Female Relative Wages, Household Specialization and Fertility

Transcription

1 Female Relative Wages, Household Specialization and Fertility Christian Siegel January 31, 2017 Abstract Falling ertility rates have oten been linked to rising emale wages. However, over the last 40 years the US total ertility rate has been rather stable while emale wages have continued to grow. Over the same period, women s hours spent on housework have declined, but men s have increased. I propose a model in which households are not perectly specialized, but both men and women contribute to home production. As the gender wage gap narrows, the time allocations o men and women converge, and while ertility alls at irst, the decline stops when emale wages are close to male s. Rising relative wages increase women s labor supply and due to higher opportunity cost lower ertility at irst, but they also lead to a reallocation o home production and child care rom women to men, and a marketization. I ind that both are important in understanding why ertility did not decline urther. In a urther quantitative exercise I show that the model perorms well in matching ertility over the entire 20th century, including the overall decline, the baby boom, and the recent stabilization. JEL codes: D13, E24, J13, J22 Keywords: Fertility, Female Labor Supply, Household Production, Intrahousehold Allocations University o Kent, School o Economics, Canterbury, Kent, CT2 7NP, UK; C.Siegel@kent.ac.uk. This is a substantially revised version o a chapter o my PhD thesis at the London School o Economics entitled Female Employment and Fertility: The Eects o Rising Female Wages. I am grateul to Francesco Caselli or his advise on this project. I would also like to thank two anonymous reerees, the editor Matthias Doepke, Zsóia Bárány, Wouter den Haan, Monique Ebell, Paula Gobbi, Moshe Hazan, Ethan Ilzetzki, Winried Koeniger, John Knowles, Rigas Oikonomou, Albert Marcet, Michael McMahon, Rachel Ngai, Silvana Tenreyro, as well as seminar and conerence participants at Bonn, Cologne, Exeter, LSE, Uppsala, the SED 2013 meeting in Seoul and the Demographic Economics Conerence at the University o Iowa or many comments and suggestions. 1

2 1 Introduction Between the 1960s and today, we have seen enormous changes to the economic and demographic structure in all Western countries. There has been a decline in total ertility rates 1 and an increase in women s market hours (see igures 1 and 2 or US data). Many authors explain both with a rise in emale wages (e.g. Galor and Weil (1996), Doepke, Hazan, and Maoz (2015)). An apparent puzzle, however, is that while emale wages and market hours have continued to grow, since the early 1970s ertility has stopped alling. 2 Understanding the underlying ertility decisions is important since they aect population growth, labor orce composition and social security systems, and thereby economic outcomes. In this paper I argue that the common driving orce behind the trends in ertility and in emale employment is the narrowing o the gender wage gap (shown in igure 3), rather than the level o emale wages per se, since it changes the division o labor within the amily. The explanation I propose is based on imperect specialization o households, such that both men and women contribute to home and market production, implying that a reallocation has a nonlinear eect. My explanation is based on the observation that men s home hours have increased, allowing women s home hours to all. In the 1960s, when the wage gap was large, the catching up o emale wages increased women s labor supply. The associated increase in the opportunity cost o women s time, who shouldered most o child care, lowered ertility, as argued by Becker (1960) or Galor and Weil (1996). But as relative wages become more equal over time, specialization in the household decreases. Consequently, male home hours increase, a ather s time at home becomes more important or raising children, and the allocation o time between home and market work becomes more evenly balanced or men and women. I show that when the complementarity between mother s and ather s time working at home is suiciently large, the marginal utility cost o having an additional child can become constant, or even all, despite women working more hours in the market. Circumstantial evidence in avor o this mechanism is provided by data on non-market hours. Using data rom Aguiar and Hurst (2007), based on the American Time Use Survey since 1965, I show the trends in hours spent working at home in igure 4. I ocus on the sum o time spent non-market work, deined by Aguiar and Hurst as home production plus obtaining goods, and basic child care. 3 The data displays a shit in household production 1 The total ertility rate (TFR) is the average number o children that would be born i all women lived to the end o their childbearing years and bore children according to the current age-speciic birth rates. 2 Most o the recent rise in the oicial total ertility rate is driven by the eects o immigration. For US-born women the increase is much less and ertility virtually lat since the late 1970s. The details on the decomposition o TFR by mothers birthplace are given in the online appendix, section A.1. 3 The measure is constructed as weekly hours spent on home production in a narrow sense, on obtaining goods, and on basic child care. Arguably, or many people child care is an activity that is more enjoyable than other orms o housework. Excluding child care rom the measure o market work does not change the qualitative trends. This can be seen in the regression results o table A-1 in the online appendix. 2

