A dynamic model of labor supply and fertility with. Ben-Porath human capital accumulation

Transcription

1 A dynamic model of labor supply and fertility with Ben-Porath human capital accumulation Minhee Kim Job Market Paper October 23, 2017 Abstract The aim of this study is to explain two remarkable changes that occurred in the female labor market between the 1970s and the 1990s using a human capital investment model: (1) labor supply of married women has been more compatible with having children and (2) married women have faced higher wage growth rates as a result of higher participation. To this end, this paper constructs a dynamic stochastic labor supply model in which fertility and investment decisions are choice variables. I incorporate the Ben-Porath human capital accumulation into the model to link the expected intermittency caused by having children to women s human capital. Specifically, women who expect to work less during child-rearing ages invest less in their human capital and face lower wages. After calibrating the model to match the 1970s cohort in the PSID data, which has lower participation and lower wages, I examine factors that might contribute to the observed changes over the two decades. From the numerical experiments, the main findings are as follows: (1) the decrease in childcare costs is the sole factor that accounts for the simultaneous increase in the labor participation and fertility rates and (2) the increased returns to experience generate higher wage growth through higher human capital investment. I emphasize the role of investment to shed light on the observed changes by comparing the model with no investment. JEL: J13, J16 and J24 KEYWORDS: Female labor participation, Fertility choice, Ben-Porath human capital, Childcare cost, Returns to experience. Department of Economics, University at Albany, mkim8@albany.edu I would like to be grateful to my advisor, John B. Jones for insightful advice and valuable discussions. Special thanks to Michael Sattinger, Yue Li, Byoung Gun Park and Ibrahim Gunay for helpful comments and discussions. I also thank all participants in Macro workshop at University at Albany in 2016 and

2 1 Introduction Not only does the presence of children have a significant effect on married women s current labor supply, but the career breaks associated with childcare influence their future wages and hence future employment as well. Specifically, mothers leave the labor market around the time of childbirth and gradually return back as their children get older (Figure 1a). Their wage rate is rarely recovered although their participation rate almost approaches to the pre-childbirth rate. This implies that a lower labor force participation (LFP) rate during childbearing and childrearing ages has a persistent effect on their earnings. Accordingly, mothers earn less and have a lower wage growth rate than married women with no kids (Figure 1b). This is known as the motherhood wage penalty, which is empirically estimated to be a 4-7 percent wage loss per child (Waldfogel 1998; Anderson et al. 2002; Budig and England 2001; Gupta and Smith 2002). Adda et al. (2017) estimate that 75% of this motherhood wage penalty arises due to the time spent out of the labor force for childbearing and childrearing. In this paper, I explore how mothers career interruption in their childrearing age reduces their subsequent wage and its growth rate. Mincer and Ofek (1982) provide that less human capital investment before the interruption period and skill depreciation during the interruption cause lower wages for intermittent workers. In particular, mothers have less incentive to invest in human capital if they expect to have career break or career discontinuity for childbearing and childrearing (Mincer and Polachek 1974; Polachek 1975; Polachek 2004). For instance, potential mothers are less likely to attend on-the-job training, take part-time education, or put in an effort to getting promoted at work. The intermittency also depreciates existing human capital and hinders accumulating additional capital, which subsequently reduces the mother s earnings. While empirical studies estimate this wage loss due to work interruption, incorporating investment decision into a model is difficult because it is forward-looking (Light and Ureta 1995; Albrecht et al. 1999; Beblo and Wolf 2002; Spivey 2005; Baum 2002; Buligescu et al. 2009; Adda et al. 2017). However, ignoring this decision cannot fully capture the career interruption effects caused by children. As such, we need to include the investment choice in the model. To this end, I construct a dynamic stochastic discrete choice model of labor supply, fertility and human capital accumulation decisions of married women. This paper makes a contribution to literature by incorporating the Ben-Porath specification to provide a link between 2

3 children and human capital investment (Ben-Porath 1967). Although the Ben-Porath model is dominant in the human capital literature, there are relatively few analyses of the life-cycle labor supply model with the Ben-Porath mechanism (Fan et al. 2013; Manuelli et al. 2012; Erosa et al. 2016). 1 Most of the previous female labor supply studies incorporate the learning-by-doing (LBD) framework in which human capital is accumulated automatically through participation. In the Ben-Porath formulation, however, human capital is generated by explicit investment as well as participation, to some extent. This investment plays a role in capturing the effects of discontinuities in women s careers from raising kids on their wages. To consider two aspects of work interruption costs suggested by Mincer and Ofek (1982), this paper also includes the human capital depreciation when career interruption occurs. The joint model of labor supply and fertility choices allows us to understand how the career decisions of married women interact with their fertility choices. Nonetheless, many of the previous studies treat the female labor participation and fertility decisions separately. Some papers propose a dynamic model of female labor supply by taking fertility decisions as given (Eckstein and Wolpin 1989; Attanasio et al. 2008; Olivetti 2006; Eckstein and Lifshitz 2011; Van der Klaauw 1996; Blundell et al. 2016; Cardia and Gomme 2013; Jones et al. 2003; Doepke et al. 2015). Other authors establish a dynamic model of fertility by taking the labor supply decisions as exogenous (Arroyo and Zhang 1997; Hotz and Miller 1993). 2 Recently, however, a growing literature has analyzed dynamic life-cycle models of joint decisions over labor supply and fertility (Francesconi 2002; Adda et al. 2017; Sheran 2007; Gayle and Miller 2006; Fehr and Ujhelyiova 2013; Erosa et al. 2016). I use the model to explain the observed changes in the female labor market from the 1970s to the 1990s. This model is calibrated by matching to the 1970s cohort using the PSID (Panel Study of Income Dynamics) data. This paper has a two-step calibration strategy similar to Olivetti (2006). First, the wage equations of a wife and a husband are estimated outside the structural model, separately. The husband s earning equation is estimated by a fixed effect model to control for unobserved heterogeneity and the wife s wage equation is estimated by a Heckman two-stage method to control the selection bias. I find a positive selection bias, suggesting that high ability women are more likely to participate. Thus, the average wage falls if more women enter the labor market. In the second step, this paper calibrates the model to 1 Fan et al. (2013) and Manuelli et al. (2012) adopt the Ben-Porath model to explain retirement, and Erosa et al. (2016) find that lower hours worked of women can account for the gender wage gap. 2 Cardia and Gomme (2013) and Van der Klaauw (1996) consider marital decisions together. 3

