Fertility and education decisions and child-care policy effects in a Nash-bargaining family model

Abstract

This paper presents development of a household Nash-bargaining model in an overlapping generation setting to analyze the intergenerational dynamics of education decisions and to analyze cooperatively bargained fertility within a family. A stronger preference by women for the welfare of children induces redistribution from women to men in exchange for higher educational investment in children, although the cost to women of child rearing is compensated by men when women care for their children. Subsequently, this paper presents description of the policy effects on the dynamics that arise from expansion of formal child-care coverage. The policy substitutes time costs that were previously borne by mothers, i.e., a policy targeted at mothers. If the education level of mothers is sufficiently high (low), then the policy lowers (raises) fertility and increases (decreases) investment in education for children in the long term. Most notable is the intermediate case: When a mother’s education level is not too high and not too low, the policy raises both the fertility rate and educational investment in daughters, while probably decreasing investment in sons. The quantity–quality tradeoff for children might not hold. The policy also raises the probability of marriage and the probability of having children.

Appendices

Appendix 1 Existence and uniqueness of a family optimum

From (12) and (13), we can obtain e _it + 1 as a function of n _t , because w ⁱ(e _it + 1) is monotonic, as

$$ {e}_{it+1}={\varphi}^i\left({n}_t\right)\kern0.5em \left(i=f,m\right), $$

(34)

where

$$ \frac{de_{ft+1}}{dn_t}=\frac{p}{\left({\gamma}^f+{\gamma}^m\right)\left(1-\sigma {zn}_t\right){w}^{f"}\left({e}_{ft+1}\right)}<0, $$

(35)

$$ \frac{de_{mt+1}}{dn_t}=\frac{p}{\left({\gamma}^f+{\gamma}^m\right){w}^{m"}(e)}<0. $$

(36)

Using (34), one can rewrite (14) as

$$ \left({\varepsilon}^f+{\varepsilon}^m\right){u}^{\hbox{'}}\left({n}_t\right)=\sigma {zw}^f\left({e}_{ft}\right)+\frac{p\left[{\varphi}^f\left({n}_t\right)+{\varphi}^m\left({n}_t\right)\right]}{2}. $$

(37)

The left-hand side of (37) is decreasing in n _t , although the right-hand side is also decreasing in n _t . Because the left-hand side is the marginal benefit of an additional child and the left-hand side is the marginal cost, a sufficient condition for the uniqueness of the optimal number of children is

$$ \left({\varepsilon}^f+{\varepsilon}^m\right){u}^{"}\left({n}_t\right)<\frac{p^2}{\left({\gamma}^f+{\gamma}^m\right)}\left[\frac{1}{w^{f"}\left({e}_{ft+1}\right)}+\frac{1}{w^{m"}\left({e}_{mt+1}\right)}\right]\left(<0\right), $$

(38)

for a given e _ft . This condition might be satisfied when the degree of concavity of u(n) or w(e) is sufficiently weak.

When condition (38) holds for relevant ranges of variables, the optimum can exist. From the assumptions for functions u(n) and w(e), the left-hand side of (37) approaches zero as n goes to infinity, although the right-hand side approaches σzw ^f > 0. Therefore, as long as (37) holds, both sides cross at one point n.

Appendix 2 Uniqueness and stability of steady states

The steady-state values of the number of children and educational investment in daughters, n and e _f , are given respectively by the following equations.

$$ n=\frac{\frac{\gamma^m+{\gamma}^f}{2}}{\frac{\varepsilon^m+{\varepsilon}^f}{n}-\sigma z{\theta}^f\ln \left(1+{e}_f\right)+p+\sigma z{\theta}^f\frac{\gamma^m+{\gamma}^f}{2}}, $$

(39)

$$ 1+{e}_f=\frac{2{\theta}^f}{p\left({\theta}^m+{\theta}^f\right)}\left[\frac{\varepsilon^m+{\varepsilon}^f}{n}-\sigma z{\theta}^f\ln \left(1+{e}_f\right)+p-\sigma z{\theta}^f\frac{\gamma^m+{\gamma}^f}{2}\right]. $$

