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Marriage and Work among Prime-Age Men WP 23-02 Adam Blandin Vanderbilt University John Bailey Jones Federal Reserve Bank of Richmond Fang Yang Federal Reserve Bank of Dallas Marriage and Work among Prime-Age Men Adam Blandin John Bailey Jones Vanderbilt University Federal Reserve Bank of Richmond Fang Yang Federal Reserve Bank of Dallas January 13, 2023* Abstract Married men work substantially more hours than men who have never been married, even after controlling for observables. Panel data reveal that much of this gap is attributable to an increase in work in the years leading up to marriage. Two potential explanations for this increase are: (i) men hit by positive labor market shocks are more likely to marry; and (ii) the prospect of marriage increases men’s labor supply. We quantify the relative importance of these two channels using a structural life-cycle model of marriage and labor supply. Our calibration implies that marriage substantially increases male labor supply. Counterfactual simulations suggest that if men were unable to marry, prime-age male work hours would fall by 7%, and if marriage rates fell to the extent observed, men born around 1980 would work 2% fewer hours than men born around 1960. Keywords: Labor supply, family structure, marriage, marital wage premium. JEL Classifications: D15, J1, J22, J31 * Email: adam.blandin@vanderbilt.edu, john.jones@rich.frb.org, fang.yang@dal.frb.org. We are grateful to Stefania Albanesi, Alex Bick, Bettina Brueggemann, Katrina Mullen, Luigi Pistaferri and seminar participants at the 2019 Arizona State University Reunion Conference, 2020 Southern Economic Association Annual Meeting, 2021 Midwest Economic Association Meeting, and 2022 Duke/FRB Richmond/UVa Jamboree, CAU Conference on Structural Approach in Public Economics, 2022 BSE Summer Forum Workshop, and 2022 SED Annual Meeting for useful comments. Yang gratefully acknowledges the support from E. J. Ourso College of Business Summer Research Funding at LSU. The opinions and conclusions are solely those of the authors and do not reflect the views of the Federal Reserve Bank of Dallas, the Federal Reserve Bank of Richmond, or the Federal Reserve System. But if anyone does not provide for his relatives, and especially for members of his household, he has denied the faith and is worse than an unbeliever. — 1 Timothy 5:8 How can you frighten a man whose hunger is not only in his own cramped stomach but in the wretched bellies of his children? You can’t scare him – he has known a fear beyond every other. —John Steinbeck, The Grapes of Wrath ‘What does a man do Walter? A man provides for his family.’ — Breaking Bad 1 Introduction A longstanding empirical finding is that the labor supply of married women, especially those with children, tends to be lower than that of women who have never been married. An immense literature has sought to understand this pattern and its implications; important contributions include Becker (1985), Becker (1988), Becker (1991), Goldin (1992), Weil and Galor (1996), Goldin (2014), Doepke and Tertilt (2016). In this paper, we consider whether there are also important interactions between marital status and labor supply among men. We are motivated by a simple observation: in the cross section, married men work substantially more than men who have never been married. For example, among men ages 20 - 54 in the Current Population Survey (CPS), currently married men work at least 30% more annual hours than men who have never been married, a gap which has remained roughly constant since 1975 (see Section 2). Since 1975, the magnitude of the marital gap in annual hours worked for men has been larger, albeit with the opposite sign, than for women. The first contribution of this project is to document that many of the additional hours worked by married men can be attributed to an increase in work in the years leading up to marriage. In particular, we regress hours worked on a set of dummy variables for distance-from-marriage, as well as individual fixed effects, on panel data from the National Longitudinal Survey of Youth 1979 (NLSY79). The regressions show that men increase their hours by roughly 13% in the ten years preceding marriage, and that this increase in hours persists for at least ten years after 1 marriage. With fixed effects included, these results raise the possibility that marriage itself, rather than persistent differences between married and never-married men, leads to higher hours of work. Our second contribution is to quantitatively assess potential explanations for why male hours of work increase prior to marriage. We develop a life-cycle model of male labor supply and saving, where men face uncertainty over wages, marital status and fertility, and use it to evaluate two categories of explanations. The first category we refer to as dynamic selection: we allow men receiving higher wage shocks, who respond by increasing their labor supply, to experience a higher likelihood of marriage. The second category we refer to as causal effects: for one or more reasons the prospect of marriage raises male labor supply. In particular, we allow marriage and children to change a man’s income and expenses and, importantly, raise the marginal utility of consumption expenditures. This “mouths-to-feed” effect leads married men to work more when married and to increase their hours of work in anticipation of starting a family. We calibrate the model using data from the NLSY79. The calibrated model closely matches the joint distribution of children and marital status over the life cycle, as well as the correlation of earnings between spouses. To discipline the strength of the selection effect, we also require the model to replicate the marginal effect of wages on marriage probabilities found in the CPS, where we instrument for individual wage changes with state-level changes. The calibrated model is able to generate marriage-related hours dynamics similar to the data, but only if the mouths-to-feed channel is sufficiently strong. Our estimates from the CPS imply that positive wage shocks are associated with positive selection into marriage. However, in the model this force is quantitatively weak. Moreover, because empirically hours increase more than wages prior to marriage, the model would require larger-than-typical labor supply elasticities for wage-based selection to be the sole explanation. Additional evidence against the selection explanation comes from the behavior of men whose marriages are preceded by pregnancies. Responses in the NLSY79 indicate that pre-marital pregnancies are more likely to be unplanned, and thus less likely to be driven by labor market shocks. Yet the data and the model both show that male hours increase by more, not less, around marriages preceded by pregnancies than around marriages where children arrive later.1 The mouths-to-feed effect is in turn a combination of several mechanisms. The one most responsible for the marriage-related increase in hours is the value men place on the consumption utility of their spouses and children. In a household with multiple members, a smaller fraction of a man’s earnings is directed toward his own personal consumption, effectively taxing his la1 The hours response to pre-marital children is also evidence against marital selection along other dimensions, such as unobserved heterogeneity in wage growth (Guvenen, 2009) or employment shocks (Kaplan, 2012). 2 bor income. But when a man internalizes the utility of other household members, the diverted expenditures also give him utility, increasing his returns to work. The model provides a framework to assess the importance of marriage for male labor market outcomes. We begin with a simple model counterfactual showing that eliminating the marriage process altogether reduces average hours worked by prime-age men by 7%. Next, we quantitatively assess a recent hypothesis by Binder and Bound (2019) that declining rates of marriage could help explain declines in male work rates in recent decades. We do so by re-estimating the marriage and family formation processes with data from the NLSY97, a longitudinal dataset similar to the NSLY79 that follows a cohort born roughly two decades later. Men in the NLSY97 cohort marry at lower rates than men in the NLSY79: for example, by age 25 only 27% of men in the NLSY97 have married, compared with 47% of men in the NLSY79. When we simulate our structural model using the NLSY97 marriage process, but leave all other parameters constant, we find that average hours worked by prime-age men fall 2.3%. For context, between 1979 and 2018, average annual hours worked among prime-age men in the CPS declined 8.4%; our results suggest that falling marriage rates could explain a sizable share of this decline. This project lies at the intersection of three literatures that are related but have nevertheless remained largely isolated from each other. The first is a series of reduced form analyses attempting to explain the “male marriage premium.” Most of these focus on the difference in hourly wages between married and never-married men (see, e.g., Korenman and Neumark (1991), Cornwell and Rupert (1997), Ginther and Zavodny (2001), Antonovics and Town (2004), Rodgers III and Stratton (2010), Budig and Lim (2016), Glauber (2018), Killewald and Lundberg (2017), and a meta-analysis by de Linde Leonard and Stanley (2015)). On average, these papers find that wages are about 10% higher for married men than for never-married men after controlling for observables. The leading causal explanation for the male marriage premium in wages is that marriage increases husbands’ productivity by allowing them to specialize in market work rather than home production (Becker, 1991). The leading non-causal explanation is that men with higher wages are more likely to marry. Our view is that this literature has not reached a firm conclusion about which explanation is more important.2 Our model includes both sorts of mechanisms: we account for specialization by allowing wages to increase with hours of work; and we account for selection by allowing the probability of marriage to depend on transitory wage shocks. Within our model, both mechanisms contribute to the increase in hourly wages around marriage, but specialization plays a somewhat larger role. 2 In a recent structural analysis, Pilossoph and Wee (2021) argue that the wage premium for married workers is due in part to different job search dynamics. 3 We are aware of two papers within this reduced form literature that emphasize differences in hours, rather than wages, between married and never-married men. Akerlof (1998) studies men in the NLSY79 and shows that after marriage they receive higher wages, work more and are less likely to abuse drugs and alcohol. Lundberg and Rose (2002) study men in the Panel Study of Income Dynamics (PSID). They show that after marriage and the birth of their children, men receive higher wages and work more. However, neither study analyzes the time path of these variables, and thus do not show that the increase in hours and wages begin prior to marriage and persists for at least a decade into marriage. Moreover, they do not attempt to quantify the channels that might generate this increase in hours, which is one of our main objectives. The second literature to which our paper contributes is a collection of structural analyses that explore the interactions of gender, marriage, children and the labor market. Becker (1985) and Becker (1991) are foundational theoretical contributions, while Greenwood, Guner and Knowles (2003) and Attanasio, Low and Sánchez-Marcos (2005) are early dynamic quantitative exercises. Some of these papers examine only the labor supply decisions of couples, and so cannot speak to differences between married and single individuals (e.g., Knowles (2013), Blundell, Pistaferri and Saporta-Eksten (2016, 2018), Alon, Coskun and Doepke (2018), Chiappori, Dias and Meghir (2018), Bick and Fuchs-Schündeln (2018) and Ellieroth (2019)). Other papers model male labor supply and earnings as exogenous (Greenwood et al., 2016; Low et al., 2017; Caucutt, Guner and Rauh, 2021). There is also existing work featuring both marital dynamics and endogenous male labor supply, such as Guner, Kaygusuz and Ventura (2012a,b), and Borella, De Nardi and Yang (2019). But to our knowledge, none of these papers have attempted to explain male labor market behavior around the time of marriage. Within this literature, our paper is most closely related to two existing papers. Siassi (2019) seeks to explain differences in income and wealth by marital status. An important difference between our paper and his is that Siassi (2019) measures a single marital gap for men and women together, while we emphasize that married men work more hours, even as married women work less. In addition, Siassi (2019) focuses on cross-sectional differences by marital status, while we also assess individual transitions near the time of marriage. The second closely related paper is Mazzocco, Ruiz and Yamaguchi (2014), who model the link between labor supply, home production, savings and marriage. Like us, they use panel data to document that hours increase in the years around marriage, though unlike us they do not document the qualitatively similar patterns in hourly wages and annual earnings. Relative to both these papers, our analysis differs in three key dimensions. First, because the two existing papers model marriage as an instantaneous shock, only our model replicates the 4 gradual pre-marital increase in hours, wages, and earnings in the data that lies at the heart of our motivation. Second, our paper disciplines the quantitative strength of the selection channel using plausibly exogenous variation in state economic conditions, which is central to our exercise of understanding why hours increase in the run-up to marriage. Third, we conduct a series of counterfactuals demonstrating that marriage is an important determinant of overall labor supply by prime-age men. Finally, we contribute to the nascent literature studying linkages between the secular decline in marriage and employment among prime-age men (Binder and Bound, 2019). In a pair of related event studies, Autor, Dorn and Hanson (2019) find that negative labor demand shocks reduce both marriage and fertility, while Kearney and Wilson (2018) conclude that fracking booms increase fertility but not marriage. Our goals are somewhat different from these latter two papers: we seek to understand how marriage and labor supply interact within a particular cohort of men (the NLSY79), taking their labor and marriage markets as given. We view our findings that marriage leads to a substantial increase in male labor supply as a complementary input into the larger project of understanding these fundamental, interconnected social transformations. The rest of the paper proceeds as follows. Section 2 documents that married men work more than single men and that their hours of work increase significantly before their first marriage. Section 3 develops a structural model that can explain these facts, and section 4 describes how the parameters of the model are set. Section 5 documents the properties of the baseline parametrized model. Section 6 uses the model to quantitatively analyze potential drivers of the marriage-related hours increase. We conclude in section 7. 