Fertility and endogenous gender bargaining power

Abstract

We develop an intra-household bargaining model to examine the feedback effect of household fertility decisions on gender bargaining power. In our model, the household balance of power is endogenously determined reflecting social interactions, i.e., the fertility choices by the other couples in society. We show the presence of multiple equilibria in fertility outcome: one equilibrium characterized by patriarchal society with a high fertility rate, and another in which women are sufficiently empowered and the fertility rate is low. In other circumstances, this study also demonstrates a positive relationship between female wage rates and the fertility outcomes. Finally, we discuss its policy implications, comparing the effects of two family policies: the child allowance and the subsidies for market childcare.

1 Introduction

Since the influential work by Konrad and Lommerud (2000), many theoretical studies on family bargaining have endogenized the power balance within a family (Vagstad 2001; Lundberg and Pollak 2003; Basu 2006; Rainer 2008). Recently, the relationship between endogenous bargaining power and fertility has begun to be analyzed in the studies of household decisions.¹ On one strand, it is depicted that the individual choices affect the future power balance of themselves. Iyigun and Walsh (2007) showed that the fertility rate is reduced by an increase in women’s bargaining power, which depends upon the premarital educational investment.² On the other strand, some authors explain that the bargaining power between sexes is influenced though social interactions. Doepke and Tertilt (2009) and Fernandez (2010) demonstrated the negative relationship between fertility and women’s autonomy in models where a man’s vote for women’s liberation may alter the balance of gender power of future couples including his offspring.³

Although their analyses are recognized as pioneering research in the field of development and population economics, they still do not consider that the fertility decisions themselves affect the economic strength of women. Fertility decisions affect the choice of women’s labor supply since childbearing necessarily keeps women away from earning activities, which in turn, leads to a wider gender gap in society. Cigno (2008, 2012) pointed out that only the existence of the prenatal period can lead women into an economically unfavorable position at cooperative marriage, followed by their lower outcome of intra-family distribution.⁴ The purpose of this paper is to construct a simple family bargaining model, taking into account the effect of having children on women’s economic vulnerability.

We set two distinct features in the intra-household decision making model. Our main feature is that gender bargaining power is endogenously determined by social norm or peer pressures, with an aim of exploring the feedback effect of fertility decisions on the bargaining power. In our model, social externalities impose certain gender roles to individuals in the marriage market.⁵ Their balance of power in the marriage market depends on the ratio of incomes earned if they have the average number of children per household. As long as the level of women’s bargaining power bears some relation to their fertility choices, it is worth investigating fertility and the balance of power between sexes in a model where both variables are endogenized interdependently.

The other feature is that the family members negotiate the distribution within themselves in the presence of conflicting parental preferences over the number of their children.⁶ Figure 1 plots the young women’s average ideal number of their children against that of young men of OECD countries in Europe (except for the unavailable data of Iceland, Norway, and Switzerland), using the cross-sectional data set of Testa (2006). The data suggest that the preferences on fertility outcomes are not necessarily the same between men and women. For example, women are likely to prefer a larger family size than men in Northern Europe.

Fig. 1

Mean personal ideal number of children of young men and women. The line represents 45°, which indicates that men and women want exactly the same number of children.

As Fig. 1 indicates, family-size preferences vary not only across countries but between sexes in one country, and thus, the traditional common preference approach is not strictly enough for the study of family behaviors including fertility choice. In order to incorporate the conflicting parental preferences into fertility analysis, we follow the bargaining analysis by Chiappori (1988; Chiappori (1992) in which family members always make efficient decisions according to a particular decision rule of their marriage.⁷

