Correction to: J Popul Econ (2019) 32:1315–1352.

It has been brought to our attention that the comparative statics derived in Section 3.3.1 of our paper and presented in Table 1 (p. 1326) are wrong.

To see this, it is easiest to first re-express eqs. (11a) through (11c) as:

$${\mathrm{R}}_{\mathrm{x}}^{\mathrm{t}}=\frac{\frac{\mathrm{\partial Q}}{\partial {\mathrm{Z}}_{\mathrm{t}}}}{\frac{\mathrm{\partial Q}}{\partial {\mathrm{Z}}_{\mathrm{x}}}}=\frac{\mathrm{w}}{\mathrm{p}}\cdotp \left({\mathrm{A}}^{\mathrm{P}}-\mathrm{a}\right)\cdotp \left(\frac{\frac{\mathrm{d}\mathrm{t}}{\mathrm{d}{\mathrm{Z}}_{\mathrm{t}}}}{\frac{\mathrm{d}\mathrm{x}}{\mathrm{d}{\mathrm{Z}}_{\mathrm{x}}}}\right)=\frac{\mathrm{w}}{\mathrm{p}}\cdotp {\mathrm{a}}_{\mathrm{w}}\cdotp {\uppsi}_{\mathrm{x}}^{\mathrm{t}}$$
(11a)
$${\mathrm{R}}_{\mathrm{x}}^{\mathrm{a}}=\frac{\frac{\mathrm{\partial Q}}{\partial {\mathrm{Z}}_{\mathrm{a}}}}{\frac{\mathrm{\partial Q}}{\partial {\mathrm{Z}}_{\mathrm{x}}}}=\frac{\mathrm{w}}{\mathrm{p}}\cdotp \left({\mathrm{T}}^{\mathrm{P}}-\mathrm{t}\right)\cdotp \left(\frac{\frac{\mathrm{d}\mathrm{a}}{\mathrm{d}{\mathrm{Z}}_{\mathrm{a}}}}{\frac{\mathrm{d}\mathrm{x}}{\mathrm{d}{\mathrm{Z}}_{\mathrm{x}}}}\right)=\frac{\mathrm{w}}{\mathrm{p}}\cdotp {\mathrm{t}}_{\mathrm{w}}\cdotp {\uppsi}_{\mathrm{x}}^{\mathrm{a}}$$
(11b)
$${\mathrm{R}}_{\mathrm{t}}^{\mathrm{a}}=\frac{\frac{\mathrm{\partial Q}}{\partial {\mathrm{Z}}_{\mathrm{a}}}}{\frac{\mathrm{\partial Q}}{\partial {\mathrm{Z}}_{\mathrm{t}}}}=\frac{\mathrm{w}\cdotp \left({\mathrm{T}}^{\mathrm{P}}-\mathrm{t}\right)}{\mathrm{w}\cdotp \left({\mathrm{A}}^{\mathrm{P}}-\mathrm{a}\right)}\left(\frac{\frac{\mathrm{d}\mathrm{a}}{\mathrm{d}{\mathrm{Z}}_{\mathrm{a}}}}{\frac{\mathrm{d}\mathrm{t}}{\mathrm{d}{\mathrm{Z}}_{\mathrm{t}}}}\right)=\frac{{\mathrm{t}}_{\mathrm{w}}}{{\mathrm{a}}_{\mathrm{w}}}\cdotp {\uppsi}_{\mathrm{t}}^{\mathrm{a}}$$
(11c)

From these equations, it is more straightforward to arrive at the corrected Table 1:

Corrected Table 1.

Comparative statics of parenting style and traditional models.