2 Evidence on Marriage and Male Labor Market Outcomes We begin with empirical evidence on the relationship between marriage and male labor market outcomes. First, we use repeated cross-sectional data from the CPS to document a large and stable gap in hours worked between married and never-married men over the last four decades. Next, we use individual-level panel data from the NLSY79 to establish a direct relationship between marriage and changes in labor market outcomes. 2.1 Marriage and Work in Cross-Sectional Data We use cross-sections for the years 1975 to 2019 taken from the Annual Social and Economic Supplement (ASEC) of the CPS (Flood et al., 2020). The ASEC includes information on both 5 Figure 1: Hours Worked by Marital Status: 1975–2019 (a) Men: log(married / never married) .5 log Difference in Average Hours Worked -.5 -.4 -.3 -.2 -.1 0 .1 .2 .3 .4 log Difference in Average Hours Worked -.5 -.4 -.3 -.2 -.1 0 .1 .2 .3 .4 .5 (b) Women: log(married / never married) 1975 1985 No controls 1995 Year 2005 2015 1975 With controls 1985 1995 Year No controls 2005 2015 With controls Source: Men and women ages 19 - 54 in the 1975–2019 waves of the CPS ASEC. Data points are logs of ten-year centered averages, except for 1975, which averages across the years 1975-80. Annual hours worked are the product of usual weekly hours in the previous year and weeks worked in the previous year. The sample includes those with zero annual hours. The solid line plots the percent difference in average annual hours worked by currently married individuals versus individuals who have never been married. The dashed line plots the difference in the estimated coefficient for married versus never-married individuals in a regression of annual hours worked on marital status and controls for education, age, race and state of residence. weekly hours and weeks worked in the previous calendar year, which allows us to construct a measure of annual hours worked. We restrict attention to the core working ages 19 - 54, and we include people with zero annual hours worked. Figure 1 documents how annual hours of work differ by marital status. Our findings are consistent with a large number of earlier studies (see, e.g., Doepke and Tertilt 2016). Figure 1a shows results for men. The solid black line with circles shows the log ratio of average annual hours worked for currently married men relative to men who have never been married. Between 1975 and 2019, average annual hours worked by married men exceeded average hours of never married men by 31 to 39 log points. Most of the gap remains after controlling for the mens’ education, age, race and state of residence (the grey dashed line with triangles) . Figure 1b shows results for women. In 1975, married women worked nearly 30 log points less than never-married women. By the 2000’s, however, the gap in the raw data had completely disappeared. After controlling for women’s observables, the gap is always negative, but nonetheless, by 2019 it was only 12 log points. Since 1985, the magnitude of the marital hours gap among men has been larger than that of women, even after controlling for observables. A clear cross-sectional relationship between marital status and hours worked is also apparent 6 Figure 2: Cross-State Variation in Marriage and Hours Worked for Men (b) 2015-2019 2400 2400 (a) 1975-1979 ND SD IA NE WY WI KS MN NH CA OK TX MS IL OH MO IN VT MI AR LACO NV DE MD CO NJ OR TN MA RI AL PANC GA VA WA AZ SC NY MT ID UT KY NM ME FA Correlation = 0.481 Average Annual Hours Worked 1800 2000 2200 Average Annual Hours Worked 1800 2000 2200 Correlation = 0.524 HI WV ND NE WYIA SDNH CO MN WI KS TXIN MD VT MO VA MT IL GA CO NJ PA WA MA AK NC OK DE NV MEOH MI OR FA AZ RI NY CASCAL LAHI AR KY TN AK MS .75 .8 Share Married .85 1600 1600 NM .7 UT ID .9 .45 .5 WV .55 .6 Share Married .65 .7 Source: Males ages 19 - 54 in the CPS ASEC. Annual hours worked are the product of usual weekly hours in the previous year and weeks worked in the previous year. The sample includes those with zero annual hours. The dashed line is the line of best fit using OLS. Share married refers to the share of men in the sample who were currently married at the time of the survey. at the state level. Figure 2 plots the share of men who are currently married in each state against statewide average annual hours of work. Figure 2a plots data from 1975 to 1979, the first five years for which the CPS micro data is available. The correlation between hours worked and share married is 0.524, and the slope is significantly positive. Figure 2b shows the same scatter plot for the years 2015 to 2019, the most recent five years of data prior to the large disruption from the pandemic. Even though both marriage and work has decreased in virtually every state, a similar positive relationship remains, with a correlation of 0.481.3 Table 4 in Appendix A shows that this cross-state pattern continues to hold after controlling for age, education, and state fixed effects. 2.2 The Dynamics of Marriage and Work in Panel Data At the individual level, do men work more when they become married? Or, alternatively, do men who eventually marry always work more, even before they are married? To answer this question, we need to move beyond the cross-sectional comparisons in the preceding subsection and make 3 In 2015, two noteworthy outliers are Utah and Idaho, with marriage rates of 69% and 68%, respectively. A likely contributing factor is that these two states have by far the largest population share that is Mormon, a religion which emphasizes the importance of marriage. 7 use of panel data. For this, we turn to the NLSY79, a longitudinal study of 12,686 individuals born between 1957 and 1964 (Bureau of Labor Statistics, 2019a). Respondents were recruited and initially interviewed in 1979, when they were between 14 and 22 years old. They were then re-interviewed annually until 1994, then biennially afterward. The dataset contains a rich collection of information on family background, including detailed information on marriage and children, and labor market outcomes. Importantly, as of their initial interview 95% of male respondents in the NLSY79 had never been married, which allows us to observe changes in labor market outcomes around the date of marriage or the arrival of a child.4 We construct a “nearly-balanced” panel of men from the NLSY79 as follows. First, we drop the military over-sample portion of the survey. This leaves us with 5,579 individuals who were originally interviewed in 1979. Second, we restrict attention to men ages 19 and older. Third, we restrict attention to men who we observe at ages 50 or later, indicating that they remained in the survey for a substantial period of time. Fourth, among the remaining men, we restrict attention to those who were interviewed at least 20 times between 1979 and 2014 (out of a possible maximum of 26 interviews). These criteria balance our desire for a fairly complete life history against our need for a sufficiently large sample. This results in a final sample size of 2,731 men. Among this sample, we also exclude observations where men were currently enrolled in formal school; in particular, observations for men with a college degree do not enter into the analysis until ages 23 or older. To begin our panel analysis, we first regress annual hours worked on a dummy for current marital status, with and without controlling for individual fixed effects. The results are displayed in Table 1. Column (1) shows that the reference group of unmarried men work on average 1,751 hours per year. Married men of the same age and in the same calendar year, work 328 hours more, a difference of 19%. Column (2) shows that adding controls for education reduces the increment to 283 hours. Column (3) makes use of the panel aspect of the data to include individual fixed effects in the controls. This further reduces the coefficient on marital status, but it remains statistically significant and economically meaningful at 99 hours. Based on these results, we conclude that a sizable share of the difference in hours worked by marital status is due to individual changes in hours that coincide with changes in marital status. Next, we develop a fuller picture of how hours evolve around marriage by regressing an4 Another candidate dataset with a long panel dimension is the PSID. Unfortunately, the PSID consistently collects detailed information only for household “heads” and “spouses.” To the extent that younger individuals live with their parents, especially prior to marriage, this interviewing scheme limits our ability to study how labor market outcomes change in the years around marriage. 8 Table 1: Predictors of Male Annual Hours Worked in the NLSY79 (1) Constant 1751.0 (21.5) Married 328.1 (10.5) Separated / Widowed 71.7 (15.1) Less than High School – (2) ∗∗∗ Some College – Bachelor’s + – Black – Hispanic – Age Cubic Year FEs Individual FEs Y Y 1808.4 (21.9) 283.0 (10.5) 86.7 (15.1) −228.6 (14.9) 25.0 (11.3) 121.7 (11.1) −296.5 (13.3) −150.3 (17.4) Y Y 0.09 42,930 0.11 42,930 R2 -adj N ∗∗∗ ∗∗∗ (3) ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ 1866.0 (19.2) 99.2 (13.1) 7.7 (17.7) – ∗∗ – ∗∗∗ – ∗∗∗ – ∗∗∗ – ∗∗∗ ∗∗∗ Y Y Y 42,930 Source: Males ages 19 - 54 in the NLSY79; see text for details. The sample includes those with zero annual hours. nual hours on a sequence of dummies corresponding to the distance in years from the man’s first marriage. Specifically, we run the following regression: hi,t,d = βddistance + βtyear + βiindividual + εi,t . (1) The terms βiindividual and βtyear are individual and year effects. The term βddistance , 10 ≤ d ≤ 10, is a “distance-from-marriage” effect, with d = −10 indicating ten years prior to the man’s first marriage, and d = 10 indicating ten years after the man’s first marriage. When running the regression, we exclude the coefficient at the time of marriage, β0distance , so that the reference group is men in the year they were first married. The regression excludes observations that are more than ten years away from the man’s year of first marriage in either direction. To control for age effects that 9 Figure 3: Labor Market Dynamics in the Years around Marriage -10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Years from First Marriage .2 (c) Annual earnings Annual Earnings rel. to Year of Marriage -.3 -.25 -.2 -.15 -.1 -.05 0 .05 .1 .15 .2 (b) Hourly wages Hourly Wage rel. to Year of Marriage -.3 -.25 -.2 -.15 -.1 -.05 0 .05 .1 .15 Annual Hours Worked rel. to Year of Marriage -.3 -.25 -.2 -.15 -.1 -.05 0 .05 .1 .15 .2 (a) Annual hours worked -10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Years from First Marriage -10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Years from First Marriage Source: Males ages 19 - 54 in the NLSY79; see text for details. The solid line plots distance-from-marriage coefficients from the individual fixed effects regression equation (1). The shaded region corresponds to 95% confidence intervals. are independent from marriage, the hours measure hi,t,d equals annual hours worked divided by the average hours of married men of the same age. For example, a value of 1.1 at age 30 indicates that individual i’s age-30 hours are 10% larger than the sample average for 30-year-old married men. Figure 3a plots the estimated coefficients of βddistance . The figure shows that, relative to married men of the same age, annual hours increase 13% from ten years before marriage to the year of marriage, with a majority of this increase occurring in the six years leading up to marriage. Importantly, ten years after marriage, mens’ relative hours are essentially unchanged from the year they were married.5 6 Figure 3b shows the results from a parallel regression of hourly wages on distance from marriage. Qualitatively, we observe a similar “S-shape” to the coefficients for hours worked, with a sharp increase in the years around marriage, and then a leveling off several years after marriage. Nonetheless, the wage and hours coefficients differ in two notable ways. First, the magnitude of 5 Although we control for age by normalizing hours relative to the average among men of the same age, we have encountered concerns that our estimated distance-from-marriage coefficients may nevertheless reflect age effects. To address these concerns, Figure 15 in Appendix A.3 displays the results of a placebo test of the regressions in Figure 3, in which we randomly scramble the age at first marriage of men in the regression sample, and use this to compute a placebo distance-from-marriage measure. If the estimates in Figure 3 reflected the effect of age rather than distance from marriage, we would expect the placebo regression to yield similar estimates for the distance-frommarriage coefficients, since the labor market outcomes and age of men are identical in the two regressions. However, the placebo estimates are virtually all insignificant. This provides confidence that our original estimates are in fact reflecting the effect of distance from marriage. 