3 All Men and Women Married Single Year Year Year Male Female Male Female Male Female Figure 1: Male and emale market hours This graph shows average (per capita) yearly market hours worked or men and women aged 20 to 60. The data is taken rom the PSID and reers to sel-reported hours worked in the previous year. In the panel on the let, all individuals o age 20 to 60 are included. The panels to the right restrict the sample to married and to single individuals respectively. Per capita hours worked are computed as the simple average o the average hours worked in 5-year age bins, in order to take out a potential composition eect due to changing cohort sizes. Total Fertility Rate Children Ever Born Year Year TFR US Born Women Oicial TFR all ever married never married Total Fertility Rate Children Ever Born Figure 2: Total ertility rate and children ever born In the let panel, the red dashed line shows the oicial total ertility rate (TFR) or the United States, taken rom the World Bank (based on national statistics and the United Nations Demographic Yearbook), and the blue solid line the author s computation o TFR or US-born women, based on the Vital Statistics o the United States combined with population estimates rom the US Census; see online appendix or details. The right panel shows children ever born to women o age 40 to 44; the blue dashed line or all women o age 40 to 44, the red solid line and the green dot-dashed line the breakdown or ever married and never married women respectively. The data is taken rom the United States Census Bureau s Fertility Data Historical Time Series and based on the CPS. 3

4 Ratio Year Figure 3: Female Relative Wages Ratio o median emale to male hourly wages, computed in a PSID subsample o men and women aged 20 to 60 who work at least one hour in paid employment and report positive earnings. rom women to men, with an overall reduction o hours worked at home. As a consequence, the ratio o married men s to married women s hours devoted to working at home rose rom 0.25 to 4 over There has also been a shit in time investments into children. Sayer, Bianchi, and Robinson (2004) ind in time diary data that the ratio o married mothers to married athers time in child care declined in all primary child care activities since mid-1960s. This is consistent with the acts documented by Robinson and Godbey (2008), who ind in time use data over evidence or convergence in activities across gender. 4 The tendencies or men and women s time spent on home production to converge have been at work throughout the last century, as documented by Ramey (2009). The importance o considering intrahousehold allocations can be seen in the right panels o igure 4. While home hours o married men have increased over time, single men s hours o home production are constant, ater an initial change between 1965 and This is consistent with the explanation I propose. Single men, who are without a emale partner, are not aected by rising emale wages, whereas married men spend more hours working at home a trend in US data that has not received much attention by researchers yet, with the notable exception o Knowles (2013). 5 Single women, on the other hand, spend less time working at home, but the decline is not as pronounced as or 4 They also report that average parental hours spent on child-care per child has been roughly constant. Ramey and Ramey (2010), on the other hand, include time spent teaching children and document a rise since the mid 1990s, especially or college educated parents. 5 Burda, Hamermesh, and Weil (2007) study time use data across various developed countries, and ind that total work, the sum o home and market work, is virtually the same or men and women. For the US, Ramey and Francis (2009) report that total work o men and women has been constant throughout the 20th century. 4

5 All Men and Women Married Single Year Year Year Male Female Male Female Male Female Figure 4: Male and emale home hours worked This graph shows average (per capita) weekly hours spent on nonmarket work and basic child care or men and women aged 20 to 60. The data is taken rom Aguiar and Hurst (2007). Per capita hours worked are computed as the simple average o the average hours worked in 5-year age bins, in order to take out a potential composition eect due to changing cohort sizes. married women, whose husbands devote more o their time to home production. 6 Distinguishing household types is also important in the ertility data. Figure 2 shows that the total ertility rate (let panel) has been rather stable over the last orty years (and even increased slightly) and that the number children ever born (CEB) or women o age (right panel) has been lat or the last 25 years. 7 However, the trends in CEB or ever-married and never-married women dier substantially. While overall ertility has been lat, there has been an increase in the ertility o never married women. However, the rise in single women s ertility is not the sole driver o the lattening out in aggregate data. To the contrary, the total igure o children ever born ollows more closely the number or ever-married women, which has been stable or the last 25 years. Understanding the evolution o married ertility is the ocus o this paper. I present a model matching the observed patterns o married ertility and hours worked through an exogenous decrease in the gender wage gap. As the main interest o this paper 6 In the Aguiar and Hurst (2007) data marital status is deined in a legal sense, and it is not possible to disentangle cohabiting rom other singles. In igure A-4 o the online appendix I show that home hours have changed more or singles with children (who are more likely to be cohabiting) than or singles without children. 7 CEB has lattened out only rom 1990 on, since it shows lie-time ertility o women aged 40-44, whereas TFR is a measure o ertility across women o all ages at a point in time. Since TFR is computed by adding up the age-speciic ertility rates o all women (in their child-bearing years), it is potentially subject to a tempo eect, which is that an increase in the age o childbearing mechanically lowers the computed total ertility rate. CEB is not aected by changes in the timing o ertility, as it measures actually completed ertility. In general, TFR and CEB ollow similar trends though. 5

