract the second equation from the first equation: \(\frac{1}{2}S = \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots\) Notice that the right-hand side of the equation is an infinite geometric series with a common ratio of 1/2. We can find the sum of this series: \(\frac{1}{2}S=\frac{\frac{1}{2}}{1-\frac{1}{2}}=\frac{\frac{1}{2}}{\frac{1}{2}}=1\) To find the sum of the original series (S), multiply both sides by 2: \(S=2\) So, each family is expected to have 2 children under the one-son policy.

Compare the Policies

To compare both policies, let's find the average number of children in each policy: One-Child Policy: 1 child per family One-Son Policy: 2 children per family Under the one-son policy, each family will have, on average, twice the number of children compared to the one-child policy.

Key concepts

One-Child Policy

The "One-Child Policy" was a major family planning strategy implemented in China in 1978. Its main aim was to control population growth by limiting Chinese couples to only having a single child. The policy emerged out of economic and social concerns that unchecked population growth could hinder China's development.

Here are some key aspects of the One-Child Policy:

• Intent and Implementation: The policy was designed to slow the rate of population growth and to alleviate social, economic, and environmental issues. It mainly applied to urban areas, while rural families sometimes received exceptions.
• Consequences: The policy led to a decrease in birth rates, allowing for economic growth and development in some sectors. However, it also resulted in certain social challenges, including an aging population and workforce shortages.
• Gender Imbalance: Cultural preferences for male children led to a skewed gender ratio, as some families would favor having a son due to traditional reasons, leading to fewer girls being born or surviving.

The One-Child Policy officially ended in 2015 when China moved to a "Two-Child Policy," allowing families to have up to two children.

One-Son Policy

The "One-Son Policy" is a theoretical model brought up to address one of the One-Child Policy's unintended side effects: gender imbalance. Given the preference for sons, this policy suggests families could continue to have children until they successfully have a boy.

Under this policy, families can potentially have many children but stop at their first son. This would affect the family size on average, as shown in the geometric series model:

• Population Impact: Each family's expected number of children would increase due to the aspiration of having at least one male child.
• Mathematical Model: The number of children in a family forms a geometric series, where the outcome of which child is born plays a crucial role. For example, the probabilities are halved each time a family tries for a son (e.g., 1/2 chance for a son, 1/4 chance for a son on the second child, etc.).
• Expected Number of Children: The sum of this geometric series results in families having, on average, two children. This is double the average under the One-Child Policy.

While this policy is not implemented, it introduces an interesting mathematical perspective on family planning and gender biases.

Population Growth

Population growth is a critical concept when evaluating family planning policies like the One-Child and One-Son policies. The growth rate of a population is affected by birth rates, death rates, and migration, but for these policy considerations, the focus is usually on birth rates.

Some pivotal points about population growth include:

• Impact of Policies: Policies like the One-Child Policy have significant effects on reducing birth rates, potentially slowing population growth substantially. Conversely, a theoretical One-Son Policy could accelerate growth by increasing the average number of children born per family.
• Balancing Growth and Sustainability: Rapid population growth can lead to challenges, such as limited resources, environmental strain, and economic pressures. Slowing growth can alleviate some of these issues but may also lead to problems with an aging population.
• Balance with Family Choices: Governments must balance population control measures with individual rights and cultural norms, ensuring that personal freedoms are not unduly compromised.

Understanding population dynamics through the lens of these policies helps grasp the broader implications of demographic changes on society and development.