6 Figure 16 decomposes the change in annual hours worked around marriage into changes in hours per workweek and changes in annual weeks worked. Each margin generates roughly 50% of the increase in annual hours. 10 Annual Hours Worked rel. to Year of Marriage (%) -.3 -.25 -.2 -.15 -.1 -.05 0 .05 .1 .15 .2 Figure 4: Hours Worked, Marriage and Children -10 -8 -6 -4 -2 0 2 4 6 Years from First Marriage 1st Child Pre-marital 8 10 1st Child Post-marital Source: Males ages 19 - 54 in the NLSY79; see text for details. The solid line refers to men whose first child arrives before his first marriage (“pre-marital”). The dashed line refers to men whose first child arrives after his first marriage (“post-marital”). The lines plot distance-from-marriage coefficients from the individual fixed effects regression equation (1). The shaded regions correspond to 95% confidence intervals. the increase is smaller for wages than for hours. From ten years before marriage until ten years after marriage, relative hourly wages only increase 10%, compared to 13% for hours; alternatively, the increase from the lowest coefficient before marriage to the highest coefficient after marriage is 12% for hourly wages, compared to 15% for hours. Second, much of the increase in hourly wages occurs after the increase in hours. In particular, roughly half of the increase in wages occurs in the year of marriage or later, while the increase in hours occurs almost entirely prior to marriage. Figure 3c shows the results for annual earnings. The picture is very roughly the sum of the coefficients for annual hours and hourly wages in Figures 3a and 3b: earnings are essentially flat from ten years before marriage to six years before marriage, then increase 18% from six years before marriage to two years after marriage, and are essentially flat afterward. We emphasize that, in a pure accounting sense, the majority of the increase in annual earnings is attributable to an increase in hours worked, rather than wages. From this perspective, understanding the “hours premium” appears to be at least as important as understanding the “wage premium” emphasized in the existing literature (see Section 1). Marital status is highly correlated with the presence of children. A natural question is to 11 ask whether the changes in labor market outcomes that we have documented are more closely related to the onset of marriage or to the arrival of children. To investigate this, we run separate regressions for men whose first child appears before their first marriage, and for men whose first child appears after their first marriage. Figure 4 displays the results, with the solid blue line corresponding to men with pre-marital children and the dashed green line corresponding to men with post-marital children. The results indicate that both groups of men experience significant increases in hours around the time of their first marriage. An important difference is that the increase in hours is more abrupt for men with pre-marital children. For example, in the five years before marriage, relative hours for men with pre-marital children increase by 16%, compared with 6% for men with post-marital children. Because the NLSY79 data also show that pre-marital pregnancies are more likely to be unplanned,7 these results suggest that marriage and children have causal effects that encourage work. 2.3 Summary The data show a strong positive relationship between marriage and market work for prime-age men. In the cross-section, married men work substantially more than never-married men. This cross-sectional relationship has been fairly stable in the US since at least the mid 1970’s, and remains after controlling for a host of observables. Panel data reveal that much of the crosssectional difference in work by marital status is driven by increases in hours around the time when men first marry, especially during the five years prior to marriage. One possible explanation for these patterns is that marriage leads men both to work more once they marry and to work more in anticipation of marriage. An alternative explanation is transitory selection, where events that increase hours of work create or coincide with an increased likelihood of marriage. Because the two explanations have very different implications for how marriage affects male labor market outcomes, we would like to measure the importance of each. The results presented here provide some clues. For example, the increase in average hours is larger than, and begins before, the increase in hourly wages that occurs around marriage. This suggests that shocks to wages are not the sole reason hours increase in the lead-up to marriage. We will revisit these findings in our quantitative analyses below. 7 The NLSY79 asked whether pregnancies were unplanned in 1982 and every other year thereafter (even when the survey was annual). The responses to this question reveal that pre-marital pregrnancies are three times as likely to be unplanned, and half as likely to be planned. In particular, among married men with one child or no children but one on the way, 72% reported that their first child was planned, 16% reported the child was unplanned, and 12% reported the child was neither planned nor unplanned. Among never-married men, 36% reported that their first child was planned, 45% reported the child was unplanned, and 19% reported the child was neither planned nor unplanned. 12 3 Model 3.1 The Life Cycle We study a life-cycle model of male labor supply and saving. The man’s age, j, is discrete. At birth, men are endowed with education level e ∈ {nc, c} (non-college, or college). Non-college men enter the model at age Jnc = 19. College men are absent from the model during ages 19-22, and enter at age Jc = 23. Regardless of education, men retire exogenously at age JR , and die at age J. 3.2 The Man’s Wage Process In each period before retirement, j < JR , men supply labor hours h j . Male earnings are given by 1+ζ me j = w j h j 1+ζ where w j denotes a base hourly wage, and h j , (2) introduces a “part-time penalty / overtime bonus” if ζ > 0. The base wage w j follows an AR-1 process with an age- and education-specific meanshifter: log w j = αe,w j + w̃ j , (3) w̃ j = ρew w̃ j−1 + ε wj , (4) ε wj ∼ N(0, σeε ), i.i.d., (5) w̃0 ∼ N(0, σew0 ). (6) In the notation above, and throughout the rest of this paper, superscripts are used to differentiate parameters, while subscripts indicate dependencies. For example, αe,w j is the mean-shifter for wages, w, for a man of age j and education e. Equations (4)–(5) give rise to the cumulative distribution function F w (w̃′ | w̃), which describes the distribution of next period’s idiosyncratic wage shock, w̃′ , given the current shock w̃. 3.3 Family Structure and Family Dynamics The structure of the man’s family is described by the triple f = (r, a, n). The first component, r, denotes relationship status. Men can be single (r = sn), engaged (r = en), married (r = mr), or divorced (r = dv). To allow for selection, we allow most of the relationship transition probabilities 13 to depend on the wage shock w̃. The second component denotes the age of any children, a ∈ {0, yc, oc, gc}, corresponding to no children, young children (ages 0-5), older children (ages 618), and grown children, respectively. We distinguish between young and older children because younger children are more expensive, imposing higher formal child care costs and discouraging spousal employment, as detailed in Sections 3.4.1 and 3.4.2. To simplify the model, we assume that all children in a household belong to the same age group. The third component denotes the number of children, n = 0, 1, ..., n. As in Cubeddu and Rı́os-Rull (2003), we treat family structure as an exogenous stochastic process. 3.3.1 Fertility Dynamics A childless man (n = 0) of age j, education e, and relationship status r will have one (young) n child next period with probability ϕ0,r,e, j . As long as his children are young, additional offspring are possible. A man with 0 < n < n young children will have n + 1 children next period with n n probability ϕn,r,e, j and n children with probability 1 − ϕn,r,e, j . We assume that once young children age, a man does not have additional children. We further assume that divorced men have no n additional children: ϕn,dv,e, j = 0. Children age stochastically and all at the same time.8 Young children, a = yc, evolve to older a children, a = oc, with probability ϕyc,e, j , and older children evolve to grown children, a = gc, a with probability ϕoc,e, j . We assume that in families with young children, the aging shock occurs after the fertility shock; this implies a newborn can age immediately into an older child. Having a grown child is an absorbing state. an ′ ′ Let φa,n,r,e, j (a , n ) denote the probability that a man of age j with education e, relationship status r, and current child status (a, n) will have child status (a′ , n′ ) next period. Collectively, our assumptions imply that: an n φ0,0,r,e, j (yc, 1) = ϕ0,r,e, j , (7a) an n φ0,0,r,e, j (0, 0) = 1 − ϕ0,r,e, j , (7b) n an a φyc,n,r,e, j (oc, n + 1) = ϕyc,e, j · ϕn,r,e, j , (7c) an a n φyc,n,r,e, j (yc, n + 1) = (1 − ϕyc,e, j ) · ϕn,r,e, j , (7d) a n an φyc,n,r,e, j (oc, n) = ϕyc,e, j · (1 − ϕn,r,e, j ), (7e) an a n φyc,n,r,e, j (yc, n) = (1 − ϕyc,e, j ) · (1 − ϕn,r,e, j ), (7f) 8 Stochastic aging indirectly captures the uncertainty inherent in the costs of children, since there is variation expost in the time it takes for children to mature. The assumption that children age at the same time simplifies the computation of the model by reducing the number of states in the child age space. 14 an a φoc,n,r,e, j (gc, n) = ϕoc,e, j , (7g) an a φoc,n,r,e, j (oc, n) = 1 − ϕoc,e, j , (7h) an φgc,n,r,e, j (gc, n) = 1. (7i) n a a We assume that ϕn,r,e, j = 0, ∀ j ≥ JR − 2, ϕyc,e, JR −2 = 1, and ϕoc,e, JR −1 = 1, which ensures that all children are grown by retirement. 3.3.2 Relationship Status Although some men are married when they enter the model, most marry later. Childless men advance up the “relationship ladder” as follows: single men become engaged with probability φe,enj (w̃), and engaged men marry with probability φemr (w̃). The dependence of the transition probabilities on wages (w̃) allows for (positive) wage selection into marriage.9 In addition, a nevermarried (single or engaged) man can have an out-of-wedlock birth, which makes marriage more likely; this “shotgun marriage” effect is motivated by the higher marriage rates observed in the first few years following an out-of-wedlock birth. We assume that a man with an out-of-wedlock birth faces “double jeopardy.” Because of the birth, his relationship status advances one stage with probability φeowb ; should this “shotgun advancement” not occur, he still faces the “regular” probability of advancing faced by childless men. For simplicity, we assume that only the first dv . Divorce out-of-wedlock birth has this effect. Once married, men divorce with probability φn,e, j is an absorbing state, i.e., we rule out re-marriage. Finally, we assume that relationships are fixed once an individual reaches retirement. This structure gives rise to the following transition probabilities for single men: 9 Being Pr j (r j = en | r j−1 = sn, n j = 0, w̃, e) = φe,enj (w̃), (8a) Pr j (r j = sn | r j−1 = sn, n j = 0, w̃, e) = 1 − φe,enj (w̃), (8b) Pr j (r j = en | r j−1 = sn, n j = 1, n j−1 = 0, w̃, e) = φeowb + 1 − φeowb φe,enj (w̃), Pr j (r j = sn | r j−1 = sn, n j = 1, n j−1 = 0, w̃, e) = 1 − φeowb 1 − φe,enj (w̃) , (8c) (8d) Pr j (r j = en | r j−1 = sn, n j > 0, n j−1 > 0, w̃, e) = φe,enj (w̃), (8e) Pr j (r j = sn | r j−1 = sn, n j > 0, n j−1 > 0, w̃, e) = 1 − φe,enj (w̃), (8f) defined as a zero-mean deviation, w̃ has little effect on the average probability of marriage. 15 and the following probabilities for married and divorced men: dv Pr j (r j = dv | r j−1 = mr, n, e) = φn,e, j, (9a) dv Pr j (r j = mr | r j−1 = mr, n, e) = 1 − φn,e, j, (9b) Pr j (r j = dv | r j−1 = dv, n, e) = 1. (9c) The transition probabilities of men who are currently engaged take the same form as those of single men, with the term φe,enj (w̃) replaced by φemr (w̃). The probability of getting married conditional on being engaged, φemr (w̃), does not vary with age: the age pattern of marriage depends solely on life-cycle changes in the rate of engagements and out-of-wedlock births. This is a concession to our data, which does not report engagement — we observe only whether an individual is married or has children. We assume further that the effect of a pre-marital birth, φeowb , is the same for single and engaged men. 3.3.3 Cohabitation Empirically, many couples begin pooling resources prior to marriage. For example, in the NLSY79, 49% of men reported cohabiting with their partner before their first marriage, with an average length of two years among those cohabiting. To capture this pattern, we allow engagement to take one of two forms, non-cohabitation, en-n, and cohabitation, en-c. For non-cohabiting couples, en-n, the man’s budget constraint and preferences are identical to those of a single man. For cohabitating couples, en-c, the man’s budget constraint and preferences are identical to those of a married man, except that the man and his partner file taxes separately rather than jointly. Throughout the paper, we will use the term “spouse” to denote partners of both cohabiting and married men. When a couple first becomes engaged, it is assigned permanently to one of the two possible engagement types, cohabiting with probability φ en-c . The probability that an engaged couple has additional children, has their children age, or transitions to marriage is independent of cohabitation. 