We show that when husbands wish to have more children than their wives do, the interaction between fertility outcome and gender bargaining power leads to multiple equilibria; one equilibrium characterized by a patriarchal society with a high fertility rate, and another in which women are sufficiently empowered, keeping their fertility rate low. Using our model, in the opposite but occasionally observed situation where wives prefer a larger family size than their husbands, it is also theoretically possible to achieve improvements in women’s labor conditions through their wage increases and an increase in fertility rate in spite of the higher opportunity cost of childrearing. This result is in contrast to the traditional opportunity cost theory that explains demographic transition by the negative relationship between women’s wage rates and their fertility.⁸ In some countries, we observe the environment in which both fertility rates and female wage rates are relatively high, and our finding can partly explain these observations.⁹ Moreover, this framework provides some new implications for the effects of different family policies; the subsidies for bought-in childcare increases the bargaining power of women, thereby the fertility rate may fall, while the child allowance has no such effect, so that it increases fertility rates.¹⁰ Our model, which includes the interactions between fertility and bargaining power nicely, properly acts to derive these policy implications.

This paper is organized as follows. Section 2 presents an intra-family decision-making model with endogenous bargaining power. Section 3 presents the main results. Section 4 discusses the effectiveness of family policies, and Section 5 contains brief concluding remarks.

2 Model

Consider an economy consisting of two types of groups, $i\in \left\{ {f,m} \right\}$, men (denoted by m) and women (denoted by f). Although preferences differ between the groups, they are all assumed to have identical preferences within their group. Two individuals out of each group form a monogamous family. After marriage, they decide on the number of children. Their conflicting preferences are resolved through cooperative bargaining based on their bargaining power. In our model, such bargaining power, in turn, will be affected by the other couple’s fertility decisions through social norm or peer pressures regarding the gender roles such as women’s participation in the labor market.

2.1 Preferences

The individual i’s utility function is

$$ \label{eq1} u_i \left( {c,n} \right)=c+v_i \left( n \right), $$

(1)

where c and n denote the level of consumption by parents and the number of their children, respectively. The subutility of v _i (n) stands for individual i’s utility perceived from n.¹¹ The husband and wife bargain over their own consumption and the number of their children to maximize the welfare function:

$$ \label{eq2} \mathop{\max }\limits_{c,n} \Omega =\theta u_f +\left( {1-\theta } \right)u_m , $$

(2)

where $\theta \in \left[ {0,1} \right]$ is the wife’s bargaining power in the household. This means that in one extreme case, θ = 0, represents the case where the household solely maximizes the husband’s utility function. In the other extreme, θ = 1, the household preference corresponds to the wife’s.

2.2 Constraints

We assume here that each individual’s total available hours are normalized to unity, and that only women take the responsibility for parental attention.¹² Because household members do not perceive utility from leisure, her hours are allocated between child care and market work as 1 = L + t, where L is the actual labor supply of women and t is the total parental attention to n children. The husband supplies inelastically one unit of time in the labor market, so that we simply denote his labor income by his wage rate, Y.

Instead of being busy with childrearing, the wife can substitute her parental attention by purchasing the market goods for childcare, x. The household’s budget constraint is c + px = wL + Y, where p is the price for bought-in childcare and w is women’s wage rates, respectively.

The number of children is given by

$$ \label{eq3} n=\phi \left( {x,t} \right), $$

(3)

where ${\partial n} / {\partial x>0}$, ${\partial^2n} / {\partial x^2<0}$, ${\partial n} / {\partial t>0}$, and ${\partial^2n} /{\partial t^2<0}$. Even though women can substitute their time for childrearing with market goods, they are not perfectly substitutable since a child requires special maternal time.¹³ Put differently, having another child requires women to leave the labor market for a certain period. In this paper, we also maintain the assumption that the household production function of childcare $\phi \left( {x,t} \right)$ is characterized by constant returns to scale (CRS).¹⁴ Under the CRS technology, the unit costs of having a child depends on relative prices only, which makes the analysis substantially simpler. The spouses minimize the cost of raising children, C = px + wt subject to Eq. 3, yielding input demand functions, $x^\ast =\hat{x}\left( {p,w} \right)n$, $t^\ast =\hat{t}\left( {p,w} \right)n$, and the fixed cost of having a child, $q\left( {p,w} \right)=p\hat{x}{+w\hat{t}=C} / n$, where $\hat{t}\left( {p,w} \right)$ and $\hat{x}\left( {p,w} \right)$ are the per unit requirements of the mother’s time and market good for childcare, respectively. Making use of the unit cost for having a child, the household budget constraint can be rewritten as follows:

$$ \label{eq4} c+qn=w+Y. $$

(4)

