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

Abstract

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.

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):

$$\hat{V}_{\hat{\theta}(\tau)}=\frac{1}{n}\left(\hat{R}(\tau)^{\prime},\hat{L}(\tau)^{\prime}\right)^{\prime}\hat{S}(\tau)\left(\hat{R}(\tau)^{\prime},\hat{L}(\tau)^{\prime}\right),$$

(10)

where,

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

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.