Schooling and Fertility

Abstract

This chapter is motivated by two of the growth facts stated in Chap. 1.

• G2—Children spend more time in school, within and across years, and less time working as an economy develops

• G3—Population growth rates may first rise but eventually experience a steady decline as economies develop

Appendices

Appendices

1.1 Maximizing Utility with Fertility and Schooling

In addition to the standard necessary conditions for optimal life-cycle consumption from Chap. 2, the choices of n_t+1 and e_t associated with maximizing (4.1) subject to (4.2), yield the following first order conditions

$$ \frac{\psi \theta}{e_t}\le {\lambda}_t{n}_{t+1}{w}_t{D}_t\gamma \overline{h} $$

$$ \frac{\psi }{n_{t+1}}={\lambda}_t\left[\eta {w}_t{D}_t{h}_t-\left(T-{e}_t\right){w}_t{D}_t\gamma \overline{h}\right], $$

where λ_t is the Lagrange multiplier.

The first equation says the marginal utility of additional child quality, measured by the child’s adult human capital, must be equated to the marginal value of consumption lost from allowing children of working age to attend school. The strict inequality holds when the marginal cost of educating children beyond the schooling received in their early years, $\overline{e}$, exceeds the marginal benefit. In this case, parents are content to set ${e}_t=\overline{e}$, i.e. to have their children begin work as soon as they are able.

The second equation says the marginal utility of additional children must be equated to the marginal value of lost consumption associated with raising a child. Consumption is lost from having an additional child because we assume the cost of children exceeds the earnings that older children bring to the household (otherwise parents would always choose the, biologically determined, maximum number of children).

1.2 Fertility in the Traditional Economy

In the traditional economy from Sect. 4.6, the household maximizes the utility function U_t =  ln (c_t − c) + ψ ln (n_t+1 w_t+1) subject to the budget constraint ${c}_t+\overline{\eta}{y}_t{n}_{t+1}={y}_t$. The first order necessary conditions for the optimal choices of c_t and n_t+1 are

$$ \frac{1}{c_t-c}={\lambda}_t $$

$$ \frac{\psi }{n_{t+1}}={\lambda}_t\overline{\eta}{y}_t, $$

where λ_t is the Lagrange multiplier. The two first order conditions imply $\overline{\eta}{y}_t{n}_{t+1}=\psi \left({\mathrm{c}}_t-\mathrm{c}\right)$. Substituting this expression into the budget constraint and solving for c_t yields (4.15a). Substituting (4.15a) into the expression $\overline{\eta}{y}_t{n}_{t+1}=\psi \left({\mathrm{c}}_t-\mathrm{c}\right)$, gives us (4.15b)

$$ {n}_{t+1}=\frac{\psi }{1+\psi}\frac{y_t-c}{\overline{\eta}{y}_t}=\frac{\psi }{\left(1+\psi \right)\overline{\eta}}\left(1-\left(c/{y}_t\right)\right). $$