2.3 Household decision making

Given the level of the gender bargaining power, the household maximizes a weighted average of the husband and wife’s utility, subject to Eq. 4. Solving the welfare maximization problem, we have the first-order condition,

$$ \label{eq5} \theta v_f^\prime \left( n \right)+\left( {1-\theta } \right)v_m^\prime \left( n \right)=q\left( {p,w} \right), $$

(5)

which depicts the fact that the cost of having a child in the household must be equal to the sum of the weighted individual marginal utilities for having a child. Equation 5 gives the fertility as a function of bargaining power and prices of childcare, $n=n\left({\theta ;p,w}\right)$.

Note that the sign representing the effect of bargaining power on the number of children can be checked by the specification of each spouse’s utility functions as follows:

$$ \label{eq6} \frac{\partial n}{\partial \theta }=-\frac{v_f^\prime -v_m^\prime }{\theta v_f^{\prime\prime} \left( n \right)+\left( {1-\theta } \right)v_m^{\prime\prime} \left( n \right)}\frac{>}{<}0\Leftrightarrow v_f^\prime \frac{>}{<}v_m^\prime . $$

(6)

Equation 6 means that an increase in the wife’s bargaining power brings the household outcomes close to her fertility goal. When she prefers a larger family size than her husband, a rise in her autonomy increases the number of children, and vice versa. We can summarize the above arguments as follows:

Lemma 1

(Property of fertility outcomes) The sign of the effect of a rise in women’s bargaining power on the fertility rate is determined by the difference in the degree of parental preferences.

For further reference, we derive $\theta =\xi \left( {n;p,w} \right)$ by solving the equation $n=n\left( {\theta ;p,w} \right)$ for θ.

2.4 Endogenous gender bargaining power

This subsection defines the gender bargaining power. In most industrial countries, women have been obtaining almost the same rights as men. For example, they can invest enough in their human capital narrowing wage gap and participate in the labor market. In the households, women increase the autonomy of their decisions on the use of household resources such as family planning. They can now also choose to divorce and win custody and control their earnings and assets. Despite the growing liberation of women, there still exists considerable consciousness of gender issues in these countries. As Cigno (2008, 2012) pointed out, a commonly cited cause of this gender inequality is the fundamental gender difference of giving birth with an inevitable women’s leave from the labor market. Moreover, the blank due to childrearing in women’s career not only limits their own economic position but shapes the social pressure on the gender role, thereby weakening the position of other young women in their marriage.

In order to describe this situation, we assume that the wife’s intra-household bargaining power, θ, is determined in the marriage market, where individuals learn their gender roles through social interactions. The degree of women’s empowerment in the marriage market depends on the ratio of their labor income compared to that of men,¹⁵ which is earned if the wife spends the same number of hours for domestic childcare as that of average one in each household. This means that the hours for market work which the society or the social norm of gender role expects women to supply is determined by subtracting the average parental attention, $\overline {tn}$, from their endowed time normalized to one, as $1-\overline {tn}$.¹⁶ The couples in the marriage market (especially women) form their family by accepting this socially determined bargaining power, i.e., the balance of power between sexes is exogenous for each family, but endogenous on the societal level, and this is the “marriage market externality” in our model.¹⁷ Hence, women’s bargaining power is represented by a continuous and differentiable function of

$$ \label{eq7} \theta =\theta \left( {\frac{w\left( {1-\overline {tn} } \right)}{Y}} \right),{\theta }'\left( \cdot \right)>0,\theta \left( 0 \right)=0. $$

(7)

The assumption of θ(0) = 0 means that women have no autonomy in the household decision making if society expects them to devote the whole endowed time to childcare.