4 match life-cycle participation, fertility and wages age profiles of the 1970s cohort to those generated by the model. The estimated and calibrated model provides a good fit to the data. The model can replicate lower participation of married women in childbearing and childrearing ages, and a decreasing fertility rate with ages. The calibrated parameters are comparable with the values in previous literature. After calibration, several factors are examined to explain the changes over the two decades. Even though many studies already account for the increase in the labor supply of women in the late twentieth century, this is the first attempt to shed light on the simultaneous increases in labor supply, fertility rates and wages in a dynamic setting. There are two remarkable changes that occurred in the 1990s. First, participation has been more compatible with having children. It has been a general belief that an increase in labor participation leads to the reduction in the probability of having a baby. Doepke et al. (2015) reveal that the reduction in the labor supply of young women induces to the Baby Boom period ( ). 3 In Figure 2, however, the participation rate has risen while the fertility rate slightly increased in the periods of our interests. In Figure 3a and 3b, we can observe the similar phenomena using the cohort base data. The LFP and fertility rates have simultaneously increased for married women aged ,5 Therefore, women were less likely to withdraw from the labor force while they gave birth and raised kids. The negative correlation between the LFP and fertility rates has significantly disappeared. The literature also documents that the fertility rate increased while the LFP rate rose after the early 1980s (Ahn and Mira 2002; Cardia and Gomme 2013). Second, as women worked more, the logarithm of hourly real wage has risen more rapidly with the ages, partly due to an increase in investment, in the 1990s (Figure 4a). The average wage growth rate increased from to over the two decades. The possible determinants include a decreased childcare cost relative to household income, increased returns to experience, a decrease in husbands income and increased education. From the numerical experiments, the key findings are following: (1) the reduction in childcare cost is the sole factor that simultaneously raises the labor participation and fertility rates 3 The wartime increased labor participation of women and it had a persistent effect, because older women had work experiences after the war. They demonstrate that young women left the labor market because of the competition with experienced women hence gave more birth. 4 The 1970s (1980s, 1990s) cohort consists of married women aged in 1972 (1982, 1990). 5 This phenomenon may be due to the delay of having a child in the 1990s, since the model starts at age 25. However, the number of children women ever had from their first child until age 44 increased from to The number of children at age 25 was 1.71 in the 1970s and 1.67 in the 1990s, but that at age 44 was 1.64 in the 1970s and 1.73 in the 1990s. 4

5 and (2) the increased returns to experience account for higher wage growth rates in the 1990s. In the learning-by-doing specification as in Attanasio et al. (2008), the increased returns to experience do not raise wages and wage growth rates. This is due to the selection effect, wherein new entrants to the labor market have lower productivity than existing participants (Attanasio et al. 2008). In the Ben-Porath specification, however, women have more incentive to invest in human capital when they participate more, which generates higher wages and wage growth rates. Thus, the higher wages and growth rates can be explained by the rise of investment, which dominates the selection effect. Furthermore, the results are presented by education status because women may react differently to observed changes depending on their education level. This paper is built as follows: Section 2 reviews the previous female labor supply and fertility literature. I present a life-cycle model in Section 3 and describe the data in Section 4. The calibration and experiments are shown in Section 5 and Section 6. In Section 7, I compare the results between the Ben-Porath specification and the learning-by-doing model. I describe the different behaviors by education in Section 8. The conclusion is provided in Section 9. 2 Previous literature: The increases in labor supply and fertility of married women from the 1970s to the 1990s This section reviews the determinants of labor supply and fertility decisions considered in the previous literature. The increase in labor supply of married women is a noticeable change in the U.S. labor market (Buttet and Schoonbroodt 2013; Goldin and Mitchell 2017). A large and growing number of studies address the question of why women s LFP rate has dramatically increased as in Figure 3a. The leading explanations include an increase in the education of women (Eckstein and Wolpin 1989; Eckstein and Lifshitz 2011; Keane and Wolpin 2009; Francesconi 2002), an increase in their relative wages (Attanasio et al. 2008; Cardia and Gomme 2013; Eckstein and Lifshitz 2011; Van der Klaauw 1996; Jones et al. 2003; Greenwood et al. 2015), increased returns to experience (Attanasio et al. 2008; Olivetti 2006; Buttet and Schoonbroodt 2013) and issues related to the household environment such as the childcare cost (Attanasio et al. 2008; Bick 2016; Sheran 2007; Cardia and Gomme 2013; Connelly 1992; Hardoy and Schøne 2015; Blau and Robins 1989; Powell 2002) or home production changes (Greenwood et al. 2005; Greenwood et al. 2015; Cardia and Gomme 2013; Jones et al. 2003). 5

