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


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.

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Fig. 1
Fig. 2


  1. This paper assumes away other forms of child care such as that provided by relatives, particularly grandparents. Heckman (1974) and Blau and Robins (1988) have emphasized the importance of such informal care.

  2. Although non-cooperative factors in family bargaining have also been suggested in the literature (Lundberg and Pollak 1993; Konrad and Lommerud 2000; Basu 2006; Iyigun and Walsh 2007a), we assume away such behaviors in this paper.

  3. It is widely perceived that unitary models are not supported empirically (Lundberg et al. 1997; Duflo 2003; Blundell et al. 2007). There are two strands in the literature of family economics: Manser and Brown (1980) and McElroy and Horney (1981) present Nash bargaining models, whereas Chiappori (1988, 1992), Apps and Rees (1997), Komura (2013), and Kemnitz and Thum (2015) present collective approaches. Basu (2006) explains the difference between a collective approach and Nash bargaining.

  4. Family bargaining and female empowerment have been analyzed recently in dynamic intergenerational settings (Basu 2006; Alesina and Giuliano 2006; Iyigun and Walsh 2007a; Doepke and Tertilt 2009; de la Croix and Donckt 2010; Voena 2015).

  5. See, for example, OECD Statistics (Social Expenditure, Public expenditure on family by type of expenditure, in % GDP, Accessed 10 June 2015). Another example is the limited accessibility to formal child care. For example, in Japan, the ratio of the number of children on a waiting list to use day nurseries (“Taiki-Jido” in Japanese) to the total number of children aged 0 to 4 was high, i.e., about 0.81%, in 2014 (Source: Ministry of Health, Labor and Welfare: and National institute of Population and Social Security Research: Both accessed 15 June 2016).

  6. Iyigun and Walsh (2007a), among others, took it for granted that inherent biological differences exist between sexes in the requirements of parental time investment. Childbearing and child rearing are sometimes distinguished in the literature (e.g., Rasul 2008).

  7. Lundberg and Pollak (1993) proposed the plausibility of separate spheres bargaining.

  8. Echevarria and Merlo (1999) do not discuss policy effects.

  9. Although the feature of the model is counterfactual, the present analysis is meant to characterize the behavior of a representative couple, as described by Echevarria and Merlo (1999).

  10. However, re-negotiations of family decisions have been studied in intertemporal maximization settings. Rasul (2008) uses the Malaysia Family Life Survey to show that spouses bargain without commitment. Mazzocco (2007) also uses US data to test intra-family commitment. Results show that non-commitment collective models might be appropriate for policy-making. Incomplete contracts models include those of Lundberg and Pollak (1993), Konrad and Lommerud (2000), and Rainer (2008).

  11. A contractual solution for marriage market externality would require written contracts involving all families who might be linked through marriage at any future date, which cannot be done in the absence of perfect foresight of future marriages (Doepke and Tertilt 2009).

  12. This assumption might not hold in reality, although it is usually made in the literature (e.g., Iyigun and Walsh 2007a).

  13. The personal ideal numbers of children for women and men older than 15 are reported for OECD countries in Europe and some other countries by OECD Family Database ( Accessed 12 Jan 2017). They are not necessarily perfectly correlated. The database also shows that the OECD average ideal numbers of children are 2.22 for men and 2.28 for women. After surveying the literature related to developing countries, Mason and Taj (1987) conclude that gender differences in fertility goals tend to be smaller in modern, gender-equal conditions. Recently, Doepke and Kindermann (2016) report that a woman’s intention is the more important for having children, although her partner’s agreement greatly increases the probability of having a child.

  14. More general specification makes it impossible to solve the solution analytically.

  15. Psacharopoulos and Patrinos (2004) present a survey of the mean return rates of investment in education for women and men. For the secondary education level, the returns to women are higher, although the mean return to men is slightly higher on the overall education level.

  16. The price of educational investment is taken to be pegged to the goods price, which is assumed to be unity without loss of generality.

  17. We assume the existence of a symmetric equilibrium. Although the first-order conditions are not described in their general format, we make use of the symmetry nature of the equilibrium in deriving these conditions.

  18. The causal relation between parents’ education levels and those of children cannot be readily derived. Black et al. (2005) conclude that high correlations are primarily attributable to family characteristics and inherited ability and not education spillovers. Currie and Moretti (2003) demonstrate that increases in maternal education over the past 30 years have had strong positive effects on birth outcomes such as birth weights. The benefits were estimated to include a reduction of $5.5 to $6 billion in health, education, and other costs. Sewell and Shah (1968) assert that when parents have discrepant levels of educational achievement, which parent’s education has greater effects on educational aspiration and achievement of children depends on the child’s gender and intelligence level and on each parent’s level of educational achievement.

  19. Although Doepke and Tertilt (2009) assume human capital stock formation from generation to generation with intergenerationally external effects, we assume away the intergenerational externalities of human capital.