2.5 Equilibrium

We are now ready to define the equilibrium for our economy.

• 1. Given the prices of goods, individuals’ wage rates, and the bargaining power, (p, w, Y, θ), the couples derive the unit cost of a child $q=p\hat{x}\left( {p,w} \right)+w\hat{t}\left( {p,w} \right)$, and then maximize their welfare function given by Eq. 2, to obtain the fertility demand function:

  $$\label{eq8a} n=n\left( {\theta ;p,w} \right) $$

  (8a)

• 2. In the marriage market, given the average maternal attention per household in society, $\overline {tn}$, the bargaining power of women is determined to be $\theta =\theta \left( {{w(1-\overline {tn} )} \mathord{\left/ {\vphantom {{w(1-\overline {tn} )} Y}} \right. \kern-\nulldelimiterspace} Y} \right)$. From the homogenous marriages, the choices of household coincide with the average fertility and the average number of hours at domestic childcare in the equilibrium, i.e., $\overline {tn} =\hat{t}n$. By substituting it into Eq. 7, we then have

  $$\label{eq8b} \theta =\theta \left( {{w\left( {1-\hat{t}n} \right)} \mathord{\left/ {\vphantom {{w\left( {1-\hat{t}n} \right)} Y}} \right. \kern-\nulldelimiterspace} Y} \right)=\psi \left( {n;p,w,Y} \right). $$

  (8b)

Finally, solving Eq. 8a and 8b allows us to derive fertility and bargaining power in the equilibrium, $\left( {n^\ast ,\theta^\ast } \right)$.

Note that $\theta =\psi \left( {n;p,w,Y} \right)$ is obviously decreasing with respect to $n \left( {{\partial \psi } / {\partial n}=-{\left( {\theta ^\prime w\hat{t}} \right)} / Y<0} \right)$. This implies that having an extra child by other families decreases the bargaining power of young women in the marriage market. It leads to a rise in the required maternal attention, and thus reduces her labor supply. Since young couples determine their balance of power reflecting the other couples’ choices, the reduction in women’s expected labor income lowers their economic positions in marriage.

3 Comparative statics

This section analyzes the effects of changes in exogenous variables on fertility and endogenous bargaining power. In the discussion below, to make our analysis simple and apparent, we assume $v_f \left( n \right)=av_m \left( n \right)$, implying that women’s preference on children is equal to men’s multiplied by a constant a > 0 for all n. As the effect of bargaining power on fertility outcome depends on the heterogeneity of the parental attitudes from Lemma 1, we study fertility and gender bargaining power in the following different cases; case 1 (a < 1) and case 2 (a > 1).¹⁸

Before investigating the total effects of changes in exogenous variables, we explore their effect on $\psi \left( {n;p,w,Y} \right)$ and $\xi \left( {n;p,w} \right)$. Given the level of fertility outcomes, by differentiating ψ with respect to p, w, and Y, we have the following:

$$ \label{eq8} \frac{\partial \psi }{\partial p}=-\frac{\theta^ \prime wn}{Y}\frac{\partial \hat{t}}{\partial p}<0, $$

(9)

$$ \label{eq9} \frac{\partial \psi }{\partial w}=\frac{\theta^ \prime }{Y}\left[ {\left( {1-\hat{t}\,\overline n } \right)-wn\frac{\partial \hat{t}}{\partial w}} \right]>0, $$

(10)

and

$$ \label{eq10} \frac{\partial \psi }{\partial Y}=-\frac{\theta^ \prime w\left( {1-\hat{t}\,\overline n } \right)}{Y^2}<0. $$

(11)

The sign of Eq. 9 is negative because an increase in the price of bought-in childcare leads to a reduction in expected female labor income due to a technical substitution of childcare, thereby lowering their bargaining power. In Eq. 10, we can find two effects according to a rise in women’s wage rates. The first term in brackets is the direct effect on expected female labor income that comes from the increased wage itself. The second term is positive, corresponding to the technical substitution effect. In sum, the expression of Eq. 10 is positive. Equation 11 shows that the women’s bargaining power is decreasing in the men’s wage rates.