6 Existing literature also emphasizes the role of an increased divorce rate (Fernández and Wong 2014) and a decreased fertility rate (Fernández and Wong 2014; Buttet and Schoonbroodt 2013). Table 2 provides a summary. Between the 1970s and the 1990s, the ratio of women s hourly income to that of men rose from 0.69 to Also, the college entrance rate of married women increased from 38.85% to 48.44%. Higher educational attainment can be a key driver of higher participation as the rise in college entrance results in increased potential earnings. However, these explanations can affect the LFP of both single and married women, but married women experience the sharp increase in the LFP alone in the late twentieth century (Leibowitz and Klerman 1995; Blau and Kahn 2005; Eckstein et al. 2016). In consequence, issues related to family and children appear more likely to account for the increase in the labor supply of married women than the reasons listed above. Eckstein and Lifshitz (2011) find that approximately 40 percent of the rise in women s labor supply cannot be explained by improved education or wages. The unexplained portion may be due to changes in household environments related to children and husbands. In the decades of interest, the fertility rate of married women has risen (Figure 3b). Some papers emphasize that the decreased participation of married women or improved maternal health causes the post-war baby boom (Doepke et al. 2015; Prados and Albanesi 2014), but there are a few studies that focus on increased fertility rate in the late twentieth century. First, Ahn and Mira (2002) suggest that positive relation between labor supply and fertility after the late 1980 s is largely due to an increase in the use of childcare services. An increase in availability of childcare service can provide incentives to have more kids while women raise their labor force participation. Second, they also provide that the rise of income effect can be another factor for the increase in the fertility rate. However, increased income of a mother may have a negative association with fertility rate because of higher opportunity costs of having a kid (Gayle and Miller 2006; Balbo et al. 2013). Improved educational attainment which results in higher earnings will lead to a reduction in the birth rate (Francesconi 2002; Caucutt et al. 2002). On the other hand, an association between other household income and having a child is ambiguous (Gayle and Miller 2006). While the positive relation arises if a child is a normal good (Hotz and Miller 1988), higher income can reduce the birth rate if parents consider the quality of children. Recent studies show that high income parents can choose to invest more in their existing children instead of having more children, a quality-quantity trade-off. (Balbo et al. 2013; Jones et al. 2008). Third, Bouvier (1991) provides that an increase in immigration 6

7 and an increasing proportion of high fertility groups such as Asian and Hispanic contribute to the rise of the fertility rates. 3 Model 3.1 Model set-up This section formulates a dynamic stochastic discrete choice model of labor supply and fertility decisions for married women. The model is built on Eckstein and Wolpin (1989), Eckstein and Lifshitz (2011) and Francesconi (2002). 6 The model begins at age 25 and decisions are taken at an annual basis. At the start of the model (age 25), marital status and education are predetermined. That is, all women have husbands and their education status is either less than a college education or a college education. I restrict for married women because they can more freely leave the labor market for children as a second earner. The wife is the sole decision maker in the household. In each period between age 25 to 59 (t), one maximizes the present value of her lifetime utility by choosing whether to work, whether to give birth, her human capital investment and consumption. Since extensive margin is a main change in the data, I focus on participation decisions (Attanasio et al. 2008). While several previous papers suppose that women imperfectly control their fertility (Hotz and Miller 1988; Sheran 2007; Erosa et al. 2016), I assume that women can give birth in the same period that they decide to. The model incorporates two dynamic considerations: children and human capital accumulation. The number of children evolves according to N t = N t 1 + n t (1) where N t is the total number of children at period t and n t is equal to one if a woman has a additional child at t. She can have 0, 1, 2 or 3 children, and N t = 3 is an absorbing state. At the beginning of every period, married women observe their realization of the shocks. And then, they take decisions by maximizing: max V t = E[St=td T t t 1 U(P t, c t, n t, N t 1, h t 1, v t )] (2) {P t,c t,i t,n t } t 6 Their models are discrete choice models, but Eckstein and Lifshitz (2011) only consider labor supply decision, while Francesconi (2002) and Eckstein and Wolpin (1989) consider the labor supply and fertility decisions together. 7