(40)

To examine local uniqueness and stability of the steady state, linearizing the dynamic system (21) and (22) around the steady state, we obtain the Jacobian matrix as

$$ J\left(n,{e}_f\right)=\left|\begin{array}{cc}\frac{2\left({\varepsilon}^m+{\varepsilon}^f\right)}{\left({\gamma}^m+{\gamma}^f\right)\left({\theta}^m+{\theta}^f\right)}& \frac{2\sigma {zn}^2{\theta}^f}{\left({\gamma}^m+{\gamma}^f\right)\left({\theta}^m+{\theta}^f\right)\left(1+{e}_f\right)}\\ {}-\frac{2{\theta}^f\left({\varepsilon}^m+{\varepsilon}^f\right)}{pn^2\left({\theta}^m+{\theta}^f\right)}& -\frac{2\sigma z{\left({\theta}^f\right)}^2}{p\left({\theta}^m+{\theta}^f\right)\left(1+{e}_f\right)}\end{array}\right|. $$

(41)

Denoting the determinant and the trace of matrix J(n, e _f ) by D(J) and T(J), we obtain

$$ D(J)=\frac{-2\left({\varepsilon}^m+{\varepsilon}^f\right)}{\left({\gamma}^m+{\gamma}^f\right)\left({\theta}^m+{\theta}^f\right)}\frac{2\sigma z{\left({\theta}^f\right)}^2}{p\left({\theta}^m+{\theta}^f\right)\left(1+{e}_f\right)}+\frac{2{\theta}^f\left({\varepsilon}^m+{\varepsilon}^f\right)}{pn^2\left({\theta}^m+{\theta}^f\right)}\frac{2{n}^2\sigma z{\theta}^f}{\left({\gamma}^m+{\gamma}^f\right)\left({\theta}^m+{\theta}^f\right)\left(1+{e}_f\right)}=0, $$

(42)

$$ T(J)=\frac{2\left({\varepsilon}^m+{\varepsilon}^f\right)}{\left({\gamma}^m+{\gamma}^f\right)\left({\theta}^m+{\theta}^f\right)}-\frac{2\sigma z{\left({\theta}^f\right)}^2}{p\left({\theta}^m+{\theta}^f\right)\left(1+{e}_f\right)}=\frac{2\left[p\left({\varepsilon}^m+{\varepsilon}^f\right)\left(1+{e}_f\right)-\sigma z{\left({\theta}^f\right)}^2\left({\gamma}^m+{\gamma}^f\right)\right]}{p\left({\theta}^m+{\theta}^f\right)\left({\gamma}^m+{\gamma}^f\right)\left(1+{e}_f\right)}. $$

(43)

We have three cases: (i) if both |1 + D(J)| > |T(J)| and |D(J)| < 1 hold, then both eigenvalues are inside the unit interval. Furthermore, the two associated manifolds are stable around the steady state. (ii) If |1 + D(J)| > |T(J)| and |D(J)| > 1 hold, then both eigenvalues are outside of the unit interval, and the associated manifolds are unstable around the steady state. (iii) If |1 + D(J)| < |T(J)|, then one eigenvalue is inside of the unit interval. The other is outside of the unit interval. One stable and one unstable manifold exist around the steady state. The steady state is (saddle-point) unstable in this case.

In this dynamic system of (21) and (22), because we have |D(J)| = 0 and 1 > T(J) under the assumption that 2(ε ^m + ε ^f) − (θ ^m + θ ^f)(γ ^m + γ ^f) < 0, only case (i) is possible.