Given the level of fertility outcomes, we can express the effect of exogenous variables on ξ by using the Shepherd’s lemma as follows:

$$ \label{eq11} \frac{\partial \xi }{\partial p}=\frac{\hat{x}}{\left( {a-1} \right)v_m^\prime }\frac{>}{<}0\Leftrightarrow a\frac{>}{<}1, $$

(12)

$$ \label{eq12} \frac{\partial \xi }{\partial w}=\frac{\hat{t}}{(a-1)v_m^\prime }\frac{>}{<}0\Leftrightarrow a\frac{>}{<}1, $$

(13)

and

$$ \label{eq13} \frac{\partial \xi }{\partial Y}=0. $$

(14)

From Lemma 1, we can interpret Eqs. 12–14 as the effects on fertility outcomes. Each of Eqs. 12 and 13 represents the negative effect on fertility because raising p or w induces an increase in the unit cost of having a child, while the fertility rate is not influenced by Y as in Eq. 14.

3.1 Case 1 (a < 1)

Case 1 represents a situation in which the men desire more children than women do. It is quite natural that women would hesitate to have many children in comparison to men considering their physical and mental strain attendant upon the frequent childbirth and the fact that the longer period of childrearing is likely to narrow the range of women’s occupational choices. This case is supported by many evidences focusing especially on developing countries, while some theoretical papers studying the conflict of preferences over fertility outcomes also assume this case.¹⁹ In order to ensure an interior equilibrium, we make an assumption on the ideal family size of the husband.

Assumption 1

Let n _i be the number of children achieved by maximizing solely the utility of individual $i\in \left\{ {f,m} \right\}$ . Then, p and w satisfy the following condition:

$$ n_m =n\left( {0;p,w} \right)<\frac{1}{\hat{t}\left( {p,w} \right)}. $$

Assumption 1 implies that the husband does not want so many children that his wife must spend all her time for childrearing, which seems to be plausible. Making use of Assumption 1, we have the following proposition.

Proposition 1

• 1. When men prefer a larger family size than their wives, there exists at least one equilibrium.

• 2. When men prefer a larger family size than their wives and there is an equilibrium such that $\left. {{\partial \psi } / {\partial n}} \right|_{\left( {\theta^\ast ,n^\ast } \right)} <\left. {{\partial \xi } / {\partial n}} \right|_{\left( {\theta^\ast ,n^\ast } \right)}$ , we have at least three equilibria of fertility and bargaining power.

Proof of Proposition 1.1

For illustrative purposes, we provide a proof to go along with the graphical analysis by using Fig. 2a, b. From Lemma 1, a > 1 and the property of fertility function, we have $n\left( {0;p,w} \right)=n_m$ and $n\left({1;p,w} \right)=n_f$ as in Fig. 2a. Under the Assumption 1, $\psi \left( {0;p,w,Y} \right)\le 1$ and $\psi \left( {1 / {\hat{t}};p,w,Y} \right)=\theta \left( {0 / Y} \right)=0$, it is obvious that ψ and ξ intersect at least once as described in Fig. 2a. □

Fig. 2

a Unique equilibrium in case 1. b Multiple equilibria in case 1.

Proof of Proposition 1.2

Since $n\left( {0;p,w} \right)=n_m$, $n\left( {1;p,w} \right)=n_f$, $\psi \left( {0;p,w,Y} \right)$ ≤ 1, $\psi \left( {1 / {\hat{t}};p,w,Y} \right)$ = $\theta \left( {0 / Y} \right)$ = 0 under Assumption 1, if there exists an equilibrium where the two curves cross at angles of ${\partial \psi } / {\partial n}<{\partial \xi } / {\partial n}$, applying the mean value theorem to $\psi \left( {n;p,w,Y} \right)$ and $\xi \left( {n;p,w} \right)$ allows us to obtain at least three equilibria. □