  20. Nash (1953) showed that, for a unique feasible outcome to be selected, a bargaining solution should have the following four properties: (i) Pareto-optimality, (ii) symmetry, (iii) independence of equivalent utility representation, and (iv) independence of irrelevant alternatives (see Zhang 2005). Proof of the existence and uniqueness of the time path of the present problem (e ft , e mt , n t ) is given in Appendix 1. Stability is examined in Appendix 2.

  21. One can assume a wage tax instead of lump-sum taxes, but the wage tax reduces the opportunity cost of child-rearing at home, thereby exerting positive effects on fertility.

  22. This assumption is not necessary. Although a female worker can care for more than one child. Female workers must be trained specifically to care for children covered by the policy expansion. We assume that such costs are financed entirely by taxes. In reality, although all workers in the (public) child-care service sector might not necessarily be women, the ratio of men is fairly low even in developed countries. For example, see the Japan Gender Equality Bureau Cabinet Office (http// Accessed 1 January 2015) and the Japanese Embassy in Norway ( Accessed 1 January 2015).

  23. Otherwise, no individuals get married. For analytical purposes, it is necessary to have at least one married couple.

  24. This wage function means that the wage rate of an individual is zero if his parents did not invest in him. We can instead assume that w(e) = θ ln(a + e) where a > 1, in which case the wage rate is positive even if his parents did not invest in him.

  25. This assumption implies that an increase in the number of children of a generation will not, ceteris paribus, raise that of the next generation by less than the original increase, i.e., ∂n t + 1/∂n t  < 1. It is noteworthy that the violation of the inequality does not render the results in the present paper invalid per se. The utility weight on the number of children, which can be equal to or greater than the utility weight on the welfare of offspring, is commonly used in the literature (e.g., de la Croix and Doepke 2003).

  26. The steady state can be unstable if the second inequality is not satisfied. However, under the assumption of perfect foresight, the system is expected to jump to the steady state if it is unstable.

  27. Condition (23) is equivalent to condition |T(J)| < 1 in Appendix 2.

  28. The positive number of children derives from the specification of the utility function. Under a general form of utility function, individuals might have no child, as shown by Hirazawa et al. (2014).

  29. Based on a “natural experiment,” Lundberg et al. (1997) report strong evidence that a substantial shift toward greater expenditures on women’s goods and children’s goods followed the transfer of a substantial child allowance to wives of the UK in the late 1970s. Attanasio and Lechene (2002) also used PROGRESA data of rural Mexico to find that a higher share of income for wives is associated, on average, with significantly higher budget shares of expenditures on clothing for children of both genders.

  30. The transition is assessed in a numerical example presented in the next sub-section.

  31. Controlling the percentages of female occupations and so on, O’Neill (2003) reports that the adjusted female–male wage ratio for ages 35–43 is 97.5% from the 2000 National Longitudinal Survey of Youth data, although the actual ratio was 82.5% in the US in 2014 (OECD Stat

  32. Even if women and men have the same preferences for children and the same wage functions, i.e., if γ f = γ m, ε f = ε m, and w f(e) = w m(e), we can obtain fundamentally equivalent results related to the policy effects. The assumption that mothers care for their children at home plays an important role in determining the long-term effects of a policy which targets mothers.

  33. Doepke and Kindermann (2016) used the Generations and Gender Programme (GGP) data to show that policies which lower the child-care burden burden for mothers by providing public child care can be more than twice as effective as policies that provide general subsidies for childbearing. However, they do not consider some aspects of educational investment, such as the tradeoff between quantity and quality of children.

  34. For derivation of (33), see Appendix 3.

  35. For the present study, following Echevarria and Merlo (1999), we assume that education is a “good” and that its price is constant: unity. Actually, de la Croix and Donckt (2010) also assume that the price of education is equal to that of consumption goods.

  36. We assume here that the new steady state is sufficiently close to the initial one. The decrease in σ shifts line L to the upper-right in Fig. 1. The fertility rate and educational investment jump to a point on the shifted line.

  37. Different from the analyses explained herein, Iyigun and Walsh (2007b) assert that asymmetries in the sex ratio in the marriage market can generate the difference in premarital investment between sexes in explanation of a rapid increase in the education of women in the USA.

  38. Therefore, the “unmarried threat” is not credible in this case.


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The author thanks two anonymous referees and the editor of this journal for their insightful comments and helpful suggestions. He is also greatly indebted to Ryo Ishida, Jun-ichi Itaya, Mizuki Komura, Kazutoshi Miyazawa, and seminar participants at The Center for Risk Research of Shiga University, the 2016 Fall Meeting of the Japan Association for Applied Economics, Hokkaido University, Nanzan University, and the Nagoya Macroeconomics Workshop for their comments.


This work was partially funded by the Japan Society for the Promotion of Science KAKENHI Grant (No. 16H03635).

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Correspondence to Akira Yakita.

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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

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


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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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).

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Yakita, A. Fertility and education decisions and child-care policy effects in a Nash-bargaining family model. J Popul Econ 31, 1177–1201 (2018).

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  • Nash bargaining
  • Fertility
  • Educational investment
  • Child-care policy

JEL classifications

  • D91
  • H53
  • J13
  • J16