6 is in understanding the recent time series o ertility, starting with the irst available time use data in 1965, I abstract rom many actors that might have been important or the ertility decline in earlier periods, such as a reduction in inant mortality or qualityquantity considerations in the wake o technological changes during the 19th century (Becker, Murphy, and Tamura (1990), Galor and Weil (2000)). However, in an extension I show that the model under parameters calibrated to perorms well in replicating the whole 20th century data. Moreover, since the ocus is on the time series, I do not model heterogeneity within a cohort. Nonetheless it is worth noting that Hazan and Zoabi (2015) ind in recent U.S. cross-sectional data that ertility is not monotonically declining in women s education, but U-shaped. For each cohort I consider a representative household that maximizes the sum o a male and a emale member s utilities. Both members can work in the market and at home. Due to complementaries in home production the household is not perectly specialized. When women s wages are lower than men s, they contribute more time to home production. A rise in emale relative wages directly increases emale labor supply and lowers emale home production, whereas more male time is devoted to home activities and less to market work. Initially, when the gender wage gap is large, there is an overall drop in home labor and a couple devotes less time to having and raising children. However, when the gender wage gap is airly small and shrinks urther, ertility stabilizes and might even increase. The reason or this dierential reaction to improvement in emale relative wages is in the degree to which households are specialized. Initially, a husband s labor supply was much higher than his wie s, and his time spent working at home much lower than hers. As a consequence, the opportunity cost o having children was largely determined by her wage (which was low compared to his), and when emale wages improved the cost o having children increased substantially. In this situation the substitution eect o higher emale wages dominates the positive income eect, leading to a all in ertility. But as relative wages become more equal, the increase in the opportunity cost o having children becomes smaller, since the cost o reallocating home production rom the wie to the husband alls. As a consequence, the substitution eect weakens with the narrowing o the gender wage gap. On top o this, with the improvement in the wie s earnings, a couple can acquire more parental time-saving inputs, and the larger her labor supply already is, the more the household gains when her wage increases. Both the rise in male home labor and the higher use o parental time-saving inputs into home production is what I ind key in explaining why the ertility decline ended. First I use a simpliied model to show analytically that the combined substitution eect becomes weaker than the income eect beore the gender wage gap has ully closed i complementaries between both parents time spent in the household are suiciently strong and i marketization is possible. Then I calibrate the ull model to study the quantitative implications, which suggest urther that the degree o marketization is important or the 6

7 timing o the ertility stabilization. To the best o my knowledge, the only previous paper that has noted the lattening out in total ertility rates and inormally suggested an explanation in terms o increased male home production is Feyrer, Sacerdote, and Stern (2008). 8 My paper ormally models and quantiies the endogenous response o male and emale hours and their implications or ertility. In a working paper Regalia and Ríos-Rull (1999) discuss the stability in ertility and use a rich model to assess the eects o changes in wage premia, including the gender wage gap, or ertility decisions. My paper diers along two margins. On the one hand, ocusing on married ertility, I assume that the number o children is a joint decision the couple takes. On the other hand, since both parents can contribute to home production, the implicit cost o having children is endogenous and a unction o both parents wages. Other papers that have studied the implications o the decline in the gender wage gap or both male and emale hours include Jones, Manuelli, and McGrattan (2003) and Knowles (2013), who highlights the rise o male home production and on whose work I build. In a calibrated lie-cycle model Attanasio, Low, and Sanchez-Marcos (2008) ind that a reduction in the gender wage gap alone cannot explain the change in labor supply o young mothers and that also a all in the cost o children is needed. However, none o these papers have explored the implications or ertility. In my model, the rise o men s time spent on home production in response to improved emale wages endogenously lowers mothers cost o having children and the ertility decline ends. 9 Galor and Weil (1996) present a uniied ramework to explain the rise in emale employment and the all in ertility that we observed until the early 1970s. Since their mechanism links ertility decisions to the market value o women s disposable time 10, it cannot explain why ertility stopped alling when emale wages continued improving. My work thereore highlights that the existing literature that assumes perect specialization within amilies, such that women shoulder all o home production or child care, has overlooked important implications o intrahousehold allocations. 2 Cross-sectional Time Use Data To support the view that the reallocation o home production rom women to men is linked to relative wages, I use data rom the American Time Use Survey (ATUS) or the 8 Feyrer, Sacerdote, and Stern (2008) explain the increase in male home hours with intrahousehold bargaining. I show that bargaining is not necessary; a change in relative wages per se implies not only a reallocation o work in the market, but also at home. 9 Attanasio, Low, and Sanchez-Marcos (2008) report that data on child care costs are not consistently available beore For that reason, they calibrate the drop in the price o market child care using their model. Since they assume that only mothers provide domestic child care, they are likely to overstate the required reduction in child care costs as they do not take into account responses in athers time. 10 Other papers linking ertility decisions to the market value o women s disposable time include Greenwood, Seshadri, and Vandenbroucke (2005) and Doepke, Hazan, and Maoz (2015), besides others. 7