3.4 Family Structure and Financial Resources Relationships and children affect a man’s financial resources in four ways. First, in larger households consumption must be spread across more individuals. We capture this effect through the use of equivalence scales that convert total consumption to per capita amounts. Second, spouses can 16 generate earnings or, if they stay home with children, substitute for costly formal child care. Third, non-grown children are costly. Men with working spouses pay for formal child care, and men who are neither married or cohabiting pay child support. Finally, couples who divorce split their wealth in half. The possibility of such a split tends to reduce the husband’s expected consumption, since, in the quantitative model, he usually has the higher earnings. 3.4.1 Spousal Earnings At the time that a man becomes engaged, Jen , his spouse draws the permanent earnings shock s̃ ∼ N(ρes w̃Jen , σes ). (10) This shock is potentially correlated with the man’s idiosyncratic wage shock at the time of engagement, w̃Jen ; we will estimate ρ s from the data. Equation (10) gives rise to the cumulative distribution function F s (s̃ | w̃). Men who are cohabiting or married combine their earnings with any earnings that their spouses receive. Given s̃, spousal earnings at age j ≥ Jen , se j , follow a two-stage process. The first stage uses a logit model to determine whether the spouse works: 1se j >0 = 1 if q j < κ 1+κ , (11) 0 otherwise s,1 κ = exp s̃ + αa,e, j , (12) q j ∼ U[0, 1], i.i.d., (13) where 1A is the indicator function for event A. Equation (13) gives rise to the distribution function F q (q). The second stage determines the earnings of spouses who work: se j = 1se j >0 · exp log(s) + max{0, βes s̃ + αe,s,2j } . (14) s,1 s,2 In the above equations, βes is a wage-scaling parameter, and αa,e, j and αe, j are mean-shifters that depend on the man’s age and education. The mean-shifter for spousal participation also depends on the age of the children. Spouses with young children are least likely to work, spouses with no (or grown children) are mostly likely, and spouses with older childer fall in between. The parameter s, which places a lower bound on the earnings of working spouses, equals the earnings cutoff we use to define working spouses in the data. In our model of spousal earnings, spouses with higher potential earnings are more likely to 17 work. While a standard Tobit model would generate a similar relationship, we found that to fit the spousal earnings data well, we needed a more flexible specification. Even though the shock s̃ is s,1 permanent, spouses will move in and out of employment as αa,e, j and q j vary over the life cycle. 3.4.2 Child Costs Married and cohabitating couples must provide care to their non-adult children. If the spouse does not work, se j = 0, she provides the child care herself at no additional cost to the family. If the spouse works, then the couple must purchase child care in the market, at a total cost of n j χa j se j . We assume that child care costs are proportional to the number of children and to the spouse’s earnings. The cost factor χa depends on the childrens’ age; older children (a = oc) are less expensive than younger children, and grown children (a = gc) impose no costs at all. We assume that men who are neither married nor cohabiting do not live with their children and instead pay child support.10 Child support equals the fraction nδa of the father’s labor income. The parameter δa equals δ for young and older children, a ∈ {yc, oc}, and zero for grown children. 3.4.3 Divorce Costs At the time of divorce, men lose half their assets. Divorced men also pay a fraction of their earnings, nδa , in child support. 3.5 Preferences Men have time-separable preferences over consumption and hours worked each period that vary with family structure, education and age: (c/η f )1−γ h1+1/ξ u f ,e, j (c, h) = N f − ψe, j , 1−γ 1 + 1/ξ (15) with γ, ξ > 0. The parameter η f is a household equivalence scale converting total household consumption, c, into the per capita amount consumed by the man. The shift term N f captures the possibility that married and cohabiting men derive additional utility from the consumption of other household members. The labor disutility shifter ψe, j varies with age and education. Future utility is discounted at the rate β ∈ (0, 1). 10 We assume that when a single father marries, his children join his new household. 18 3.6 Total Income and Taxes A household’s total income, y, equals the sum of male earnings me, spousal earnings se (for married and cohabiting men) and capital income: y j = me j + se j + (R − 1)k j , (16) where k denotes the household’s assets, and R is the constant gross rate of return. It faces payroll and income taxes: T (me + se, k; f ) = τ ss (me + se) + T finc (y), 2 T finc (y) = τ 0f + τ 1f yτ f y, (17) (18) where τ ss is the payroll tax rate, and T inc (·) is the income tax function. We allow the parameters of T inc (·) (τ 0f , τ 1f , and τ 2f ) to differ by marital status and the number of dependent children. 3.7 Recursive Formulation The state vector for an age- j man consists of the man’s education level (e), assets (k j ), wage deviation (w̃ j ), relationship status (r), age and number of children (a j , n j ), and spousal earnings shocks (s̃ and q j ). We denote the non-existent spousal earnings shocks of single and divorced men with the placeholders s̃sn and qsn . We will continue to use f = (r, a, n) as a compact index of family structure. 3.7.1 Single The Bellman equations for a working-age ( j < JR ) single man, r = sn, is11 ( V j (e, k, w̃,sn, a, n, s̃sn , qsn ) = max ′ c,k ,h Z u f ,e, j (c, h) + β w̃′ ∑ ′ ′ an ′ ′ φa,n,sn,e, j (a , n ) (19) (a ,n ) × Pr j sn | sn, n′ , n, w̃′ , e V j+1 e, k′ , w̃′ , sn, a′ , n′ , s̃sn , qsn + Pr j en | sn, n′ , n, w̃′ , e ) Z Z × V j+1 e, k′ , w̃′ , en, a′ , n′ , s̃, q′ dF q (q′ ) dF s (s̃ | w̃′ ) dF w (w̃′ | w̃) , s̃ q′ s.t. c + k′ ≤ Rk + wh1+ζ (1 − nδa ) − T (wh1+ζ , k; f ), 11 Although (20) the exposition can be simplified by using the conditional expectation operator, explicitly writing out the integrals illustrates the model’s stochastic structure more clearly. 19 log w = αe,w j + w̃, (21) and the borrowing constraint k′ ≥ kmin . (22) The man’s expected continuation value depends on the evolution of his wage and family structure. A single man can father children or have his children age. He can stay single or become engaged: with probability Pr j r′ | sn, n′ , n, w̃′ , e , his relationship status evolves from sn to r′ ∈ {sn, en}. Engagement can occur with cohabitation or without: the continuation value conditional on becoming engaged is itself an expectation over the value of engagement with and without cohabitation: V j+1 e, k′ , w̃′ , en, a′ , n′ , s̃, q′ = φ en-cV j+1 e, k′ , w̃′ , en-c, a′ , n′ , s̃, q′ + (1 − φ en-c )V j+1 3.7.2 (23) e, k′ , w̃′ , en-n, a′ , n′ , s̃, q′ . Engaged The Bellman equation for a working-age man who is engaged and not cohabiting, r = en-n, is ( V j (e,k, w̃, en-n, a, n, s̃, q) = max ′ c,k ,h Z Z u f ,e, j (c, h) + β w̃′ q′ ∑ ′ ′ an ′ ′ φa,n,en,e, j (a , n ) (24) (a ,n ) × Pr en-n | en-n, n′ , n, w̃′ , e V j+1 e, k′ , w̃′ , en-n, a′ , n′ , s̃, q′ ) ! + Pr mr | en-n, n′ , n, w̃′ , e V j+1 e, k′ , w̃′ , mr, a′ , n′ , s̃, q′ dF q (q′ ) dF w (w̃′ | w̃) , s.t. equations (20)-(22). The expected continuation value for an engaged man includes the possibility of marriage. The Bellman equation for a working-age man who is engaged and cohabiting, r = en-c, is identical to the above Bellman equation except that: (i) the budget constraint includes spousal earnings and the taxes paid on these earnings; (ii) with children now residing with their father, child support costs proportional to the man’s earnings are replaced with child care costs proportional to the earnings of the spouse. This yields 20 ( Z Z u f ,e, j (c, h) + β V j (e,k, w̃, en-c, a, n, s̃, q) = max ′ c,k ,h w̃′ ∑ ′ ′ q′ an ′ ′ φa,n,en,e, j (a , n ) (25) (a ,n ) × Pr en-c | en-c, n′ , n, w̃′ , e V j+1 e, k′ , w̃′ , en-c, a′ , n′ , s̃, q′ ) ! dF q (q′ ) dF w (w̃′ | w̃) , + Pr mr | en-c, n′ , n, w̃′ , e V j+1 e, k′ , w̃′ , mr, a′ , n′ , s̃, q′ s.t. c + k′ ≤ Rk + wh1+ζ + se(1 − nχa ) − T (wh1+ζ , k; f en-c ) − T (se, 0; f en-c,sp ), se satisfies equations (11)-(14), (26) (27) equations (21)-(22). Cohabiting couples pool their labor income together, but each partner files taxes as an individual. We capture this by replacing f with f en-c and f en-c,sp in the tax functions.12 Consistent with our assumption that spouses bring no wealth into the relationship, we assign all of the cohabiting couple’s asset income to the man. 3.7.3 Married The Bellman equation for a working-age married man, r = mr, is ( V j (e, k,w̃, mr, a, n, s̃, q) = max ′ c,k ,h Z u f ,e, j (c, h) + β w̃′ ∑ ′ ′ an ′ ′ φa,n,mr,e, j (a , n ) (28) (a ,n ) 1 ′ ′ ′ ′ dv × φn,e, j V j+1 e, k , w̃ , dv, a , n , s̃sn , qsn 2 dv + (1 − φn,e, j) Z q′ V j+1 ) e, k′ , w̃′ , mr, a′ , n′ , s̃, q dF q (q′ ) dF w (w̃′ | w̃) , ′ s.t. c + k′ ≤ Rk + wh1+ζ + se(1 − nχa ) − T (wh1+ζ + se, k; f ), se satisfies equations (11)-(14), (29) (30) equations (21)-(22). d With probability φn,e, j the man divorces, keeping half of the household assets. The spousal earn- ings shocks for a divorced man, s̃sn and qsn , are placeholder values. With probability (1 − φ dn,e, j ) 12 We assume that the man, who is most likely to have the higher income, will be the partner who claims the children as tax dependents. This means that f en-c = (sn, n, a) and f en-c,sp = (sn, 0, 0). 21 the man remains married. Married couples file taxes jointly. 3.7.4 Divorced The Bellman equation for a working-age divorced man, r = dv, is ( V j (e, k, w̃,dv, a, n, s̃sn , qsn , ) = max u f ,e, j (c, h) + ′ (31) c,k ,h Z β w̃′ a ′ ∑′ ϕa,e, j (a )V j+1 e, k′ , w̃′ , dv, a′ , n, s̃sn , qsn ) dF w (w̃′ | w̃) , (a ) s.t. equations (20)-(22). Divorced men face the same budget constraints as singles. Their Bellman equation is much simpler, however, due to our assumptions that: (i) divorced men never remarry; and (ii) divorced men have no additional children. 3.7.5 Retired Retired men ( j ≥ JR ) do not work, and if they are married or cohabiting, their spouses do not work. Their only income comes from their assets and from Social Security benefits received by the man (b1,e ) and his spouse (b2,e ). All children are grown in retirement, implying that there are no child care costs or child support payments due. Finally, relationships do not change in retirement, eliminating any uncertainty due to them. The resulting Bellman equation is completely deterministic: V j (e, k, r) = max ′ c,k u f ,e, j (c, 0) + βV j+1 (e, k′ , r), s.t. c + k′ ≤ Rk + b1,e + b2,e 1r∈{mr,en-c} − T (b1,e + b2,e 1r∈{mr,en-c} , k; f ), (32) (33) equation (22), VJ+1 ≡ 0. 4 (34) Model Parameters We set the parameters of our model in three steps. First, we set a number of parameters to values consistent with the broader literature. In the second step, we estimate the stochastic processes 22 for fertility, relationships, wages and spousal earnings, which can be identified outside our model, from the data. While the principal dataset used in this step is the NLSY79, we also utilize state variation in the CPS. The final step of our estimation process is to use the model to estimate the discount factor and the age-varying component of the disutility from work. We set the discount factor so that the mean asset holdings at age 50 generated by the model match those in the NLSY79. We set the work disutility parameters to match life-cycle labor supply profiles. 