With respect to:
Partial derivative of: In model: $$\frac{\boldsymbol{\partial}}{\boldsymbol{\partial}{\boldsymbol{A}}^{\boldsymbol{P}}}$$ $$\frac{\boldsymbol{\partial}}{\boldsymbol{\partial}{\boldsymbol{A}}^{\boldsymbol{P}}}$$ $$\frac{\boldsymbol{\partial}}{\boldsymbol{\partial}{\boldsymbol{A}}^{\boldsymbol{P}}}$$
$${\boldsymbol{R}}_{\boldsymbol{x}}^{\boldsymbol{t}}$$ Parenting $$\frac{\mathrm{w}}{\mathrm{p}}{\uppsi}_{\mathrm{x}}^{\mathrm{t}}\frac{\partial {\mathrm{a}}_{\mathrm{w}}}{\partial {\mathrm{A}}^{\mathrm{P}}}$$ $$\frac{\mathrm{w}}{{\mathrm{p}}^2}{\uppsi}_{\mathrm{x}}^{\mathrm{t}}\left(\frac{\partial {\mathrm{a}}_{\mathrm{w}}}{\mathrm{\partial p}}\mathrm{p}-{\mathrm{a}}_{\mathrm{w}}\right)$$ $$\frac{\uppsi_{\mathrm{x}}^{\mathrm{t}}}{\mathrm{p}}\left({\mathrm{a}}_{\mathrm{w}}+\mathrm{w}\frac{\partial {\mathrm{a}}_{\mathrm{w}}}{\mathrm{\partial w}}\right)$$
Traditional 0 $$-\left(\frac{\mathrm{w}}{{\mathrm{p}}^2}{\uppsi}_{\mathrm{x}}^{\mathrm{t}}\right)$$ $$\left(\frac{\uppsi_{\mathrm{x}}^{\mathrm{t}}}{\mathrm{p}}\right)$$
$${\boldsymbol{R}}_{\boldsymbol{x}}^{\boldsymbol{a}}$$ Parenting $$\frac{\mathrm{w}}{\mathrm{p}}{\uppsi}_{\mathrm{x}}^{\mathrm{a}}\frac{\partial {\mathrm{t}}_{\mathrm{w}}}{\partial {\mathrm{A}}^{\mathrm{P}}}$$ $$\frac{\mathrm{w}}{{\mathrm{p}}^2}{\uppsi}_{\mathrm{x}}^{\mathrm{a}}\left(\frac{\partial {\mathrm{t}}_{\mathrm{w}}}{\mathrm{\partial p}}\mathrm{p}-{\mathrm{t}}_{\mathrm{w}}\right)$$ $$\frac{\uppsi_{\mathrm{x}}^{\mathrm{a}}}{\mathrm{p}}\left({\mathrm{t}}_{\mathrm{w}}+\mathrm{w}\frac{\partial {\mathrm{t}}_{\mathrm{w}}}{\mathrm{\partial w}}\right)$$
Traditional N/A N/A N/A
$${\boldsymbol{R}}_{\boldsymbol{t}}^{\boldsymbol{a}}$$ Parenting $$\frac{\uppsi_{\mathrm{t}}^{\mathrm{a}}}{{\mathrm{a}}_{\mathrm{w}}^2}\left(\frac{\partial {\mathrm{t}}_{\mathrm{w}}}{\partial {\mathrm{A}}^{\mathrm{P}}}{\mathrm{a}}_{\mathrm{w}}-{\mathrm{t}}_{\mathrm{w}}\frac{\partial {\mathrm{a}}_{\mathrm{w}}}{\partial {\mathrm{A}}^{\mathrm{p}}}\right)$$ $$\frac{\uppsi_{\mathrm{t}}^{\mathrm{a}}}{{\mathrm{a}}_{\mathrm{w}}^2}\left(\frac{\partial {\mathrm{t}}_{\mathrm{w}}}{\mathrm{\partial p}}{\mathrm{a}}_{\mathrm{w}}-{\mathrm{t}}_{\mathrm{w}}\frac{\partial {\mathrm{a}}_{\mathrm{w}}}{\mathrm{\partial p}}\right)$$ $$\frac{\uppsi_{\mathrm{t}}^{\mathrm{a}}}{{\mathrm{a}}_{\mathrm{w}}^2}\left(\frac{\partial {\mathrm{t}}_{\mathrm{w}}}{\mathrm{\partial w}}{\mathrm{a}}_{\mathrm{w}}-{\mathrm{t}}_{\mathrm{w}}\frac{\partial {\mathrm{a}}_{\mathrm{w}}}{\mathrm{\partial w}}\right)$$
Traditional N/A N/A N/A

The discussion of the comparative statistics (p. 1326) remains unaffected by this correction. A corrected Appendix for the paper can be found here.

* We thank Linfeng Fan at the School of Labor and Human Recourses of Renming University of China for carefully going through our derivations and making us aware of this mistake, and Tiffany Ho for verifying our corrections in this errata.