Time with sons and daughters

Abstract

We present a model of parental investment in child quality in which the effectiveness—objectively or as perceived by the parents—of parental childcare depends on the sex of the child. In particular, the time of the same-sex parent is more productive than that of the opposite-sex parent. When parents have equal wages, efficiency considerations dictate that a parent spends more time with a same-sex child than with an opposite-sex child, but parents allocate the same total time to boys and girls, and costs of raising a boy are the same as raising a girl. When wage rates differ, and the mother is the lower-waged parent, it is cheaper to produce child quality of girls than of boys. We show that many of the empirical results in terms of a different time allocation pattern, total amount of time invested in a child, expenditures on child consumption goods, and family size and composition can be explained by this technological difference and the gender wage gap, without relying on parental preferences for girls versus boys. Our analysis is largely diagrammatic.

Appendix

Proof of Remark 1

Inspection of the FOC’s from (1) yields

$$ \frac{w_{w}}{w_{h}} =\frac{\partial F^{c}(t_{w},t_{h})/\partial t_{w}}{ \partial F^{c}(t_{w},t_{h})/\partial t_{h}} $$

(16)

$$ w_{j} =\frac{\partial Q(F^{c}(t_{w},t_{h}),x^{c})/\partial F^{c}}{\partial Q(F^{c}(t_{w},t_{h}),x^{c})/\partial x^{c}}\frac{\partial F^{c}(t_{w},t_{h})}{\partial t_{j}},\quad j=w,h $$

(17)

The FOC’s for cost minimization of Q(F(t _w , t _h ), x ^c) are

$$ \begin{aligned} w_{j}-\lambda ^{c}\frac{\partial Q\left( F^{c}(t_{w},t_{h}),x^{c}\right) }{ \partial F^{c}}\frac{\partial F^{c}(t_{w},t_{h})}{\partial t_{j}} &=0,\quad j=w,h \\ 1-\lambda ^{c}\frac{\partial Q\left( F^{c}(t_{w},t_{h}),x^{c}\right) }{\partial x^{c}} &=0 \end{aligned}$$

Conditions (16)–(17) above can be recovered from these FOCs. Finally, minimizing the cost of t _q yields FOCs

$$ w_{j}-\mu \left( \partial F^{c}(t_{w},t_{h})/\partial t_{j}\right) =0,\quad j=w,h $$

Thus condition (16) is necessary for all three problems. $\square$

Proof of Lemma 2

Marginal costs$\square$

Let $C\left( w_{ss},w_{os},t_{q}\right)$ be the cost function of quality time. We want to know the change in marginal cost as w _ss switches from w _w to w _h and as w _os switches from w _h to w _w . That is, dw _ss  = w _h  − w _w , dw _os  = w _w  − w _h . By the FOC’s of the cost minimization problem, $\mu =w_{os}/\left( \alpha f^{\prime }\left( t_{os}\right) \right) =w_{ss}/f^{\prime }\left( t_{ss}\right)$ and by the Envelope Theorem, $\partial C/\partial t_{q}=\mu .$ We want to determine the sign of $\frac{\partial C}{\partial t_{q}\partial w_{ss}}dw_{ss}+\frac{\partial C}{\partial t_{q}\partial w_{os}}dw_{os}.$ We need to sign

$$ \Updelta =\left( \frac{\partial C}{\partial t_{q}\partial w_{ss}}-\frac{ \partial C}{\partial t_{q}\partial w_{os}}\right) dw_{ss} $$

where dw _ss  > 0. By the FOC’s we can write

$$ \begin{aligned} \frac{\partial C}{\partial t_{q}\partial w_{ss}} &=-\frac{w_{os}f^{\prime \prime }\left( t_{os}\right) }{\alpha \left( f^{\prime }\left( t_{os}\right) \right) ^{2}}\frac{\partial t_{os}}{\partial w_{ss}} \\ \frac{\partial C}{\partial t_{q}\partial w_{os}} &=-\frac{w_{ss}f^{\prime \prime }\left( t_{ss}\right) }{\left( f^{\prime }\left( t_{ss}\right) \right) ^{2}}\frac{\partial t_{ss}}{\partial w_{os}} \end{aligned} $$

By Shepard’s Lemma, Young’s Theorem and concavity of the cost function in wage rates $\partial t_{os}/\partial w_{ss}=\partial t_{ss}/\partial w_{os}>0,$ and we can factor out μ. To sign $\Updelta,$ we need to compare ($-f^{\prime \prime }\left( t_{os}\right) /f^{\prime }\left( t_{os}\right) )$ to ($-f^{\prime \prime }\left( t_{ss}\right) /f^{\prime }\left( t_{ss}\right) )$ where t _os  < t _ss as our starting point is the cost-minimizing time inputs to produce a certain level of quality time for a girl. If $-f^{\prime \prime }\left( t_{os}\right) /f^{\prime }\left( t_{os}\right) >-f^{\prime \prime }\left( t_{ss}\right) /f^{\prime }\left( t_{ss}\right)$ marginal cost is higher to produce the same quality time for a boy as for a girl. $\square$

Proof of Proposition 7

Consider a family with a boy. Let B be the optimal child quality and $x_{B}^{P}$ the optimal parental consumption. Denote the marginal cost of B by $p_{B}.$ Next, add a fixed amount of full income denoted by I _B , so that $\max_{x^{p},Q^{b}}U\left( x^{p},Q^{b}\right)$ subject to the budget constraint (bc) $x^{p}+p_{B}Q^{b}=w_{h}+w_{w}+I_{B}$ yields $\left( x_{B}^{p},B\right).$ Note that $I_{B}=p_{B}B-\psi ^{b}\left( B\right).$ The same child quality for a girl, Q ^g = B, has a lower cost. Construct a new bc with parental consumption $x_{B}^{p}+I_{G}=w_{h}+w_{w}-\psi ^{g}\left( B\right)$ and B on the bc. Note that $I_{G}=\psi ^{b}\left( B\right) -\psi ^{g}\left( B\right)$ and hence the bc through the point where Q ^g = B on the actual bc of parents with a girl is given by $x^{p}+p_{B}Q^{g}=w_{h}+w_{w}+I_{G}.$ Since preferences for (x ^p, Q ^c) are independent of child sex, and given that goods are normal, the optimal choice for parents of a girl on this bc is $\left( x_{G}^{p},G\right) \gg \left( x_{B}^{p},B\right).$ ²¹ While the optimal bundle for parents with a girl will not in general be $\left( x_{G}^{p},G\right),$ this inequality together with strictly convex preferences rules out the possibility that the optimal Q ^g ≤ B. $\square$