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The quality–quantity trade-off: evidence from the relaxation of China’s one-child policy


This paper uses the exogenous variation in fertility introduced by China’s family planning policies to identify the impact of child quantity on child quality. We find that the number of children has a significant negative effect on child height, which supports the quality–quantity trade-off theory. Our instrumental quantile regression approach shows that the impact varies considerably across the height distribution, particularly for boys. However, the trade-off is much weaker if quality is measured by educational attainments, suggesting that the measurement of child quality is also crucial in testing the quality–quantity trade-off theory.

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  1. The claim is supported by our sensitivity analysis conducted in Section 5.

  2. Wu and Li (2012) indeed find that mothers with fewer children have higher calorie intake.

  3. For example, Short and Zhai (1998) show that the median amount of cash subsidies accounts for less than 2 % of household income.

  4. Out of the 932 ever-married women in our sample, 46 of them lost at least one child before 1993. Seventy-seven percent of the deceased children died before age 2. To examine whether families with deceased children are comparable to other families, we regress our quantity and quality measures on a dummy variable ( = 1 for families who lost at least one child and 0 otherwise) and a series of family and community characteristics. When the number of children is used as the dependent variable, the coefficient on the deceased-child dummy is − 0. 107 ( SE = 0. 137) for the boy sample and − 0. 002 ( SE = 0. 181) for the girl sample. When child height is used as the dependent variable, the coefficient on the deceased-child dummy is − 0. 168 ( SE = 0. 142) for the boy sample and − 0. 079 ( SE = 0. 159) for the girl sample. Consequently, we do not distinguish these families from others.

  5. All these multiple births are twins. Children from families with twins are shorter than others. While it is possible to use twin births as an instrument for family size, the small number of twins makes it impractical for us. Given the small number of families with twins, excluding children from these families is unlikely to have any significant effect on our results.

  6. The dependent variable is the log of annual household income and the independent variable includes paternal height, maternal height, maternal age and its squared, paternal age and its squared, and the province of residence dummies. The coefficient on paternal height is 0.010 ( SE = 0. 004) and on maternal height is 0.011 ( SE = 0. 005).

  7. According to the Chinese Compulsory Education Law, the official school entry age is 6, but in some areas where the school system cannot cope with the demand, the school entry age can be set at 7. Using the CHNS data, we find that the school enrollment rate is only about 40 % for 6-year-olds but increases to 76 % for 7-year-olds, suggesting that the norm is 7. Therefore, we focus our analysis on 7–16-year-olds.

  8. We have to exclude children whose mother did not participate in the 1993 SEMW as we do not know how many siblings they had.

  9. Admittedly, none of these variables are perfect measures of family wealth. However, as pointed out by Angrist and Pischke JS (2008), including a proxy of wealth in the regression could still be an improvement over not controlling for wealth at all.

  10. The reason for using the height of adult men is that most men live in the villages where they were born, but many women left their home village after marriage. Based on the information extracted from the 1993 SEMW, the median distance between a married woman’s house and her mother’s house was 5 km, and more that 75 % of married women lived at least 2 km from their mother’s house. The evidence suggests that more than half of married women do not live in their home village.

  11. Adding additional control variables has no impact on our conclusion.

  12. To check whether our results are mostly driven by these 30 large families, we run regressions excluding the 111 children from these families. The IV A estimate of the Q-Q trade-off effect is − 0. 527 ( SE = 0. 178) for boys and − 0. 614 ( SE = 0. 227), which are comparable to the estimates using the entire sample.

  13. To check whether our results are sensitive to excluding these observations, we have also run regressions using the full sample with missing-father dummy as an additional control variable. When paternal height is used as the dependent variable, the coefficient on the number of children is 0.368 ( SE = 1. 022) for the boy sample and − 0. 005 ( SE = 1. 157) for the girl sample. When maternal height is used as the dependent variable, the coefficient on the number of children is 0.873 ( SE = 0. 822) for the boy sample and − 0. 554 ( SE = 1. 091) for the girl sample.