8 Table 1: Cross-sectional regressions o time spent on home production (1) (2) (3) (4) (5) (6) (7) (8) Nbr. o Kids in Hh (7.01) (8.00) (7.34) (8.15) (7.22) (8.35) (7.98) (8.06) (Nbr. o Kids in Hh) (2.02) (2.35) (2.11) (2.39) (2.03) (2.40) (2.34) (2.36) Own market hours (0.28) (0.36) (0.28) (0.36) Spouse s market hours (0.30) (0.39) (0.30) (0.39) Own wage (0.28) (0.29) (0.29) Spouse s wage (0.22) (0.23) (0.23) Own earnings (0.01) (0.01) Spouse s earnings (0.01) (0.01) Own age (0.72) (0.81) Spouse s age Constant (0.72) (0.82) (17.51) (23.96) (8.41) (8.96) (23.07) (18.96) (22.53) (9.16) Race Dummies No No No No Yes Yes No No Sample All Empl All Empl All Empl Empl Empl Observations R Standard errors in parentheses p < 0.10, p < 0.05, p < 0.01 year 2011, provided by Hoerth, Flood, and Sobek (2013). The ATUS provides data on the time use o only the respondents themselves but not o other amily members. It is thereore not possible to directly study how home production chores are split within a household. However as the ATUS respondents are sampled rom individuals who completed the Current Population Survey (CPS), it is possible to link the respondents time use data to the previous CPS data which include labor market characteristics o both the respondent and their partner. The lag between the ATUS and CPS interviews spans between 2 and 5 month, with an average o around 3 month. I restrict the sample to married or cohabiting couples in which both partners are o age 21 to 65, and drop households with amily businesses. To compute hourly wages or both partners, I divide weekly earnings by usual hours worked. As in igure 4, I deine time spent working at home as the sum o home production, purchasing goods, and caring or household members. 11 Then I regress individuals home work on their own and their partner s labor market characteristics, which come rom the earlier CPS sample, and a set o other controls including number o children in the household. Denoting individual 11 In the ATUS, the respondents answer how many minutes they have spent on the various activities over a 24-hour period. 8

9 i s time spent on household production by h i, I estimate h i = β 0 + β 1 b i + β 2 b 2 i + γ o Xi o + γ s Xi s + error i where b i are the number o children living in the household, and Xi o individual i s own and Xi s their spouses labor market characteristics, such as market hours worked, hourly wages, and weekly earnings (all taken rom the previous CPS). Table 1 shows the results or dierent speciications. The results o column 1 imply that individuals who work longer hours in the market tend to spend less time on home production. However, ceteris paribus, they spend more time on home production i their partner works more. The relationship is robust to include wages or age as urther covariates, as seen in columns 2 and 5 respectively. Moreover, columns 3, 4 and 6 show that individuals tend to devote less time to household production i they have higher hourly wages or weekly earnings, but more i their spouse earns more. Using wages in the analysis restricts the sample to households in which both the respondent and their spouse are working and wages are observed. Due to a potential sel-selection into labor market participation, the result or the interdependence between home production time and spouses wages are potentially not representative or the overall population. However, the results or the relationship with market hours worked (column 1) and with earnings (column 3) include households in which a spouse is not in employment. Excluding observations with zero market hours or earnings to restrict the sample to couples in which both are employed, gives the results in columns 7 and 8. For the correlation between home hours and market hours, or earnings, whether to include individuals out o employment, or not, does not matter much or the results. To summarize, table 1 highlights that married individuals time allocated to home production depends not only on their own labor market characteristics, but also on their spouses. 12 These results are in line with the conjecture that higher emale wages, or higher emale market hours, are associated with a reduction in women s time in home production and an increase in their husband s home hours. To the extent that better educated women earn higher wages, this is consistent with Hazan and Zoabi (2015) s inding that married men s time spent providing childcare at home increases with mothers education. However Hazan and Zoabi do not explore the implications o athers time or the ertility decision. In the next section I propose a model in which both parents time matters, and provide a link between the rise in male home production and the lattening out o ertility over time. 12 Splitting the regression sample by the gender o the respondent does not change the general pattern; see online appendix A.4. 9