4.1 Parameters Taken from Other Studies Table 2 displays the values for parameters taken from other studies. The first panel of the table shows that non-college and college men enter the model one year after their modal graduation ages, 19 and 23, respectively. They retire at age JR = 65 and die at age J = 80. We set the utility curvature parameter, γ, to 0.738, following Imai and Keane (2004). Estimates of this parameter vary widely (see the discussion in De Nardi, French and Jones (2010)). With separable utility, however, a value of γ greater than 1 would in a static model imply that the income effects of a wage change dominate the substitution effects. Given that many of the younger individuals in our model live nearly hand-to-mouth, consistent with large income effects, using γ > 1 would imply that young men would sometimes respond to wage increases by working fewer hours. This would rule out by construction the hypothesis that the higher hours of married men are due to their higher wages. Because we want to explore this hypothesis as an alternative to the mouths-to-feed mechanism, setting γ to a value less than 1 is appropriate. Sensitivity analyses in Section 5.3 show that the effects of marriage on hours are robust to this parameter. Our choice of the Frisch elasticity, ξ = 0.75, lies in the middle of a wide range (Keane and Rogerson, 2012). In a recent paper, Bick, Blandin and Rogerson (2022), applying the approach for two-earner households developed by Bredemeier, Gravert and Juessen (2019), find elasticities ranging from 0.51 to 1.07. We set the gross interest rate R to 1.02, a standard value. Our formulation of the equivalence scale η f comes from Citro and Michael (1995): 1, ηf = (2 + 1 a<gc if r ∈ / {mr, en-c}, · 0.7n)0.7 , (35) if r ∈ {mr, en-c}. Grown children, who are assumed to live outside the household, do not enter the formula. We set N f , which scales the utility from per capita consumption, to 2 for married and cohabiting men and 1 for the rest; we are effectively assuming that married and cohabiting men receive utility from their spouses’ consumption. The literature provides little guidance for setting N f . As 23 Table 2: Parameters Taken from Other Studies Description Demographics Starting age, non-college Starting age, college Retirement age Terminal age Preferences Coefficient of RRA Frisch elasticity Equivalence scale Consumption utility shifter, married or cohabiting Consumption utility shifter not married or cohabiting Budget Interest rate Child care cost per young child Child care cost per older child Child support cost per non-grown child, non-resident fathers Part-time wage penalty Borrowing limit Government Income Tax SS tax SS benefit, man, non-college SS benefit, man, college SS benefit, spouse, non-college SS benefit, spouse, college Parameter Value Jnc Jc JR J 19 23 65 80 γ ξ ηf N f :r∈{mr,en-c} 0.738 Imai and Keane (2004) (see text) 0.75 Various (see text) eqn. (35) Citro and Michael (1995) 2 See text N f :r∈{mr,en-c} / 1 R χy χo δa 102% 28% 7% 1.7% ζ 0.400 $0 kmin τ 0f , τ 1f , τ 2f τ ss b1,nc b1,c b2,nc b2,c Target/Source 5.2% $20,570 $29,520 $15,620 $21,810 See text Borella, De Nardi and Yang (2019) Borella, De Nardi and Yang (2019) See text Aaronson and French (2004) Guner, Kaygusuz and Ventura (2014) 2013 value See text See text See text See text Note: Child care and child support costs are expressed as fractions of earnings and are per child. Quantities are expressed in 2013 dollars. we show in Section 6.2 below, the model is able to match the run-up in hours around the date of marriage only if N f increases upon marriage.13 13 A plausible alternative specification would be to set N f = η f , i.e., equal to the equivalence scale, which would 24 Spouses who work surrender a fraction of their earnings to pay for formal child care. Using Borella, De Nardi and Yang’s (2019) estimates, we set χa to 28% and 7% of her earnings, per child, for young and old children, respectively. Men who are neither married nor cohabiting pay child support costs equal to δa = 1.7% of their earnings for each non-grown child. We take this number from the NLSY79.14 We set ζ , the parameter governing the part-time wage penalty, to 0.4, following Aaronson and French (2004). At this value, a person working half-time suffers a 25% decrease in wages. We set kmin to $0, ruling out unsecured borrowing. We set the income tax parameters to the values reported by Guner, Kaygusuz and Ventura (2014, tables 10 & 11), who estimate tax rates as a function of income, marital status and number of children, using administrative data. The Social Security tax rate τ ss equals 5.2%, the value in effect in 2013. Our estimates of Social Security benefits, b1e and b2e , are based on microsimulation estimates from Purcell, Iams and Shoffner (2015, Table 3), adjusted for real wage growth.15 4.2 Parameters Estimated Outside the Behavioral Model We estimate three sets of parameters outside the behavioral model: (i) the parameters governing the stochastic process for male wages; (ii) the parameters determining the probability that a spouse works and her earnings when working; (iii) the parameters determining the stochastic process for family structure. 4.2.1 The Male Wage Process Using equation (2), we compute the hourly wage term w j for an individual with annual earnings me j and annual hours worked h j as 1+ζ w j = me j /h j . (36) We estimate the wage process for men from equations (3)-(5) in two stages, following French (2005). First, we run an individual fixed effects regression of log wages on a quadratic in age and allow the number of children to affect the utility weight on consumption. Imposing this alternative would lead marriage and children to have an even larger causal effect on labor supply. 14 The NLSY79 data show that 28.73% of men with non-resident children pay child support, which as a fraction of earnings has a median value of 9.4%. These data also reveal that men with non-resident children have an average of 1.6 such children. Dividing the product of the first two numbers by the third gives us (0.2873 · 0.094)/1.6 = 0.017. 15 We use Purcell, Iams and Shoffner’s (2015) alternative specification, which assumes that the real wages of noncollege and college graduates grow at annual rates of 0.7% and 1.6%, respectively. Because these estimates are for people born between 1965 and 1979, on average 11.5 years younger than those in the NLSY, we deflate them by 11.5 years of real wage growth. (We also convert the numbers into 2013 dollars using the CPI.) 25 Table 3: Parameters for Male Wages and Spousal Earnings Description Male Wages Mean-shifter, age quadratic Autocorrelation Standard deviation, innovation Standard deviation, initial value Spousal Earnings Dependence on husband’s wages Standard deviation, innovation Effect of children on employment Effect of man’s age on spousal employment, quadratic Effect of man’s age on spousal earnings, quadratic Effect of spousal shock on earnings Parameter Value, Non-College Value, College Graduates α wj ρw σε σ w0 (−0.040, 0.066, −0.069) 0.932 0.150 0.130 (−0.624, 0.112, −0.0114) 0.937 0.142 0.171 ρs σs αa,s,1j : yc, oc αa,s,1j 0.017 0.032 (−0.853, −0.271) (0.684, 0.069, −0.185) 0.070 0.278 (−1.110, −0.332) (0.999, 0.0999, −0.322) α s,2 j (1.318, 0.057, −0.100) (1.720, 0.057, −0.101) βs 19.99 2.41 Note: Superscripts are used to distinguish parameters, while subscripts are used to distinguish dependencies. We have omitted education (e) subscripts, as every parameter varies by education level. All parameters with the age subscript j utilize a quadratic in age. See sections 4.2.1-4.2.2 and Appendix B for details. a control for the national unemployment rate during January of that calendar year. The estimated coefficients for the quadratic in age, along with the average fixed effect, provide us with values for αe,w j . In the second stage, we calculate a residual wage for each individual, which is the difference between the log of his actual wage and the predicted wage αe,w j (plus the estimated unemployment effect). We then generate the covariance matrix for the first four lags of this residual, which we use to estimate the autocorrelation term ρew , the innovation standard deviation σeε , and the initial standard deviation σew0 . The details of this procedure are described in Appendix B, and the estimates are shown in the first panel of Table 3. 4.2.2 Spousal Earnings The spousal earnings process in equations (10)-(14) requires estimates of: the dependence of the spouse’s permanent earnings shock, s̃, on the husband’s wage (ρes ), along with the standard deviation of this shock’s innovation (σes ); the parameters of the mean-shifter for the probability s,1 that a spouse works (αa,e, j ); the parameters of the mean-shifter for the earnings of working spouses s (α s,2 j ); and the relative importance of the permanent shock for spousal earnings (βe ). We assume 26 that both mean-shifters contain a quadratic polynomial in the husband’s age, and that the meanshifter for spousal employment contains coefficients for the presence of young or old children (the base case is no children or grown children). We also set the floor for earnings to s = $3, 630 and censor all spousal earnings in the data below this level.16 The bottom panel of Table 3 presents our parameter estimates. We estimate these parameters using the simulated method of moments, targeting age profiles for the share of spouses who work, the mean of log earnings among working spouses, the correlation of spousal earnings and male wages, and the standard deviation of log earnings among working spouses. We construct separate age-employment profiles for spouses with young and older children (determined by the age of the oldest child). Appendix C shows the model’s fit of spousal employment and earnings. Consistent with the data, the model predicts that the employment rate for women with young children is about 20 percentage points less than the rate for women with no children. The employment rate for women with older children lies between these cases, but is closer to the childless rate. In contrast, among spouses who work, earnings are close to invariant, at least on average, over the number of children. Appendix C shows that spousal earnings are quite volatile, leading to large estimated values of the scaling parameter β s . In contrast, our estimated values of ρ s , which links the spousal earnings shock s̃ to male wages, are relatively small, so that much of the variation in spousal earnings is specific to the spouse. As a result, in our model the correlation between male wages and spousal earnings (among workers, conditional on age, education and number of children) never exceeds 0.3 and is often much lower. This is consistent with the observed correlations that we target. Our spousal earnings process thus generates positive, though quantitatively modest, assortative matching, at least along the intensive margin. 4.2.3 Family Structure Dynamics The probabilities governing the dynamics of relationships and children in equations (7)-(9) are modeled as a set of logistic probabilities. We estimate these probabilities to match family demographics over the life cycle in the NSLY79, along with a moment capturing the effect of wages on the probability of marriage in the CPS. We include the latter moment because in the model, the probability of “moving ahead” in a relationship depends in part on wages: the transition probabilities φeen (w̃ j ) and φemr (w̃ j ) vary with the idiosyncratic wage shock w̃ j . Estimating these effects from the data requires variation in wages that is exogenous to other determinants of marriage. 16 This is the annual earnings from working ten hours per week for 50 weeks at $7.26/hour, which is the median federal minimum wage over this time period. 27 Although such variation is hard to find in the NLSY79, in the CPS we can instrument for an individual’s wage with the average wage in his state of residence. Appendix D.2 presents detailed results. The estimated coefficient in the CPS regression (Appendix Table 7) is 0.0144, which implies that a 10% increase in wages increases the probability of getting married by 0.144 percentage points. For context, the baseline probability of marriage in the estimation is 2.5%, implying that a 10% increase in wages increases the probability of getting married by 5.6%. We assume that in our logistic transition probabilities, the coefficient on the wage shock w̃, which we denote by θ , is the same at all stages of a relationship, and we set θ so that the wage coefficient on a simulated version of the CPS regression matches the observed coefficient. Appendix D describes our specification and estimation procedure in more detail, and Appendix Table 6 provides the parameter estimates. 4.3 Parameters Set within the Behavioral Model We set the discount factor β to 0.99 by matching mean asset holdings at age 50. We allow the disutility of work to depend on education and age through the parameter ψe, j . We set ψe, j so that the life-cycle profiles of hours for each education group generated by the model match those found in the data. To limit the number of parameters estimated, we assume that ψe, j is a quadratic function of age: ψe, j = ψe,0 (1 + ψe,1 · j + ψe,2 · j2 ). Appendix E shows that our estimates imply that the disutility from work is lowest in the middle of a man’s career and highest at its beginning and end. 5 Properties of the Baseline Model 5.1 Life-Cycle Patterns Figure 5a compares the distribution of marital status over the life cycle in the model (markers) and data (lines). Figure 5b compares the rates of fatherhood (at least one child) by marital status over the life cycle in the model and data.17 In both figures, the model fits the data closely. Especially notable is the model’s fit of the incidence of fatherhood among the newly-married (men in their first year of marriage), which indicates that the model does a good job of capturing the effects of pre-marital children on relationship transitions. Appendix D.1 shows that the model does a good job of matching the life-cycle patterns of fatherhood. 17 The profiles shift downward slightly at age 23 because that is the age at which college-educated men enter our sample. 28 Figure 5: Marital Status and Fatherhood by Age: Model and Data (a) Marital status shares 100 Share of Population (%) 40 60 80 20 0 0 20 Share of Population (%) 40 60 80 100 (b) Fatherhood share (one or more children) 20 30 Age Never-Married, Model Married, Model Divorced, Model 40 50 20 Never-Married, Data Married, Data Divorced, Data 30 40 Age Never-Married, Model Newly Married, Model Ever Married, Model 50 Never-Married, Data Newly Married, Data Ever Married, Data Source: Sample consists of men ages 19 - 50. Ages 19 - 22 include only men without a four-year college degree; ages 23 - 50 include all education groups. Never-married men are those who at the age in question have yet to marry; newly-married men are those in their first year of marriage. Data correspond to the NLSY79. Model results are author calculations; see section 4.2.3 and Appendix D for details. Figure 6 presents comparisons for two additional life-cycle profiles in the model and data. Figure 6a displays the life-cycle profiles for average hours worked. In the data, mean hours roughly double from age 19 until ages 35 - 40, at which point they begin to gradually decline. In the model, disutility of work is a quadratic in age, and the corresponding parameters are chosen to provide a tight fit to this profile. Figure 6b displays the life-cycle profiles for mean (net) assets. While the intertemporal discount factor β is chosen to match mean assets at age 50, assets at other ages are untargeted. Prior to age 50, mean assets are lower in the model than data, but the two series track each other fairly closely. 