Figure 2b illustrates one particular subdivision of this case with the equilibria of E ₁, E ₂, and E ₃.²⁰ Starting with a women’s bargaining power between θ ₁ and θ ₂, fertility outcome and bargaining power converge to the equilibrium of E ₁. If the women’s initial bargaining power lies between θ ₂ and θ ₃, they now converge in E ₃. Thus, the low fertility rate with a strong women’s say and the high fertility rate with low bargaining power are locally stable, while E ₂ is unstable. These two stable equilibria characterized by fertility and bargaining power are partly consistent with the empirical evidence. Gustafsson (1992) found a negative relationship of fertility and women’s economic activities, using the micro-data of Germany, where men want a larger number of children as in Fig. 1. On the other hand, it is not observed in her results with the micro-data of Sweden in which the relationship of preferences is opposite.

Let us now consider the effect of a change in women’s wage rate. Suppose that the original economy is at a high fertility equilibrium, E ₃, and there is an increase in the female wage rates. This will cause the curve representing $\psi \left( {n;p,w,Y} \right)$ to shift upward and the curve representing $\xi \left( {n;p,w} \right)$ downward from Eqs. 10 and 13. As a result, these shifts lead to a decrease in the number of children. Although the decline in fertility rate due to endogenous gender bargaining power is similar to the outcome in Iyigun and Walsh (2007), the economic mechanism behind these two results are different.²¹ In our model, because of the heterogeneity in the spouses’ preferences for family size, an increase in women’s wage rates causes not only the negative effect due to the higher opportunity cost of having a child, but another negative effect of enhanced bargaining power of women, whose desired number of children is smaller than those of men.

These results in case 1 propose that the bargaining power between sexes may turn out to be a key factor in better understanding the demographic transition and shed new light on intra-household decisions, especially the hidden issue of family planning.

3.2 Case 2 (a > 1)

Many evidences indicate that men demand a larger family size than women do, which correspond to the case 1 of our analysis. Thus, most existing theoretical works on the bargaining over fertility outcomes are based on the situation in case 1. However, some evidences also identify the women’s larger ideal number of their children than that of men in both developed and developing countries. According to such empirical studies, this case can be found in the societies characterized by specific cultural factors rather than the universal factors including the pain of giving birth.²² Since it could well capture some aspects of the real world, the other case also needs to be examined. As to the effects of a rise in the female wage rate on fertility, we have the following proposition.

Proposition 2

When women prefer a larger family size than their husbands, an increase in female wage rate raises the fertility rate if and only if

$$ \label{eq14} \hat{t}<\left( {a-1} \right)v_m^\prime \frac{\partial \psi }{\partial w}. $$

(15)

Proof

See Appendix A.□

To illustrate the result of Proposition 2 more simply, specifying the male parental utility function to be log-linear form, $v_m \left( n \right)=\ln n$, enables us to rewrite Eq. 15 as follows:

$$ \label{eq15} \hat{t}<\frac{\left( {a-1} \right)}{n^\ast }\frac{\partial \psi }{\partial w}. $$

(16)

From Eq. 16, the smaller per unit time for domestic childcare, the larger the difference between men and women’s fertility goals, the smaller the number of their children, and the larger the positive partial effect of the wage rate on women’s bargaining power, the more likely that a rise in female wage rate will increase fertility outcomes.

Intuitively, an increase in w again raises not only the opportunity cost of childcare but women’s expected labor income, providing them with an advantage over intra-household distribution. Therefore, when women want more children than men (a > 1), the sign of the total effect of a rise in w on the number of children in equilibrium depends on the relative magnitude of the negative effect due to a higher opportunity cost and the positive effect due to the decision making over fertility that better reflects women’s preferences. If the former effect is overwhelmed by the latter, the fertility outcome rises in spite of a higher opportunity cost of childrearing. This finding is not led by the traditional result of an income effect but through a rise in women’s bargaining power, since the increased fertility outcome cannot be achieved with a fixed level of bargaining power. This theoretical result implies that gender bargaining power is one of the factors that explain both a high fertility rate and a high level of women’s autonomy over decisions in the household, such as those observed in developed countries where the mothers’ burden of raising children began to be shared in many ways with their husbands.