8 where d 1 is a discount factor and E is the expectation operator given information at t. P t indicates whether the woman works, c t is consumption, n t is an indicator whether she gives birth and i t is the human capital investment level. h t v t is a preference shock. 1 is accumulated human capital at period t-1. The consumption is divided evenly between spouses in each period because this is a unitary household. The household budget constraint is given by: y w t P t + y h t = c t + g(n t 2, n t 1, n t, P t ) (3) where y w t is the wife s earnings and yt h is the husband s earnings. g(n t 2, n t 1, n t, P t ) is a childcare cost function. It depends on the number of children at t-2 (N t 2 ), whether there are newborn children at t-1 and t (n t 1 and n t ) and the mother s current work status (P t ), according to g(n t 2, n t 1, n t, P t )=(f 0 + f 1 P t )N t 2 +(f 10 + f 11 P t )n t 1 +(f 20 + f 21 P t )n t. (4) The parameters f 0, f 10 and f 20 are costs incurred per child regardless of whether the mother works or not. Mothers spend additional costs to purchase the paid child care services when they work (f 1 > 0, f 11 > 0, f 21 > 0). The cost also depends on the children s ages. The literature finds that small children generate higher childcare expenditures (Ribar 1995; Gayle and Miller 2006). However, instead I suppose the costs of having an infant (n t ) or a toddler (n t 1 ) differ from those of older children because this model does not include the children s ages. The toddler and the infant spend higher childcare expenditure (f 0 < f 10 < f 20, f 0 < f 10 < f 20 ). The budget constraint implies that there is neither saving nor borrowing. That is, households spend all their net income in each period. Although the assumption is extreme, it is standard in the previous studies on female labor supply, for tractability (Eckstein and Wolpin 1989; Eckstein and Lifshitz 2011; Van der Klaauw 1996; Francesconi 2002; Sheran 2007; Keane and Wolpin 2009; Yamaguchi 2016). The saving decision is excluded because incorporating decisions related to children makes female labor supply models more complicated (Yamaguchi 2016). I discuss about the saving assumption more in Section 3.4. According to Eckstein and Wolpin (1989) and Francesconi (2002), if the utility function is linear and additive in consumption, the life-time utility maximization problem is equivalent to a life-time wealth optimization problem modified by the non-monetary value of work and children. 8

9 The flow utility function is given by: U t = (a 1 + v t )P t + c t + a 2 P t c t + a 3 P t h t 1 + a 41 N t + a 42 N 2 t + a 5c t n t + a 6 n t t. (5) The first four parts of the function describe the utility related to women s current work. The cost of working depends on household earnings (a 2 < 0) and their previous human capital (a 3 < 0). The model also allows the number of children including the current birth to affect the utility (a 41 > 0, a 42 < 0), but this is not depending on women s participation. A value of a 5 can produce a relationship between consumption and giving birth. Also, the disutility of giving birth increases with the mother s age (a 6 < 0). In the model, women can have birth until age 59, but a negative value of a 6 discourages them from giving birth when they get old. An old woman gives little birth without the restriction of the birth periods to certain ages. All women have working husbands. This is not an abnormal assumption because the PSID data show that 96% of married men worked in the 1970s and the 1990s. 7 The husband s income equation is: ln(yt h )=g 0 + g 1 edu + g 2 t + g 3 edu t + g 4 t 2 + e 1t. (6) The husband s earnings equation consists of wife s characteristics because of the positive correlation of age and education in the marriage market which is known as assortative mating, and the temporary component. Since the fraction of couples with the same education is 74.1% (Table 3), the wife s age and education can be proxy variables for husband s age and education as in Van der Klaauw (1996) and Sheran (2007). Furthermore, the husband s income increases with the spouse s age, but grows at a different rate depending on the wife s education. Altuğ and Miller (1998) find that the coefficient on an interaction term between age and education is positive, implying that the education effect increases with age. Due to assortative mating, a similar relationship appears with the wife s age and education in the husbands earnings. Figure 5 shows that the income of husbands married to more educated women increases more rapidly with age than that of husbands of less-than-college-educated women, in both the 1970s and the 1990s. Women s earnings depend on the current human capital (h t ), human capital investment (i t ), education (edu) and her earning shock (e 2t ). In each period, women make investments in their future human capital following the Ben-Porath model if they work. Higher investment 7 The ratio is 95% in Van der Klaauw (1996) 9

10 rates, however, lead to lower current earned income. Thus, women who plan to work less in the future may choose lower investments. The wife s earnings are given by: ln(y w t )=b 0 +(1 i t )R t h t + b 3 edu + e 2t (7) where R t is a human capital rental rate that for simplicity is normalized to 1. The parameter b 3 captures returns to education, and schooling raises the earnings (b 3 > 0). e it, i = 1, 2 is the standard zero-mean, finite variance, serially independent error. This paper also assumes that the husband s income shock (e 1t ) is positively correlated with the wife s earning shock (e 2t ) as Attanasio et al. (2008). 3.2 Human capital accumulation One of the key features in this model is the human capital accumulation process. In learningby-doing model, the current labor participation augments human capital, and hence has impacts on future wages and future labor supply decisions (Eckstein and Wolpin 1989; Attanasio et al. 2008; Olivetti 2006). The presence of a child can affect human capital accumulation through non-participation. In the Ben-Porath model, however, individuals accumulate their human capital not only through labor participation but also through explicit investment. This specification can have an additional consideration for children in that women who plan to have discontinuities in their careers during child-rearing ages choose lower investment, which translates into lower wages (Mincer and Polachek 1974; Polachek 1975; Polachek 2004). This investment is defined as training at work. There are two types of human capital investment: formal education and training at work. While education is enhancing general skills, training is developing a specific skill. Also, training requires some portion of time (Song and Jones 2007). Therefore, lower investment for women implies that they have less efforts to enhance specific skills due to their lower expectation over future careers. Once women work, they spend a fraction of time, i t, investing in human capital and spend the rest of it, 1 i t at pure work. I specify the human capital production function as in Heckman et al. (1998): h t =(1 d 2 )h t 1 + b 1 i b i t where d 2 is the depreciation rate and P t 1 hb h t 1 + b 2(1 P t 1 ) (8) 1 represents the previous participation decision. The parameter b 1 reflects the ability to learn, because it measures the extent to which human capital 10