5.2 Marriage, Children and Male Labor Market Outcomes Figure 7 shows average hours of work by marital status for men ages 19 - 50. The model generates a cross-sectional distribution of hours similar to that in the data. In both the data and the model, never-married men work the least (1,496 hours in the data and 1,647 hours in the model), married 29 Figure 6: Mean Hours Worked and Assets by Age: Model and Data (a) Mean hours worked (b) Mean assets 10 5 4 2500 Assets Average Male Hours 3000 5 2000 1500 3 2 1 1000 20 25 30 35 40 45 50 20 25 30 35 40 Age Model 45 50 55 60 Age Data Data Model Source: Sample consists of men ages 19 - 50. Ages 19 - 22 include only men without a four-year college degree; ages 23 - 50 include all education groups. Data correspond to the NLSY79. Model results are author calculations; see s 4.3 and Appendix E for details. 2,500 Figure 7: Hours of Work by Marital Status: Data and Model 0 Mean Annual Hours Worked 500 1,000 1,500 2,000 2160.5 2068.8 1856.4 1834.7 1647.3 1496.2 Never-Married Married Data Divorced Model Source: Men ages 19 - 50. Data results refer to NLSY79. Model results are authors’ calculations. See text for details. men work the most (2,160 hours in the data and 2,068 hours in the model), and divorced men have hours in between (1,856 hours in the data and 1,834 hours in the model). The next test for the model is the extent to which it can reproduce observed labor market outcomes around the time of marriage or (first) childbirth. To replicate the panel analysis in 30 Section 2.2, we first normalize each man’s hours, wages and earnings by their age- and educationconditional averages. We then perform the fixed effect regression described in equation (1).18 We begin by assessing the model’s ability to replicate observed hours dynamics. Figure 8a shows that the model generates most of the marriage-related hours growth found in the data: from ten years before marriage to ten years after, hours increase 10% in the model versus 13% in the data. Both model and data generate a gradual increase in hours in the years before marriage, although the increase begins somewhat earlier in the data. Consistent with the data, in the model hours change very little once marriage occurs. Figure 8: Hours of Work by Distance from Marriage or First Child: Model and Data (b) Distance from date of first child 0.2 0.2 0.1 0.1 0 0 Hours Hours (a) Distance from date of marriage -0.1 -0.2 -0.1 -0.2 -0.3 -10 -5 0 5 -0.3 -10 10 -5 Years from First Marriage Data -95% CI +95% CI 0 5 10 Years from First Kids Model Data -95% CI +95% CI Model Source: Data results are from regressions using NLSY79 data; see section 2.2 for details. Model results are authors’ calculations; see text for details. It is natural to ask whether the effect of marriage on hours is primarily a response to the birth (or expected birth) of children. Figure 8b shows that hours do in fact rise steadily as the date of the first child approaches, and that the model generates a similar pattern. Another way to assess the role of children is to compare men whose first child arrives on or before the year of marriage, and is thus “pre-marital,” with men whose first child arrives after the year of marriage, and is thus “post-marital.” Figure 9 shows that in both the data and the model, the run-up in hours prior to marriage is larger for men with pre-marital children, and the increase in hours after marriage is larger for men whose children are all post-marital. Figures 8 and 9 thus suggest that both marriage and children are positively associated with male labor supply, and that our model replicates much of this relationship. 18 The regressions on the simulated data exclude year effects, as they have no counterpart in the model. 31 Figure 9: Hours of Work by Distance from Marriage and Timing of First Child: Model and Data (b) First child post-marital 0.2 0.2 0.1 0.1 0 0 Hours Hours (a) First child pre-marital -0.1 -0.2 -0.1 -0.2 -0.3 -10 -5 0 5 -0.3 -10 10 -5 Years from First Marriage Data -95% CI +95% CI 0 5 10 Years from First Marriage Model Data -95% CI +95% CI Model Source: Data results are from regressions using NLSY79 data; see Section 2.2 for details. Model results are authors’ calculations; see text for details. Pre-marital children are those born in the year of marriage or before. Post-marital children are those born after the year of marriage. Figure 10a shows the model’s implications for wages around the time of marriage. The model does a good job of matching the wage growth observed prior to marriage. In the model, wages rise in the run-up to marriage because higher wages increase the probability of marriage and because the increase in hours prior to marriage generates an increase in wages through the part-time wage penalty / overtime bonus. A shortcoming of the model is that it does not generate wage increases after the time of marriage. This could reflect the absence of human capital dynamics, such as learning-by-doing, that allow higher hours in one year to raise wages in subsequent years. Figure 10b presents the earnings trajectory. Since log earnings are the sum of log hours and log wages and the correlation of hours and wages is fairly weak, the dynamics of mean log earnings are roughly the sum of the profiles for log hours and log wages. Overall, the model generates a 17% increase in earnings from ten years before marriage to ten years after marriage, compared with 21% in the data. In particular, the model almost perfectly matches the 18% increase in earnings prior to marriage, although the increase begins a bit earlier in the data. On the other hand, the model underpredicts the rise in earnings after marriage, because wages do not increase after marriage in the model. 5.3 Sensitivity Analyses We now assess the sensitivity of our results to two key parameters, namely the coefficient of relative risk aversion (γ) and the Frisch elastiticity of labor supply (ξ ), halving or doubling each 32 Figure 10: Wages and Earnings by Distance to Marriage: Model and Data (b) Earnings 0.2 0.2 0.1 0.1 Earnings Wage (a) Wages 0 -0.1 -0.2 0 -0.1 -0.2 -0.3 -10 -5 0 5 -0.3 -10 10 -5 Years from First Marriage Data -95% CI +95% CI 0 5 10 Years from First Marriage Model Data -95% CI +95% CI Model Source: Data results are from regressions using NLSY79 data; see section 2.2 for details. Model results are authors’ calculations; see text for details. parameters from its benchmark value. Figure 11 shows that over this range our qualitative results do not depend on specific parameter values, and the majority of the quantitative results are robust as well. Figure 11a displays the change in hours around marriage for different values of γ. As γ increases from 0.375 to 1.5, the run-up in hours prior to marriage declines somewhat, from 10% to 7%, while the change in hours after marriage switches from a 2% decline to a 2% increase. The net result is that the total change in hours around marriage varies little within this parameter range. To interpret these results, it is helpful to rewrite the marginal utility of consumption as γ−1 −γ c . MUc = N f η f Under our calibration, when a unmarried man marries, N f increases from 1 γ−1 to 2, while η f increases from 1 to roughly 1.6. For any γ > 0, N f η f will increase, raising the marginal utility of consumption and encouraging work. If the couple then has children, η f grows even larger, but N f remains at 2. Noting that most children occur after marriage, it follows that γ−1 after marriage the product N f η f falls when γ is less than one and rises when it is greater than one. This is indeed consistent with Figure 11a, which shows hours falling after marriage when γ = 0.375 or 0.738 (the benchmark value), but rising when γ = 1.5. The effects of changing ξ , shown in Figure 11b, are more straightforward to interpret. As the Frisch elasticity increases from 0.375 to 1.5, the hours responses grow in size, but their signs remain the same. 33 Figure 11: Hours of Work by Distance from Marriage: Effects of Preference Parameters (b) Frisch elasticity, ξ 0.2 0.2 0.1 0.1 0 0 Hours Hours (a) Coefficient of relative risk aversion, γ -0.1 -0.2 -0.1 -0.2 -0.3 -10 -5 0 5 -0.3 -10 10 Years from First Marriage Benchmark CRRA=0.375 -5 0 5 10 Years from First Marriage CRRA=1.5 Benchmark Elasticity=0.375 Elasticity=1.5 Source: Model results are authors’ calculations. See text for details. 6 Quantitative Results: Why Do Married Men Work More? 6.1 The Role of Selection into Marriage In our model, men with higher wages move up the relationship ladder more quickly. (The magnitude of this effect is set to match the relationship between wages and marriage found in our IV regressions on CPS data; see Section 4.2.3.) This implies that married men have higher wages in part because of selection. Moreover, because men respond to positive wage shocks by working more, selection also leads married men to have higher hours. To assess the quantitative impact of selection on labor market dynamics around marriage, Figure 12 presents results from a specification where the only link between family structure and the labor market is selection. Specifically, in this alternative specification, spouses and children have no effect on preferences or resources: η f = N f = 1, ∀ f , and se j = χa = δa = 0.19 Under this specification, the main link between family structure and the labor market is selection. Figure 12a shows that the pre-marital increase in wages in the selection-only specification (red line with squares) is much smaller than the one found in the baseline model (blue line with stars). This implies that the larger increase in wages in the benchmark model was not driven primarily by selection; instead, it was driven mainly by the part-time wage penalty, where higher 19 In constructing this specification, we assume further that all men face the same tax schedule, using parameter estimates from Guner, Kaygusuz and Ventura (2014) that average across all unmarried men. We likewise assume that married men now keep all of their wealth upon divorce. 34 Figure 12: Labor Market Dynamics around Marriage: Benchmark and Selection-Only Models (b) Annual hours worked 0.2 0.2 0.1 0.1 0 0 Hours Wage (a) Hourly wages -0.1 -0.2 -0.1 -0.2 -0.3 -10 -5 0 5 -0.3 -10 10 -5 Years from First Marriage Benchmark Selection only 0 5 10 Years from First Marriage Stronger selection only Benchmark Selection only Stronger selection only Source: Model results are authors’ calculations. In the selection-only models, spouses and children have no effect on husbands’ consumption or the utility they receive from it. In the stronger selection-only model, the effect of wages on relationship transition probabilities is over three times its baseline value. See text for details. hours lead to higher wages. This channel is weaker in the selection-only specification, where hours do not increase very much in the run-up to marriage (see Figure 12b). To account for the possibility that our estimates understate the degree of wage selection in effect, we introduce a “stronger” selection-only alternative, where we manually increase the coefficient θ until the selection channel generates the full pre-marital increase in wages observed in the data. At this value of θ , a 10% increase in wages raises the probability of getting married by 19.6%, which is over three times as strong as the elasticity we estimate from the data. Figure 12b compares the change in hours worked around marriage in the benchmark model, the selection-only model, and the stronger selection-only model. Neither selection-only specification generates a substantial increase in hours prior to marriage. In the baseline selection-only model, hours increase less than 1% prior to marriage. In the stronger selection-only model, hours increase less than 2% prior to marriage, far below the increase seen in either the benchmark model or the data. For intuition, note that in the data and baseline model, the run-up in hours prior to marriage is larger than the run-up in wages. If the increase in hours were solely a response to higher wages, the underlying uncompensated labor supply elasticity would have to be well in excess of 1. The uncompensated elasticity in our benchmark specification, however, is well below 1. The benchmark Frisch elasticity is ξ = 0.75 (in the neighborhood of many empirical estimates), and the uncompensated elasticity is even smaller, especially when income effects are sizable, as is often the case for young men with little wealth. The benchmark assumption of a modest wage elasticity is also consistent with the data shown in Figure 3, where hours remain more or less 35 constant after marriage, even as wages continue to rise. The very fact that the relative magnitudes of the wage and hours changes are so different before and after the date of marriage argues against a simple selection story. To summarize, the results shown in Figure 12 imply that wage selection plays a secondary role in generating the hours dynamics observed around the time of marriage. The degree of wage selection that we estimate is modest, and even if it were significantly larger, matching the observed hours dynamics would require large uncompensated wage elasticities. The larger hours increase associated with pre-marital chidren (see the discussion of Figure 9) also suggests that wage selection cannot be the sole explanation. If selection were the only mechanism, the increase in hours associated with pre-marital children, who are more likely to be unplanned and thus uncorrelated with labor market shocks, would be smaller than the increase associated with post-marital children. 