4 Policy analysis

In this section, we discuss the policy implications of our model. Most developed countries are facing low fertility rates and thus the government’s aim at maintaining sustainable demographic structures (Ilmakunnas 1997; Cigno and Werding 2007). Following Apps and Rees (2004), we examine the effect of a revenue-neutral increase in subsidy for bought-in childcare financed by a reduction in child allowance so that we can compare the effects of two policies on the fertility outcome.

Consider an economy which is characterized by two stable steady-state equilibria (Fig. 2b). Now, assume that we are on point E ₁ in Fig. 2b, where the fertility rate is low. The government collects the lump-sum tax for the purpose of running the family policies. Denote the policies by T, g _S , and g _T where T is the lump-sum tax levied on each household and g _S and g _T are the subsidy for bought-in childcare allocated per unit of good and child allowance received per child, respectively. For simplicity, we assume that the price of bought-in childcare is normalized to unity in the analysis below.

Introducing two policies implies that the cost minimization problem of childcare becomes

$$ \label{eq16} \mathop{\min }\limits_{x,t} \widetilde{C}=\left( {1-g_S } \right)x+wt-g_T n, $$

(17)

subject to Eq. 3. Since, under the assumption of CRS, the unit cost of having a child is constant regardless of the number of children, recalculating the cost minimization problem above allows us to reduce the budget constraint as follows:

$$ \label{eq17} c+\left[ {\widetilde{q}\left( {1-g_S ,w} \right)-g_T } \right]n=w+Y-T, $$

(18)

where $\widetilde{q}\left( {1-g_S ,w} \right)$ = $\left( {1-g_S } \right)\widetilde{x}\left( {1-g_S ,w} \right)$ + $w\widetilde{t}\left( {1-g_S ,w} \right),\widetilde{x}\left( {1-g_S ,w} \right)$, and $\widetilde{t}\left( {1-g_S ,w} \right)$ represent the unit cost of having a child, market childcare per child, and maternal attention per child after policy implementation, respectively. Given the balance of power within a household, it maximizes its welfare function of Eq. 2, subject to Eq. 18, so that they derive their fertility outcomes as $n=n\left( {\theta ;w,g_T ,g_S } \right)$. Since the marriages in society are homogenous, we can then derive the number of children and the level of woman’s bargaining power in the equilibrium by solving both $n=n\left( {\theta ;w,g_T ,g_S } \right)$ and $\theta =\theta \left( w\left({1-{\hat{t}n}} \right)/Y\right.=\psi \left( {n;w,Y,g_S } \right)$.

The government is providing economic support for households’ childrearing with g _S and g _T , financing them by lump-sum taxes. To focus on the policy effects without tax distortion, we have made the conventional assumption that they are financed by a lump-sum tax on the same household. The budget constraint is given, in household terms as

$$ \label{eq18} T=\left( {g_S \widetilde{x}+g_T } \right)n^\ast . $$

(19)

The government implements a policy shift with an exogenous change in g _T so that they can balance their budget constraint by adjusting the level of g _S .

We can then obtain the following proposition by using the total derivative of fertility with respect to g _T .

Proposition 3

A revenue-neutral increase in the subsidy for bought-in childcare by a reduction in child allowance may reduce the fertility rate.