11 investment contributes to future earnings (Heckman et al. 1998). This paper has a further consideration on the depreciation related to non-participation. The parameter b 2 measures the depreciation in human capital associated with previous nonparticipation (b 2 < 0). The learning-by-doing effect is captured through P t 1. If married women did not work at t-1, the investment is equal to zero, i t 1 = 0, and human capital is h t =(1 d 2 )h t 1 + b 2. I assume that the exponent b i lies between 0 and 1, so that all working women invest at least some portion of their earnings. The parameters d 2 and b 2 both reflect human capital depreciations but they work differently in the model. The parameter d 2 represents general human capital depreciation incurred regardless of whether one works or not. However, the parameter b 2 reflects a cost of work interruption due to specific skill obsolescence. Inserting into the earnings function in equation (7) yields: ln(y w t )=b 0 +(1 i t )((1 d 2 )h t 1 + b 1 i b i t 1 hb h t 1 + b 2(1 P t 1 )) + b 3 edu + e 2t. (9) This model does not explicitly include experience. However, since work experience facilitates human capital accrual, the returns to experience (RTE) can be defined as the change in current earnings due to a change in lagged human capital: RTE t ln(yw t ) h t 1 =(1 i t )[(1 d 2 )+b 1 b h i b i t 1 hb h 1 t 1 ]. (10) The returns to experience are endogenously determined. Married women who made higher human capital investments in the previous period (i t 1 ) have higher current returns to experience at any value of the current investment (i t ). However, higher current investment (i t ) reduces not only the current wages but also the returns to experience. The returns to experience fall in human capital (h t 1 ) because the wage function is concave in that variable ( RTE t h t 1 < 0). 3.3 Solving the model The dynamic programming solution to the optimization problem is obtained by a process of backward recursion from the age of the last period. At each age, the value function and optimal decision rules are solved given the current state vector and the value function for the next period. The state vector is defined as: W t = {h t, N t 2, n t 1, edu, e 1t, e 2t, v t }. (11) 11

12 V t (W t ) is the value function, defined recursively as V t (W t )= max P t,c t,i t,n t U t ( )+d 1 E[V t+1 (W t+1 W t )]. (12) The solution to the problem is obtained from two stages. In the first stage, each individual maximizes her value by choosing human capital investment levels when they work. A nonworking woman does not invest in human capital. V t (W t )=max i t U t ( )+d 1 E[V t+1 (W t+1 W t, P t = 1, n t )]. (13) In the second step, given the optimal human capital investment, they choose the participationbirth alternative that yields the highest value: V t (W t )=max[v 11 t ( ), V 10 t ( ), V 01 t ( ), V 00 t ( )] (14) V 11 t (W t )=max U 11 t ( )+d 1 E[V t+1 (W t+1 W t, P t = 1, n t = 1)] (15) V 10 t (W t )=max U 10 t ( )+d 1 E[V t+1 (W t+1 W t, P t = 1, n t = 0)] (16) V 01 t (W t )=max U 01 t ( )+d 1 E[V t+1 (W t+1 W t, P t = 0, n t = 1)] (17) V 00 t (W t )=max U 00 t ( )+d 1 E[V t+1 (W t+1 W t, P t = 0, n t = 0)] (18) t ( ) represents the value when a woman chooses to both work and give birth. V 10 ( ) is the V 11 value when she chooses to work, but not to give birth. Vt 01 ( ) is the value when she chooses not to work, but to give birth. Vt 00 ( ) is the value for no work and no birth. All alternatives are mutually exclusive. Existing studies solve the model by assuming that the value function after the terminal period is zero, V T+1 = 0 (Eckstein and Lifshitz 2011; Francesconi 2002). However, if women consider their consumption after the retirement, the value after the last period can be positive and they may be more willing to work more in their later ages. Thus, this paper supposes that the terminal value function is V T+1 = z ln y w T+1. The solution method is presented in Appendix A.1. t 12