6.2 The Labor Supply Effects of Marriage and Family Structure Marriage and family structure affect male labor supply through multiple channels. Having additional household members implies that a husband/father consumes only a portion of his household’s consumption expenditures (as η f > 1). This effectively taxes the man’s earnings, yielding the usual combination of income and substitution effects. Conversely, the utility shifter N f implies that the husband receives utility from the consumption of other family members, which, all else equal, raises the shadow price of his earnings. Finally, other family members generate standard wealth effects: working spouses contribute earnings, while children impose child care costs if their mothers work. We refer to the net sum of these family-related effects as the “mouths-to-feed” effect. To assess the quantitative importance of each of these mechanisms, Figure 13 presents the hours trajectories generated by three alternative specifications. In the first alternative (black line with triangles), we set N f = 1, ∀ f . This implies no altruism within a family: husbands/fathers do not receive utility from the consumption of other family members. Under this specification, marriage leads hours of work to fall. Recall that the flow utility from consumption equals 1 N f 1−γ (c/η f )1−γ . If η f is increasing in family size while N f is not, marriage and children ef- fectively tax the man’s earnings, and with γ < 1, hours will fall. This point is reinforced by the second alternative (pink line with squares), where we set η f = 1, ∀ f , resulting in “full economies of scale.” With larger families imposing no consumption costs, but still generating altruism, the hours run-up associated with marriage is even larger than in the baseline. In the third alternative specification (solid red line), we assume that spouses have no earnings 36 of their own and that children impose no direct child care costs, se j = χa = 0. This implies that families have no direct impact on the household budget, except through changes in the income tax function T inc (y). Under this specification, hours rise more than under the benchmark (starred blue line). For intuition, note that spouses always weakly expand the households’ budget sets. Among working spouses (se j > 0), the quantity se(1 − nχa ) is always positive: the cost of each child is at most χy = 28% of their mother’s earnings, and the largest possible number of children is n = 3. Meanwhile, non-working spouses have no income by construction and care for any children present. It follows that turning off the direct budget effects of families will on average tighten household budget constraints, which raises the marginal utility of consumption and induces husbands/fathers to work more. Figure 13: Hours of Work by Distance to Marriage: Benchmark and Alternative Specifications 0.4 Hours 0.2 0 -0.2 -0.4 -10 -5 0 5 10 Years from First Marriage Benchmark No altruism Full economies of scale No family budget effects Source: Model results are authors’ calculations. “No altruism” shows hours path for specifciation where N f = 1, ∀ f . “No family wealth effects” shows hours path for specification where spouses and children have no direct budgetary effects (se(1 − nχa ) = 0). See text for details. The results from these alternative specifications imply that, through the lens of the model, men increase their labor supply around the time of marriage primarily due to altruism toward spouses and children (or its observational equivalent), and not primarily as a response to changes in family resources. The logic behind our finding is straightforward. The assumption that additional household members impose an earnings tax — η f increases in household size — is an inherent feature of equivalence scales.20 With γ < 1, this tax discourages work. Moreover, as 20 Introductions to the theory and construction of equivalence scales include Cowell and Mercader-Prats (1999) and Lewbel and Pendakur (2008). 37 long as spouses cover the complete cost of child care, out of their earnings or through home production, married or cohabiting men enjoy higher household incomes. This too should discourage work. This leaves the size-dependent shifter N f as the only feasible mechanism to generate a marriage-related increase in hours. Our study is not the first to employ this mechanism (see, e.g., Blundell et al. (2016) and Fan, Seshadri and Taber (2019)), but our results provide novel evidence in its support. Taking Stock. The results presented thus far lead us to three conclusions. First, the model generates a reasonable fit of male hours dynamics around the time of marriage. Second, reverse causality, i.e., selection into marriage on the basis of transitory wage shocks, cannot by itself explain the patterns observed in the data. Third, the mouths-to-feed mechanism appears to be an important driver of male labor supply. In particular, we find that the increase in male labor supply in the run-up to marriage is largely due to married men internalizing the consumption utility of their family. 6.3 The Aggregate Impact of Marriage on Male Labor Supply We now investigate the quantitative impact of marriage on aggregate labor supply via three model experiments that counterfactually change the frequency of marriage. We begin with an extreme experiment in which we eliminate the marriage process altogether. Specifically, we set the probability of engagement and marriage to zero, and we convert men who entered the baseline simulations as married or cohabiting into singles. Figure 14 compares the hours profile from this experiment (“No marriage & cohabitation”) to that of baseline specification. Averaging across ages 19 - 50, eliminating the marriage process reduces male work by 133.5 hours per year, a 7% reduction. The aggregate effects are largest after age 35, when most men have married. For example, among 45-year-old men, eliminating the marriage process reduces hours worked by 180 hours, or 8.7%. By way of comparison, in Siassi’s (2019) framework, the feature most akin to our mouthsto-feed mechanism is a stronger bequest motive for individuals with children.21 Removing this feature causes his estimate of the proportional earnings (not hours) gap to shrink by 2.3-2.8 percentage points (Siassi, 2019, Table 5), less than half the magnitude of our results. In the model, cohabitation increases male labor supply in a manner similar to marriage. To understand the relative importance of cohabitation and marriage, we run a second experiment in which we eliminate marriage but not cohabitation. Specifically, we set the probability of marriage 21 Because Siassi (2019) employs the GHH flow utility function (Greenwood, Hercowitz and Huffman, 1988), which has no wealth effects, his framework contains no direct counterpart to our mouths-to-feed mechanism. 38 to zero and convert men who began life married to singles, but we leave unchanged the probability of pre-marital cohabitation. Figure 14 also displays the hours profile from this experiment (“No marriage”). Averaging across ages 19 - 50, eliminating marriage but retaining cohabitation reduces male work by 100.6 hours, or 5.3%. This indicates that cohabitation has a sizable effect on aggregate hours worked (7.0 − 5.3 = 1.7%), but that the effect of marriage is roughly three times as large. Figure 14: Hours of Work with and without Marriage Average Male Hours 2400 2200 2000 1800 1600 1400 1200 20 25 30 35 40 45 50 Age Benchmark Model No Marriage & Cohabitation No Marriage NLSY97 Marriage Rates Source: Model results are authors’ calculations. In the ”No Marriage & Cohabitation” specification, initial singles never marry or cohabit, and men who would otherwise enter the simulations as cohabiting or married are converted to singles. In the “NLSY97 Marriage Rates” specification, the relationship process is re-estimated using NLSY97 data. See text for details. Finally, we quantitatively assess a recent hypothesis by Binder and Bound (2019) that declining rates of marriage could help explain declines in male work rates in recent decades. To do this, we replace the baseline relationship processes with processes estimated from the NLSY97 (Bureau of Labor Statistics, 2019b), a longitudinal dataset comparable to the NSLY79 that follows a cohort born between 1980-1984, roughly two decades after the NLSY79 cohort, which was born between 1957-1964. Men in the NLSY97 cohort marry at lower rates than men in the NLSY79: for example, by age 25, only 27% of men in the NLSY97 have married, compared with 47% of men in the NLSY79. Because the NLSY97 is based on a younger cohort than the NLSY79, the life-cycle moments we use to estimate the relationship process (see Section 4.2.3) are observed in the NLSY97 only 39 through age 32. We therefore add to these moments imputed values for the share never-married between ages 33 - 50, assuming that the gap in marriage rates between the NLSY97 and NLSY79 remains constant from age 32 onward: 79 79 97 share97 j = share j + (share32 − share32 ), j ∈ {33, ..., 50}. (37) This imputation implies that, averaging across ages 19 - 50, 50.5% of men in the NLSY97 are never married, compared with 33.7% of men in the NLSY79. With the new moments thus assembled, we re-estimate the relationship process for the NLSY97 cohort. Figure 14 displays the hours profile from this experiment (“NLSY97 marriage rates”). The lower marriage rates in the NLSY97 reduce prime-age male hours worked by 43 hours, or 2.3%. For context, the CPS data behind Figure 1 above show that between 1979 and 2018, average annual hours worked by prime-age men declined 8.4%. Our results therefore suggest that recent declines in marriage rates, if exogenous, can account for a sizable share of the overall decline in prime-age male hours over recent decades. 7 Conclusion Married men work substantially more hours than men who have never been married. Panel data reveal that much of this gap is accounted for by an increase in work at the individual level in the years leading up to marriage. Two potential explanations for this increase are: (i) men hit by positive labor market shocks are more likely to marry; and (ii) the prospect of marriage increases mens’ labor supply. We quantify the relative importance of these two channels using a structural life-cycle model of marriage and labor supply. A version of the model calibrated to life-cycle marriage and fertility moments in the NLSY79 replicates the marriage-related hours dynamics observed in the same dataset. The model provides a framework to evaluate the link between marriage and labor market outcomes. Through the lens of the model, selection into marriage based on labor market shocks explains only a small part of the increase in hours around marriage. Instead, the primary driver of the increase in hours is an altruistic “mouths-to-feed” effect, wherein men internalize the utility that their spouses and children receive from consumption. Counterfactual experiments within the model show that marriage is an important determinant of aggregate hours worked by prime-age men. In particular, when we feed in lower rates of marriage corresponding to later cohorts from the NLSY97, average hours worked by prime-age men fall substantially. 40 Our findings highlight several promising areas for future research. First, our analysis focuses on the impact of marriage on male labor market outcomes in the US. This raises the prospect that differences in marriage rates could explain cross-country differences in male labor market outcomes. While Bick and Fuchs-Schündeln (2018) study the labor market implications of crosscountry differences in the taxation of married couples (as opposed to singles), to our knowledge, no one has considered marriage itself as a source of variation. Another important open question surrounds the joint determination of family structure (relationships and fertility) and labor supply (Gayle and Shephard, 2019; Caucutt, Guner and Rauh, 2021). Our analysis takes the life-cycle process for family structure as given. Endogenizing family structure would allow researchers to study the potentially complex interactions between household formation and labor market outcomes. For example, it could be the case that a decline in wages for a subset of men makes them less attractive potential spouses, which reduces marriage rates, which in turn (via the “mouths-to-feed” channel) reduces labor supply. Through the lens of our current model, the reduction in hours worked would look like the result of a decline in male labor supply, even though a decline in labor demand was the ultimate cause. 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Weil, David N, and Oded Galor. 1996. “The Gender Gap, Fertility, and Growth.” American Economic Review, 86(3). 44 For Online Publication: Appendices A Supplemental Estimates of the Relationship between Marriage and Male Labor Market Outcomes A.1 Supplemental Estimates: State Average Hours Worked Table 4: Predictors of State Average Male Annual Hours Worked in the CPS Constant Share Married Average Age (1) (2) (3) 1450.0∗∗∗ (36.6) 706.2∗∗∗ (62.1) – 1443.3∗∗∗ (56.0) 768.7∗∗∗ (84.3) 0.4 (0.5) 0.3 (14.1) 11.1 (11.7) 6.0 (11.7) −23.1 (15.1) Y 1537.9∗∗∗ (74.5) 609.3∗∗∗ (123.1) 0.0 (0.3) −5.4 (8.1) 4.6 (6.8) 3.7 (6.8) 4.2 (9.1) Y Y 0.42 459 0.83 459 Share Less than High School – Some College Share – Bahelor’s + Share – Share Black – Year FEs State FEs R2 -adj N 0.22 459 Source: Men ages 19 - 54 in the 1975-2019 waves of the CPS ASEC. The sample includes individuals with zero annual hours. The unit of observation is a 5-year average of average hours for a given state. There are nine five-year time periods in our sample: 1975-1979, 1980-1984, ..., 2015-2019; the data include all 50 states and the District of Columbia; together this implies 459 observations. 