Next, from (21) and (22), one obtains

$$ {n}_{t+1}=\frac{1}{\frac{p}{\left({\gamma}^m+{\gamma}^f\right){\theta}^f}\left(1+{e}_{ft+1}\right)+\sigma z}. $$

(44)

The fertility rate and mother’s education level in each period must satisfy condition (39), which is presented as a dotted line L in Fig. 1. Letting (n ^∗, e _f ^∗) be the stable steady state, the dynamics can be shown by arrows in Fig. 1. Assuming that the initial condition is given as $\left({n}_{t_0-1},{e}_{ft_0}\right)$ in period t ₀ and that the initial point is not on line L, the fertility rate in period t ₀ jumps to a point corresponding to ${e}_{ft_0}$ on line L; then the system converges to the steady state S along the line. The steady state is unique and stable if it exists.

Appendix 3 Condition for marriage

In the steady state, we have

$$ {\displaystyle \begin{array}{l}{V}^f\left({e}_f,{e}_m\right)={c}_f+{\varepsilon}^fu(n)+\frac{\gamma^f}{2}\left[{V}^m\left({e}_m,{e}_f\right)+{V}^f\left({e}_f,{e}_m\right)\right]\\ {}={w}^f\left({e}_f\right)-\frac{{zn w}^f\left({e}_f\right)+\frac{p\left({e}_m+{e}_f\right)}{2}n}{2}\\ {}-\frac{\Big[\left({\varepsilon}^f-{\varepsilon}^m\right)u(n)+\frac{\gamma^f-{\gamma}^m}{2}\left[{V}^m\left({e}_m,{e}_f\right)+{V}^f\left({e}_f,{e}_n\right)\right]}{2}\\ {}+{\varepsilon}^fu(n)+\frac{\gamma^f}{2}\left[{V}^m\left({e}_m,{e}_f\right)+{V}^f\left({e}_f,{e}_m\right)\right]\\ {}={w}^f\left({e}_{mt}\right)-\frac{zn_t{w}^f\left({e}_{ft}\right)+\frac{pn\left({e}_m+{e}_f\right)}{2}}{2}+\frac{\left({\varepsilon}^f+{\varepsilon}^m\right)u(n)}{2}\\ {}+\frac{\gamma^m+{\gamma}^f}{4}\left[{V}^m\left({e}_m,{e}_f\right)+{V}^f\left({e}_f,{e}_m\right)\right],\end{array}} $$

(45)

$$ {\displaystyle \begin{array}{l}{V}^m\left({e}_m,{e}_f\right)={c}_m+{\varepsilon}^mu(n)+\frac{\gamma^m}{2}\left[{V}^m\left({e}_m,{e}_f\right)+{V}^f\left({e}_f,{e}_m\right)\right]\\ {}={w}^m\left({e}_m\right)-\frac{{zn w}^f\left({e}_f\right)+\frac{p\left({e}_m+{e}_f\right)}{2}n}{2}\\ {}-\frac{\left({\varepsilon}^m-{\varepsilon}^f\right)u(n)+\frac{\gamma^m-{\gamma}^f}{2}\left[{V}^m\left({e}_m,{e}_f\right)+{V}^f\left({e}_f,{e}_n\right)\right]}{2}\\ {}+{\varepsilon}^mu(n)+\frac{\gamma^m}{2}\left[{V}^m\left({e}_m,{e}_f\right)+{V}^f\left({e}_f,{e}_m\right)\right]\\ {}={w}^m\left({e}_{mt}\right)-\frac{zn_t{w}^f\left({e}_{ft}\right)+\frac{pn\left({e}_m+{e}_f\right)}{2}}{2}+\frac{\left({\varepsilon}^m+{\varepsilon}^f\right)u(n)}{2}\\ {}+\frac{\gamma^m+{\gamma}^f}{4}\left[{V}^m\left({e}_m,{e}_f\right)+{V}^f\left({e}_f,{e}_m\right)\right],\end{array}} $$