13 3.4 Risk neutral and no savings assumption In the model, women do not have incentives to insure against the future uncertainty with the assumption of a linear utility function in consumption. Although this is the standard setting in the female labor supply literature, I need to discuss how participation decision would change if the saving decision is added to the model. Including the ability to save still does not change women s decisions if utility is still linear. On the other hand, if the model includes the assumption of risk aversion and no ability to save, women work more in their younger ages for future uncertainty because the participation is the only device for consumption smoothing. Including savings with the risk aversion also can raise the participation, but the magnitude of the increase is less than the model without savings because savings can be another device to smooth consumption. However, it appears that including these assumptions has little effect on participation decisions and hence human capital accumulation of married women during the child-rearing periods. According to Attanasio et al. (2005), women do not increase participation to smooth consumption during the child-rearing periods, until age 35, due to childcare costs. They also find that women do not save during this period even though the model allows the ability to save. This supports that including the assumptions of savings and risk aversion has little effect on the participation decision. 4 Data This paper utilizes two cohorts of married women from the PSID (Panel Study of Income Dynamics) data. The data construction is described in Appendix A.2. First, the 1970s cohort consists of married women aged in 1972 (born between 1945 and 1949). I track this cohort for 37 years, until They spent their child-rearing periods in the 1970s. The baseline model is fit to the life-cycle age profiles of this cohort. This paper also tracks the 1990s cohort, consisting of married women aged in 1990 (born between 1963 and 1967). 8 These women were in their late twenties and early thirties in the 1990s. I track this cohort for 21 years, until The 1970s cohort includes women aged 25-59, but the 1990s cohort includes women aged because the oldest age of the 1990s cohort in 2011 is 48. These two cohorts are used to explore the changes in the LFP, fertility rate and wage profiles between the 1970s and the 1990s. 8 I also generate the 1970s cohorts and the 1990s cohorts by changing the base year. The results maintain the facts that the LFP and fertility rates increase together between two cohorts. 13

14 Figure 10 shows the timeline for the two cohorts. Table 5 reports the sample statistics of 1970s cohort and 1990s cohort by 10-year age groups. Furthermore, continuously married women have different behaviors from married women who maintained their marriage life for short periods. If women have higher probability of getting divorced, they might work more and accumulate human capital more. Previous studies find that higher divorce rates can raise the LFP rate of married women (Fernández and Wong 2014). Thus, the final sample consists of women who were married at least ten years during the sample periods. I do not restrict the age of getting married. The LFP rate is defined as the fraction of women who work at least 100 hours per year as in Attanasio et al. (2008). The fertility rate is defined as the fraction of married women who have a newborn baby at age t. Real values are calculated using the CPI (the base year 1982) from the BLS web site. 9 5 Calibration 5.1 Calibration strategy In Table 9, I report the set of external, calibrated and estimated parameters for the baseline economy. A two-step strategy is similar to Olivetti (2006). First, I take some of the external parameters from the existing literature and estimate some using the PSID (d 1, d 2, b c 0, bl 0, bc 2, b l 2, b i, b h, s e1, s e2, r e1e2, g 0, g 1, g 2, g 3, g 4 ). The parameters of the husband s and wife s income equations are estimated using the fixed effects model for unobserved heterogeneity and the Heckman two-stage method for selection bias, separately. In the second step, I follow the calibration procedure as in Buttet and Schoonbroodt (2013). Thus, the remaining parameters in the utility function and childcare cost function, (a 1, a 2, a 3, a 41, a 42, a 5, a 6, b c 1, bl 1, f 0, f 1, f 10, f 11,f 20, f 21, z), are calibrated to the 1970s cohort of the PSID by matching the following moments (see Table 10): (1) the average labor force participation rates of married women at all ages combined and by 5-year age groups (8 moments), (2) the average fertility rates of married women at all ages combined and by 5-year age groups (8 moments), (3) women s wages at all ages combined and by 5-year age groups (8 moments),

15 (4) the gender wage gap at all ages combined (1 moment), (5) the average labor force participation rates by education (2 moments), (6) the ratio of childcare costs to household income (1 moment), (7) the ratio of childcare costs to household income for newborn mothers (1 moment). The system is over-identified because the 16 parameters are calibrated by matching 29 moments. To fit the data, I minimize the sum of squared errors (SSE) between the moments simulated by the model and the moments from the data as in Buttet and Schoonbroodt (2013). Let q be a vector of parameters, m a j be the 29 1 vector of moments from the actual data and m s j (q) be the 29 1 vector of simulated moments. W is the weighting matrix, equals to the variance on the diagonal and zero otherwise. I minimize the objective function with respect to q. 10 The objective function is [m a j m s j (q)]0 W[m a j m s j (q)]. (19) Table 9 summarizes the parameter values used in the baseline economy External parameters Some parameters (d 1, d 2, s e1, s e2, r e1e2, b i, b h ) are taken from the previous literature. First, I suppose that the discount factor, d 1 equals This value is consistent with other studies, which set their discount factors between 0.8 and 0.96 (Attanasio et al. 2008; Sheran 2007; Adda et al. 2017; Francesconi 2002). Second, the depreciation rate, d 2 is set to following Blandin (2015). This represents general human capital depreciation due to the skill obsolescence or aging that is independent of the work decision. There is no consensus on the magnitude of this depreciation rate. Some papers assume that no human capital depreciation occurs when individuals work (Attanasio et al. 2008; Sheran 2007; Heckman et al. 1998), but others report that the value is between and (Huggett et al. 2006; Blandin 2015; Polachek et al. 2015; Prados and Albanesi 2014). I also assume that the standard deviations of both the wife s and husband s income shocks are 0.13 and the correlation between two shocks is 0.25, as in Attanasio et al. (2008). The parameters b i and b h are set to 0.85 and I follow the neutrality assumption b i = b h from the Ben-Porath (1967) s original specification for a tractable reason. According to Heckman et al. (1998), the conditions 0<b i <1 and 0appleb h apple1 guarantee a concave human capital 10 I use the Simplex method to find the parameters. 15