  14. The covariance matrix of the estimates \(\hat \theta (\tau )\) is provided in the Appendix

  15. The norm is defined as age − 6. The coefficient on the number of siblings is − 0. 119 ( SE = 0. 090) for the IV approach and − 0. 072 ( SE = 0. 031) for the OLS approach.


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We are grateful to three anonymous referees and the editor Junsen Zhang for valuable comments.

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Correspondence to Haoming Liu.

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Responsible editor: Junsen Zhang

Appendix: The covariance matrix the IVQR estimator

Appendix: The covariance matrix the IVQR estimator

The covariance matrix of the estimates \(\hat {\theta } (\tau )=\left (\hat {\alpha }(\tau ),\hat {\beta }(\tau )\right )\) is provided by Chernozhukov and Hansen (2008):



$$\begin{array}{@{}rcl@{}} \hat{S}(\tau) &=& \tau(1-\tau)\frac{1}{n}\sum\limits_{i=1}^{n}\Psi_{i} \Psi_{i}^{\prime},\;\; \mathrm{where}\;\; \Psi_{i} = \left[\begin{array}{l}Z_{i}\\X_{i}\end{array}\right],\\\hat{\epsilon_{i}}(\tau)&=&Y_{i}-D_{i}\hat{\alpha}(\tau)-X_{i}^{\prime}\hat{\beta}(\tau)-Z_{i}^{\prime}\hat{\eta}(\tau) \\\hat{J}_{\theta}(\tau)&=&\frac{1}{n}\sum\limits_{i=1}^{n}\frac{K\left(\hat{\epsilon_{i}}(\tau)/h(\tau)\right)}{h(\tau)}\Psi_{i} \Psi_{i}^{\prime}\\\hat{J}_{\alpha}(\tau)&=&\frac{1}{n}\sum\limits_{i=1}^{n}\frac{K\left(\hat{\epsilon_{i}}(\tau)/h(\tau)\right)}{h(\tau)}D_{i}\Psi_{i}^{\prime} \\\hat{H}(\tau)&=&\hat{\bar{J}}_{\eta}(\tau)^{\prime} \hat{A}(\alpha(\tau)) \hat{\bar{J}}_{\eta}(\tau) \\\hat{A}(\alpha(\tau)) & =& \left[ \hat{\bar{J}}_{\eta}(\tau) \hat{S}(\tau) \hat{\bar{J}}_{\eta}(\tau)^{\prime}\right]^{-1} \\\hat{R}(\tau)&=&\left[\hat{J}_{\alpha}(\tau)\hat{H}(\tau)\hat{J}_{\alpha}(\tau)^{\prime}\right]^{-1}\hat{J}_{\alpha}(\tau)\hat{H}(\tau) \\\hat{L}(\tau)&=&\hat{\bar{J}}_{\beta}(\tau)-\hat{\bar{J}}_{\beta}(\tau)\hat{J}_{\alpha}(\tau)^{\prime}\hat{R}(\tau),\end{array} $$

where K( · ) is the Gaussian kernel density function, h(τ) is the bandwidth of the kernel that is given by \(h(\tau )=1.06\times \min \left (\hat {\sigma }_{\hat {\epsilon }(\tau )},\; \frac {\mathrm {Interquartile\;range\;of}\hat {\epsilon }(\tau )}{1.349}\right)n^{-\frac {1}{5}}\), \(\left [\hat {\bar {J}}_{\beta },\hat {\bar {J}}_{\eta }\right ]^{\prime }\) is a partition of \( \hat{J}_{\theta}(\tau)^{-1} \) such that \( \hat{\bar{J}}_{\beta} \) is a dim(β) × dim(β, η) matrix, and \( \hat{\bar{J}}_{\eta} \) is a dim(η) × dim(β, η) matrix.

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Liu, H. The quality–quantity trade-off: evidence from the relaxation of China’s one-child policy. J Popul Econ 27, 565–602 (2014).

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  • Education
  • Fertility
  • Quantile regression
  • Quality–quantity trade-off

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  • J13