Proof

We can show that the total derivative of fertility with respect to g _T is

$$ \label{eq19} \frac{dn}{dg}_T =-\frac{\left( {a-1} \right)v_m^\prime n}{\left[ {1+\theta \left( {a-1} \right)} \right]v_m^{\prime\prime} }\frac{\partial \psi }{\partial g_S }\Delta^{-1}, $$

(20)

where Δ < 0 is the marginal cost to the government of an increase in g _S (see Appendix B). From ${\partial \psi } / {\partial g_S }>0$, due to the increase in women’s labor income by technical substitution into bought-in childcare, this expression depends on the difference between men and women’s preferences of family size.²³ From the concavity of v _i (n), in the case of a < 1, Eq. 20 implies that the fertility rate falls with a reduction in g _T .□

The intuition behind this result is as follows. While the child allowance will merely cut off the cost of having a child, the subsidies for bought-in childcare have a positive effect on the women’s empowerment as well as a negative one on the unit cost of a child. The subsidy for bought-in childcare triggers a technical substitution in childcare, leading to higher female labor income and, thus, an increase in women’s bargaining power in the economy. Regarding the latter effect, a subsidy decreases the cost of childrearing through a reduction in the price of bought-in childcare, so that the parents can then afford to have more children. Because child allowance and subsidies for bought-in childcare have the same effect on unit cost of a child as they offset each other, an increase in g _S financed by reduction in g _T makes the household decisions more favorable for women.

In the absence of the preference conflict between spouses (a = 1), the model in our model is essentially the same as the traditional analyses focusing on the household responses to changes in prices and income. Hence, a policy shift toward the subsidy has no effect on fertility outcomes. However, in the case that women’s desired family size is relatively small (a < 1), if the negative effect of bargaining power on fertility outcome is sufficiently large, the introduction of a subsidy may result in an unexpected negative effect on fertility rates. In other words, the introduction of a subsidy policy aiming at a higher fertility rate may itself accelerate the decline in childbirths.

As we can see from the discussion above, it is obvious that both the conflicting preferences and the gender bargaining power determine the effectiveness of the policies. From the viewpoint of the governmental target to boost the fertility rates, the former is preferable since the effect of child allowance on fertility is not affected by the bargaining effect, while the latter may end up in reducing fertility rates. If governments call for improvements in women’s autonomy in the decision making within their families in parallel with higher fertility outcomes, the coordination of these policies will be urgently needed.

5 Conclusion

In this paper, we formulate a simple intra-household decision-making model over fertility outcomes, where its bargaining power depends not only on the relative prices of time of family members but also on the relevant social pressure on gender roles, taking the advantage of the tractability in the model of Basu (2006).

We develop a model to show the observed patterns of bargaining power and fertility, i.e., the economy with unempowered women and a high fertility rate, and that with empowered women and a low fertility rate. This type of multiple equilibria is likely to emerge when women prefer a small number of children compared to their husbands. This result partly corresponds to the evidence presented in Fig. 3.

Fig. 3

GEM and TFR.

Figure 3 shows this simple relationship between the total fertility rates (TFR) of 2000–2005 and women’s say, namely, the Gender Empowerment Measure (GEM), using cross-sectional data based on the 2007–2008 Human Development Report (UNDP 2007). Apart from the different prices and other economic circumstances among these countries, we see low-GEM countries indicate high TFR while high-GEM countries do the opposite. More formal empirical evidence in support of this result can be found in Feyrer et al. (2008). They observed three phases in women’s statuses: the earliest phases of women’s economic position with the most domestic work followed by intermediate phase of improved female labor market opportunities with their household status lags. In the final phase, women commit more and more of their time to the labor market and the number of their children is the fewest.

Under the opposite but plausible assumption of parental preferences, the model also gives a potential explanation for the fact that a raise in female wage rates may cause the higher fertility rate despite the higher opportunity cost of having a child.²⁴ The results above are derived from the conflict in preferences and the feedback effect on the gender bargaining power due to the actual fertility outcomes of other couples in the society. This suggests that the effectiveness of family policy is determined not only by the economic situation but also by the intra-household decision process.

Although we have examined the policy effects on fertility, the extension to the analysis of optimal policy introducing the taxation on labor income may be a promising direction. This also relates to the debate between individual taxation and joint taxation for household since these policies can affect the economic relationships among family members.