16 function. They estimate b i = b h to be and 0.871, by education level. Blandin (2015) also sets both parameters to 0.7 which is the average value of the previous literature. The magnitudes of these values are in the range of the values in the existing studies, (Heckman et al. 1998; Blandin 2015; Huggett et al. 2006) Estimated parameters I estimate the husband s and wife s wage equations separately. 11 The parameters of these equations are estimated outside the structural model to reduce the computational burden of dynamic programming. The husband s hourly wage is expressed as a function of the wife s age, age squared, education and the interaction between age and education (Equation 6). The coefficients using the fixed effects model to control the unobserved heterogeneity are presented in Table 6. For the 1970s cohort, the estimates of the returns to wife s education (g 1 ) and the coefficient of the wife s age (g 2 ) are estimated to 8.91% and 1.05%, respectively. For the 1990s cohort, these values are 17.7% and 5.84%, implying that husband s skill premium has risen due to the assortative matching. The positive interaction term (g 3 > 0) finds that husbands who marry college graduates have higher wage growth rates, consistent with Figure 5. From those estimated results, the parameters g 0, g 1, g 2, g 3 and g 4 are set to 2.150, , , and , separately. In this paper, I use the Heckman two-steps model to control the selectivity bias in women s wage equation (Heckman 1979). The exclusion restrictions which affect the participation, but do not enter the wage equation are the number of children, the age of the youngest child and spouse s income. Table 7 shows the estimated coefficients for the wife s wage equation by education for the 1970s and the 1990s. The results reveal that return to a college education is 42.2% and return to an experience is 7.93%. The parameters b l 2 and bc 2 reflect wage losses from career interruptions. Previous estimates of the negative effect of work interruptions on wages vary significantly from 3.3% to 33.0% depending on measures and data (Mincer and Ofek 1982; Light and Ureta 1995; Albrecht et al. 1999; Beblo and Wolf 2002; Spivey 2005; Baum 2002; Jacobsen and Levin 1995). 12 A career interruption indicator equals to one if women participate less than 1,500 hours a year following Olivetti (2006). In the 1970s and the 1990s, 39% and 29% of married women have career interruptions, separately. The estimates show that a career 11 I include 2 years more in the sample, because the cohort sample is small. 12 Mincer and Ofek (1982) estimate it 3.3%-7.6%. The effect in Spivey (2005) is 12-15%, whereas Light and Ureta (1995) find an effect of 23%. 16

17 interruption of one year reduces subsequent wages by 14.7% for college educated women and 9.2% for less than college educated women. From the estimated results, the parameters, b c 2 and b l 2, are set to and Also, the positive coefficients (l) on the Heckman correction term in the 1970s and the 1990s sample suggest that high ability women are more likely to participate. This provides that the average wage may fall if more women enter the labor market. This paper supposes that individuals with different educational attainment have different learning abilities (b c 1 and bl 1 ) and depreciation rates (bc 2 and bl 2 ). Both parameters increase in education as results in Polachek et al. (2015) and Buttet and Schoonbroodt (2013). The estimated results also reveal that women with college education have higher costs for work interruptions (see Table 7). Furthermore, I suppose that two educated groups have different constant terms (b c 0 and bl 0 ), which are like TFP. In this case, the value of b 3 is given by the difference in constant terms between two education groups, b c 0 b l 0. bc 0 and bl 0 are set to and Calibrated parameters I report the calibrated parameters in Table 9. The first panel of Table 9 presents the calibrated values of the utility function (see also equation (5)). Previous estimated results provide guidance on these parameters (Francesconi 2002; Van der Klaauw 1996; Eckstein and Wolpin 1989; Eckstein and Lifshitz 2011; Sheran 2007): First, the parameter a 1, which reflects the disutility of labor participation, is set to Previous studies also estimate a negative value (Eckstein and Wolpin 1989; Eckstein and Lifshitz 2011; Francesconi 2002; Van der Klaauw 1996). Second, the sign of the interaction between consumption and labor supply has varied in previous studies. While Attanasio et al. (2008) suggest a positive value, Francesconi (2002) and Eckstein and Lifshitz (2011) estimate a negative one. I suggest that the value of consumption falls with participation (a 2 =-0.337), implying that consumption and employment (leisure) are substitutes (complements). Thus, the marginal utility of consumption, 1 + a 2 P t + a 5, is lower when working than when not working. Likewise, the marginal disutility of work is higher when consumption is higher. This also implies that wives with higher earning spouses are less likely to participate. The parameter a 3 represents the utility of work related to previous human capital. If a 3 is negative, the disutility of labor participation increases with total human capital (Francesconi 2002; Eckstein and Lifshitz 2011). If it is positive, there is habit persistence in working (Sheran 17