(46)

where we use (16) and (17). From (45) and (46), we obtain

$$ {\displaystyle \begin{array}{l}{V}^f\left({e}_f,{e}_m\right)={w}^f\left({e}_f\right)-\frac{znw^f\left({e}_f\right)+\frac{p\left({e}_m+{e}_f\right)}{2}n}{2}+\frac{\left({\varepsilon}^f+{\varepsilon}^m\right)u(n)}{2}\\ {}+\frac{\gamma^m+{\gamma}^f}{4}\sum \limits_{t=0}^{\infty }{\left(\frac{\gamma^m+{\gamma}^f}{2}\right)}^t\left[{w}^m\left({e}_m\right)+\left(1- zn\right){w}^f\left({e}_f\right)-\frac{p\left({e}_m+{e}_f\right)n}{2}+\left({\varepsilon}^m+{\varepsilon}^f\right)u(n)\right]\\ {}={w}^f\left({e}_f\right)-\frac{znw^f\left({e}_f\right)+\frac{p\left({e}_m+{e}_f\right)}{2}n}{2}+\frac{\left({\varepsilon}^f+{\varepsilon}^f\right)u(n)}{2}\\ {}+\frac{1}{2\left[2/\left({\gamma}^m+{\gamma}^f\right)-1\right]}\left[{w}^m\left({e}_m\right)+\left(1- zn\right){w}^f\left({e}_f\right)-\frac{p\left({e}_m+{e}_f\right)n}{2}+\left({\varepsilon}^f+{\varepsilon}^m\right)u(n)\right],\\ {}\end{array}} $$

(47)

$$ {\displaystyle \begin{array}{l}{V}^m\left({e}_m,{e}_f\right)={w}^m\left({e}_m\right)-\frac{znw^f\left({e}_f\right)+\frac{p\left({e}_m+{e}_f\right)}{2}n}{2}+\frac{\left({\varepsilon}^m+{\varepsilon}^f\right)u(n)}{2}\\ {}+\frac{\gamma^m+{\gamma}^f}{4}\sum \limits_{t=0}^{\infty }{\left(\frac{\gamma^m+{\gamma}^f}{2}\right)}^t\left[{w}^m\left({e}_m\right)+\left(1- zn\right){w}^f\left({e}_f\right)-\frac{p\left({e}_m+{e}_f\right)n}{2}+\left({\varepsilon}^m+{\varepsilon}^f\right)u(n)\right]\\ {}={w}^m\left({e}_m\right)-\frac{znw^f\left({e}_f\right)+\frac{p\left({e}_m+{e}_f\right)}{2}n}{2}+\frac{\left({\varepsilon}^m+{\varepsilon}^f\right)u(n)}{2}\\ {}+\frac{1}{2\left[2/\left({\gamma}^m+{\gamma}^f\right)-1\right]}\left[{w}^m\left({e}_m\right)+\left(1- zn\right){w}^f\left({e}_f\right)-\frac{p\left({e}_m+{e}_f\right)n}{2}+\left({\varepsilon}^m+{\varepsilon}^f\right)u(n)\right].\end{array}} $$

(48)

For men to marry, both conditions V ^f(e _f , em _f ) > w ^f(e _f ) − T/2 must hold, where T = (1 − σ)znw ^f(e _f ). From (47), it follows that

$$ {\displaystyle \begin{array}{l}\frac{\gamma^m+{\gamma}^f}{2}\left[{w}^m\left({e}_m\right)+\left(1- zn\right){w}^f\left({e}_f\right)\right]+\left({\varepsilon}^m+{\varepsilon}^f\right)u(n)\\ {}>\left(1-\frac{\gamma^m+{\gamma}^f}{2}\right)\sigma {znw}^f\left({e}_f\right)+\frac{p\left({e}_m+{e}_f\right)n}{2}\end{array}} $$

(49)

where

$$ \left(1- zn\right){w}^f\left({e}_f\right)=\left(1-\sigma zn\right){w}^f\left({e}_f\right)-T $$

(50)

The left-hand side of (49) is the utility derived from having children. The right-hand side represents the cost of rearing children. Similarly, using (48), it can be shown that condition V ^m(e _m , e _f ) > w ^m(e _m ) − T/2 for women is also reduced to (49).