45 A.2 Supplemental Estimates: Placebo Test for Distance-from-Marriage Regressions Figure 15: Placebo Test: Labor Market Dynamics in the Years around Marriage -10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Years from First Marriage .2 (c) Annual earnings Annual Earnings rel. to Year of Marriage (%) -.3 -.25 -.2 -.15 -.1 -.05 0 .05 .1 .15 .2 (b) Hourly wages Hourly Wage rel. to Year of Marriage (%) -.3 -.25 -.2 -.15 -.1 -.05 0 .05 .1 .15 Annual Hours Worked rel. to Year of Marriage (%) -.3 -.25 -.2 -.15 -.1 -.05 0 .05 .1 .15 .2 (a) Annual hours worked -10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Years from First Marriage -10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Years from First Marriage Source: Males ages 19 - 54 in the NLSY79, see text for details. The solid line plots distance-from-marriage coefficients from the individual fixed effects regression equation (1). The shaded region corresponds to 95% confidence intervals. The results in this table correspond to a placebo test of the results in Figure 3, in which age of first marriage was randomly reassigned among individuals in the regression sample. A.3 Supplemental Estimates: Decomposing the Distance-from-Marriage Effects on Hours Figure 16: Dynamics of Weekly Hours and Annual Weeks Worked in the Years around Marriage -10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Years from First Marriage (c) Annual weeks worked Annual Weeks Worked rel. to Year of Marriage -.3 -.25 -.2 -.15 -.1 -.05 0 .05 .1 .15 .2 (b) Hours per workweek Hours Worked per Week rel. to Year of Marriage -.3 -.25 -.2 -.15 -.1 -.05 0 .05 .1 .15 .2 Annual Hours Worked rel. to Year of Marriage -.3 -.25 -.2 -.15 -.1 -.05 0 .05 .1 .15 .2 (a) Annual hours worked -10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Years from First Marriage -10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Years from First Marriage Source: Males ages 19 - 54 in the NLSY79, see text for details. The solid line plots distance-from-marriage coefficients from the individual fixed effects regression equation (1). The shaded region corresponds to 95% confidence intervals. 46 B Estimating the Male Wage Process Our approach closely follows French (2005). We begin by running an individual fixed effects regression on log wages. The coefficients from this regression, along with the average fixed effect, give us predicted wages, αe,w j . Subtracting αe,w j (along with an adjustment for aggregate bi, j }i, j . Next, we unemployment) from observed wages produces a panel of wage residuals, {w assume that the stochastic process for the wage residuals is bi, j = w̃i, j + w̄i, j , w (38) w̃i, j = ρew w̃i, j−1 + εi,wj , (39) iid εi,wj ∼ N(0, σeε ), iid (40) w̄i, j ∼ N(0, σew̄ ), (41) w̄i, j ⊥⊥ w̃i, j+s , ∀t, s. (42) bi, j : We can then back out ρew from the autocorrelations of w ρew = bi, j , w bi, j+3 ) cove (w . bi, j , w bi, j+2 ) cove (w (43) With ρew in hand, the standard deviation of the innovation εi,wj follows from s σew = C bi, j , w bi, j+2 )(1 − (ρew )2 ) cove (w . (ρew )2 (44) Model Fit of Spousal Employment and Earnings The parameters of the spousal earnings process are set to match the age profiles of spousal employment, the mean and standard deviation of log earnings among working spouses, and the correlation of spousal earnings and male wages. Because the NLSY79 does not provide earnings information for unmarried partners, the simulated profiles and the data targets both use data only for the years the couples are married. Figure 17 compares the model’s predictions of spousal employment to those found in the NLSY79. The model replicates the lower rates of employment among women with children, es- 47 Figure 17: Spousal Employment by Age, Education and Age of Children, Model and Data (b) College Share with Positive Sp. Earnings Share with Positive Sp. Earnings (a) Less than college 1 0.8 0.6 0.4 0.2 20 25 30 35 40 45 50 55 1 0.8 0.6 0.4 0.2 60 20 25 30 35 Age Data, no kids Model, no kids 40 45 50 55 60 Age Data, young kids Model, young kids Data, old kids Model, old kids Data, no kids Model, no kids Data, young kids Model, young kids Data, old kids Model, old kids Source: Data are for spouses of married men ages 19 - 50 (less than college) or 23 - 50 (college graduates) in the NLSY79; see text for details. Model results are authors’ calculations; see text for details. Figure 18: Spousal Earnings (If Employed) by Age, Education and Age of Children, Model and Data (a) Less than college (b) College 11 Mean of log Sp. Earnings Mean of log Sp. Earnings 11 10.5 10 9.5 10.5 10 9.5 20 25 30 35 40 45 50 55 60 20 25 30 Age Data, no kids Model, no kids Data, young kids Model, young kids 35 40 45 50 55 60 Age Data, old kids Model, old kids Data, no kids Model, no kids Data, young kids Model, young kids Data, old kids Model, old kids Source: Data are for spouses of married men ages 19 - 50 (less than college) or 23 - 50 (college graduates) in the NLSY79; see text for details. Model results are authors’ calculations; see text for details. pecially young children. Figure 18 provides the corresponding comparison for the logged earnings of employed spouses. In contrast to employment, spousal earnings vary relatively little by family composition — in the model the only differences are (miniscule) selection effects. The first row of Table 5 shows the standard deviation of these earnings, again conditional on working. This statistic helps pin down βes , the coefficient on the shock s̃ in the spousal earnings 48 equation, and σes , the volatility of this shock. While we target standard deviations on an ageby-age basis, the data moments are noisy, and we therefore report just the unconditional average across ages. Spousal earnings are quite volatile: the standard deviation of logged earnings ranges between 0.65 and 0.68. Table 5: Standard Deviation of Spousal Earnings and Correlation with Male Wages, Model and Data Statistic Less than College Data Model College Graduates Data Model Standard deviation, ln(se j ) 0.6514 0.6526 0.6773 0.6784 Correlation (ln(se j ), w̃ j ) No children Young children Older children 0.1896 0.1927 0.1895 0.1681 0.0992 -0.0166 0.0851 0.0859 0.0803 0.2946 0.1631 0.1088 Note: Data are for married men (and spouses) ages 19 - 50 (less than college) or 23 - 50 (college graduates) in the NLSY79 and are restricted to observations with positive values; see text for details. Model results are authors’ calculations; see text for details. Standard deviations and correlations are calculated age-by-age; reported above are unconditional averages. Our final target is the correlation between male wages and spousal earnings, conditional on both parties working, as a function of their childrens’ age. This moment, along with the earnings’ variance, helps identify ρes and σes , the coefficient on male wages and the idiosyncratic variation, respectively, in the distribution of the shock s̃. Table 5 shows that among couples with no children, the correlations are less than 0.17 for those with a college degree and less than 0.30 for those without. The correlation coefficients for couples with children are even lower. D D.1 Estimating the Dynamics of Family Structure Main Estimates n , the probability of a new child being born given existing Fertility dynamics are governed by φn,r, j children n, relationship status r, and man’s age j. New children stop arriving once the existing a children age. Let φyc, j denote the probability that all the young children of an age- j man bea the probability that all the old children become grown. All of these come old children; and φoc, j probabilities are modeled as logistic functions of a quadratic polynomial in the man’s age. 49 Table 6: Parameters for Family Dynamics Parameter Name Parameter Value, Non-College Value, College n φsn,0, j n φmr,0, j n φsn,1, j n φmr,1, j n φsn,2, j n φmr,2, j a φyc, j a φoc, j (−3.704, 0.145, −0.995) (−0.414, −0.106, −0.077) (−2.344, 0.187, −1.387) (−1.115, 0.027, −0.120) (−1.644, 0.075, −0.858) (−1.113, −0.109, 0.141) (−4.439, 0.212, −0.344) (−14.963, 0.602, −0.608) (−4.568, 0.177, −0.997) (−2.173, 0.215, −1.462) (−2.002, −0.175, 0.156) (−1.294, 0.069, −0.329) (−3.357, −0.234, −9.920) (−2.235, 0.084, −0.751) (−5.262, 0.264, −0.355) (−8.512, 0.033, 0.462) Relationship Dynamics (logistic coefficients) Engagement φ jen Marriage φ mr Impact of wage shock θ Effect of pre-marital child φ owb Divorce, n < 3 φn,d j Divorce, n = 3 φn,d j (−2.154, 0.037, −0.766) 0.0 0.221 −0.195 (−1.799, −0.281, 0.500) (−1.632, −0.154, 0.125) (−1.528, −0.145, 0.269) 0.0 0.199 3.160 (−4.401, 0.063, −1.171) (−14.996, 1.237, −3.530) Relationship Dynamics (probabilities) Initial relationship distribution (sn, en) Fraction cohabiting φ en-c (0.811, 0.108) 0.5 (0.699, 0.115) 0.5 Children (logistic coefficients) First child, pre-marital First child, married Second child, pre-marital Second child, married Third child, pre-marital Third child, married Young children age Old children age Note: Superscripts are used to distinguish parameters, while subscripts are used to distinguish dependencies. All parameters with the age subscript j utilize a quadratic in age. See section 4.2.3 and Appendix D for details. The parameters governing relationship dynamics are φ en j (w̃), the probability of a single man of age j becoming engaged; φ en-c , the probability that a engaged man cohabits with his partner; φ mr (w̃), the probability of an engaged man becoming married; φ owb , the effect of an out-ofwedlock birth on the probability of a relationship advancing; and φn,d j , the probability of a married man of age j with n children becoming divorced. The probabilities of becoming engaged and getting divorced depend on the man’s age via a quadratic polynomial. The probability of transtioning from engagement to marriage is age-invariant; because we do not observe the engagement status of unmarried men, we simplify the model by assuming that all the age-related variation prior to marriage is captured in the rate at which engagements form. Data limitations also lead us to assume that the effect of a pre-marital birth (φeowb ) is the same for single and engaged men, and that cohabitation does not effect the rate at which engaged couples marry. (We set the fraction of 50 engaged couples who cohabit, φ en-c , to 1/2 for all ages and education levels, consistent with the cohabitation fraction in the NLSY79.) Because the divorce probability is notably higher for men with three or more children in the data, we estimate two sets of divorce probabilities, one for men with less than three children, and a second for men with three children. We estimate these parameters separately for each education group, using the simulated method of moments. Specifically, we target the following empirical age profiles: the share of men who are never married and who have n children, for n ∈ {0, 1, 2, 3+}; the share of men who have ever been married and who have n children, the share of divorced men who have n children, and the share of newly married men who have had at least one child. The latter age profile is informative about the effect of pre-marital children on marriage probability. We also target the coefficient on male wages in a regression of marriage transition probabilities; see section 4.2.3 and Appendix D.2 for details. Table 6 shows the resulting parameter estimates. Figures 5a and 5b in the main text and Figure 19 immediately below show model fits. Figure 19: Number of Children by Marital Status, Model and Data (a) Never married 100 Share of Population (%) 40 60 80 20 0 0 20 Share of Population (%) 40 60 80 100 (b) Ever married 20 30 0 Children, Model 1 Child, Model 2+ Children, Model Age 40 50 20 0 Children, Data 1 Child, Data 2+ Children, Data 30 0 Children, Model 1 Child, Model 2+ Children, Model Age 40 50 0 Children, Data 1 Child, Data 2+ Children, Data Source: Sample consists of men ages 19 - 50. Ages 19 - 22 include only men with less than a four -ear college degree; ages 23 - 50 include all education groups. Never-married men are those who at the age in question have yet to marry. Data correspond to the NLSY79. Model results are author calculations; see section 4.2.3 and Appendix D for details. 51 D.2 Supplemental Estimates: State-Level Wages and Marital Transitions Table 7 shows results from a linear probability regression of marriage on male wages. We transform wages using the inverse hyperbolic sine function. This allows for a (near-) logarithmic relationship when wages are positive, but also accomodates values of zero. The first column of the table shows the results from an OLS regression. The estimated coefficient on wages is 0.0092. The second column shows the results for an IV regression where we instrument for each individual’s wages with the average wages in his state of residence. The F-statistic for state-level wages in the first-stage regression is 254.69, highly significant and indicative of instrument relevance. Instrumenting for wages causes the coefficient to increase to 0.0144. This is the coefficient value we target when estimating our model of relationship dynamics. Table 7: Predictors of State-Level Marital Transitions OLS Constant asinh(wage) Age Age2 /100 College + New child Year FEs State FEs R2 -adj N 0.0206∗∗ (0.0082) 0.0092∗∗∗ (0.0008) 0.0065∗∗∗ (0.0014) −0.0279∗∗∗ (0.0096) 0.0124∗∗∗ (0.0030) 0.5300∗∗∗ (0.0102) Y Y 0.10 26,290 2SLS 0.0252∗∗ (0.0110) 0.0144∗ (0.0083) 0.0059∗∗∗ (0.0017) −0.0250∗∗∗ (0.0107) 0.0089∗∗∗ (0.0062) 0.5288∗∗∗ (0.0104) Y Y 0.10 26,290 Source: 1982-2019 waves of the CPS ORG. Sample is men ages 19 - 54. 52 E Work Disutility Figure 20 shows how the estimated disutility of working, ψe, j , varies across the life cycle. As discussed in Section 4.3, for each education group, our estimate of work disutility is the quadratic function that allows the model to best fit the life-cycle hours profiles found in the NLSY79. Disutility of working by education Figure 20: Work Disutility by Age and Education 12 10-7 10 8 6 4 20 25 30 35 40 45 50 55 Age Less than college Source: Author calculations. See Section 4.3 for details. 53 College 60