18 2007). However, there is little theoretical evidence about the value of this parameter (Francesconi 2002). In this baseline economy, a 3 is set to so that higher human capital increases the disutility from work, perhaps because higher wages increase the value of non-participation. In this model, the past human capital has two effects: it reduces the current utility through a 3, but increases the current wages. Eckstein and Wolpin (1989) argue that because women have increased their experience, the increased wage effect must dominate the decrease in utility effect. Also, flow utility increases with the number of children (a 41 = 0.146), but at a decreasing rate (a 42 = ). The parameter a 5 is negative, -0.02, so the value of consumption falls with a birth, which is similar in magnitude to Francesconi (2002). The negative value of a 6 ( ) reflects that mothers utility from newborn babies falls with their age because of their physical limitations. In the wife s earning equation, the parameter b 1 is set to and for college graduates and less than college educated women, respectively. Furthermore, six parameters of the childcare cost function are calibrated (f 0, f 1, f 10, f 11, f 20, f 21 ). The parameters f 0, f 10 and f 20 reflect necessary childcare costs for all mothers. However, the parameters, f 1, f 11 and f 21 are additional childcare costs incurred when working and using paid care. Their values are provided in Table Baseline life-cycle profiles Using the parameters in Table 9, I generate life-cycle profiles for the labor force participation rate, fertility rate, wages and human capital investment. The matched statistics overall and by age groups are shown in Table 10. The statistics use the weights across ages in the 1970s cohort in Table 4. The fit between observed data and moments is good because the minimum distance for the quadratic form is As shown in Figure 7, the predicted employment profiles well approximate the data profiles of the 1970s cohort. The overall labor force participation rate from the data is 70.6%, and the statistic from the model is 70.3%. The average LFP rate, fertility rate and wage rate are constructed using the distribution of ages in the PSID (see Table 4). As in the data, the model predicts that the LFP rate of college educated women is higher than that of married women without college education. The model also precisely replicates lower participation of women during the childbearing and rearing periods. The average fertility rate is equal to 8.6% from the data, and 8.8% from the model. The fertility rate also decreases with age, as in the data. After age 40, few women give birth. 18

19 The log hourly wage of married women is from data and from the model. The predicted wage profile decreases after age 50 due to lower investment and depreciation rate than the wage profile in the pooled data. However, this is consistent with the wage profile after controlling fixed effects even though I match the pooled data. This implies that women with higher earnings more remain in the labor market in their later ages. Furthermore, the relative wage (married women/married men) from the data is 69.0%, and from the model is 72.1%. The standard deviation in earnings in the baseline economy is The average childcare cost to household income is also a target in the model. The data show that the ratio of childcare cost to household income is 17.3% for the 1970s cohort, the cost ratio for newborn mothers is 20.1%. In the model, the childcare cost share equals 17.2% and 20.9%, separately. The distribution for education in the model is exogenous, and sets to that in the data, which show that 38.85% of married women entered college in the 1970s. The model generates life-cycle human capital investment profiles of married women. Human capital is accumulated through time investment. According to Ben-Porath (1967), individuals invest substantially when they are young and monotonically reduce investment as they age and their remaining work lives shorten. In Figure 8a, the investment falls with ages and women make the minimum investment at their later ages. Furthermore, mothers have lower investment than the averaged married women due to their lower participation, 46.9%. Thus, they have less incentive to invest in human capital, which is consistent with Mincer and Polachek (1974) and Polachek (1975). Women with smaller expectation over future participation have lower investment even while at work because they understand that their returns depend on how much they work in the future. A feature of the Ben-Porath model is a difference between observed wages and human capital due to investment in equation (7), suggested by Heckman et al. (1998). This generates a different gap in human capital and observed wage between mothers and non-mothers. Figure 8b shows the gap in human capital and wages between mothers and non-mothers from a simulated profile. Mothers do not invest as much as non-mothers during child-rearing periods. At younger ages, the wage gap between the two groups is less than the human capital difference because of lower investment of mothers. Mothers devote their time at work more to the current earnings rather than human capital accumulation, at any level of existing human capital. This implies that the observed wage difference between two groups, which is the wage penalty of motherhood, does not fully capture the human capital gap. Furthermore, the human capi- 19

20 tal gap and wage gap rise together because lower investment of mothers accumulates human capital less than non-mothers. 6 Numerical experiments In this section, I examine several factors that may play a key role in explaining the changing behaviors of married women between the 1970s and the 1990s cohorts, to assess their effects on labor supply, fertility and wages: (1) childcare costs, (2) the returns to investment, (3) husbands earnings and (4) college entrance rates. They are factors that mainly considered to shed light on the increased participation in the previous literature. Figure 9a-9f reveal the simulated life cycle profiles of the average wages of married women from the above experiments. However, the change in average wages depend on the change in the composition of married women. If fewer (more) capable workers enter the labor market due to the changes in experimental factors, average wages decrease (increase), although the wages of existing participants may be constant or higher. Therefore, this paper separately presents the wages of participants who were already in the labor market prior to the experiments (the dotted lines shown in Figure 9a-9f). Furthermore, Table 11 presents LFP rates, wages and fertility rates both overall and for each education group because college educated married women may respond differently to the experiments from the less than college educated women. 6.1 Decrease in childcare costs Evidence on how childcare costs have changed A decrease in childcare costs can be a key determinant of the increase in the labor participation and fertility of married women (Attanasio et al. 2008; Bick 2016; Sheran 2007; Cardia and Gomme 2013; Connelly 1992; Hardoy and Schøne 2015; Blau and Robins 1989; Powell 2002; Buttet and Schoonbroodt 2013). As the primary caregivers, mothers have an incentive to work more if they can purchase child care services at cheaper prices, and the lower costs of children also stimulate potential mothers to have more kids. However, there is little systematic evidence on the changes in childcare costs between the 1970s to the 1990s in the data. The datasets such as the Consumer Expenditure Survey (CEX) and the Survey of Income and Program Participation (SIPP) provide information since the 1980s. The PSID data include cost 20