10 3 The Model 3.1 Assumptions To explain the time series o ertility as well as o male and emale time allocations or married individuals, I propose a parsimonious model. For tractability I assume that each cohort can be represented by a representative married couple household that decides on how to allocate male and emale time and on a continuous number o children to have. In the main text I ocus on a version in which the couple lives or one period only, such that their choices are static. In the online appendix I present a lie-cycle version o this model and show quantitatively that the general conclusions regarding ertility and time allocations remain the same. The advantage o the simple model presented in the main text is that it allows or an analytical characterization o the mechanism that links ertility to women s relative wages Agents and Households All agents, men and women, derive utility rom the consumption o a market good (c), rom having children (b), and rom leisure, which is the time spent neither working in the labor market (n) or at home (h). Each individual is endowed with one unit o time. Agents dier in terms o their gender (g {m, }) and their age (j). I assume that all economic active men and women live in couple households, ormed by one man ( husband ) and one woman ( wie ). I apply a model o collective household behavior, as introduced by Chiappori (1988). The male and the emale partner have their own preerences, and derive elicity u m and u respectively. Since they have children together, they solve a joint maximization problem. In particular, the couple household solves a Pareto program with relative weight θ attached to the husband s and 1 θ to the wie s utility. As will be seen later, under the model assumption made the spouses relative time spent on home production is independent o the bargaining weight, but solely based on comparative advantage. As a consequence, θ has no eect on the implicit cost o having children. I thereore restrict attention to the unitary model with θ = throughout Fertility Child-care requires more home production (x), which could be done by the ather, the mother, or both. 13 A urther input to home production are goods acquired in the market (e). Although I reer to this home-labor saving input as home appliances, in a broader sense this could include paid domestic help, such as hiring nannies. I assume that the 13 The ormer assumption is similar to Erosa, Fuster, and Restuccia (2010), the latter is as in Knowles (2007) and similar to Greenwood and Seshadri (2005). 10

11 amount o the home good needed is (1) x(b) = κ 0 + κ 1 b κ 2 with κ 0, κ 1, κ 2 > 0, which implies x (b) > Household Production Following Olivetti (2006) and in particular Knowles (2013), I assume that the home good is produced using a technology that is consistent with substitution among household member s time and home appliances according to (2) x = e γ H 1 γ, with 0 γ < 1, where the home labor input is (3) H = ( z m h 1 m ) + z h 1 1 1, and z m and z are the male and emale home labor productivities. The parameter > 0 is the inverse o the elasticity o substitution between male and emale inputs. For inite, this deinition o H as a CES aggregator allows or substitutability between male and emale inputs to home production as long as both z m > 0 and z > 0. With γ > 0, this technology allows, to some degree, or a marketization o inputs to home production. 14 Moreover, this home production unction is consistent with the indings by Ramey (2009). From regressing the home capital-labor ratio on appliances prices, Ramey inds that the production unction is more or less consistent with a Cobb- Douglas technology over home capital and labor Preerences Agents derive utility rom consuming the market good, rom having children, and rom leisure time. In particular, assume that preerences are additively separable and given by 15 (4) u(c g, n g, h g, b) = log(c g ) + φ l (1 n g h g ) 1 η 1 1 η + φ b b 1 σ b 1 1 σ b where c g, n g, and h g are speciic to a spouse g {m, }, but the number o children b is common to the couple. 14 In a structural transormation ramework, Ngai and Pissarides (2008) study substitutions between home and market production, but do not distinguish between male and emale labor. 15 Adding utility rom children in this additive orm is generalizing Galor and Weil (1996) and Greenwood and Seshadri (2005), who assume ln(b). 11

12 Note that in this model setup individuals derive utility rom leisure, i.e. the time spent not working in the market or at home, and both spouses contribute to home production due to complementaries. 16 I assume that there is a subsistence level in consumption o the home produced good, which is increasing in the number o children living in the household (b). As a simpliying assumption, ollowing Knowles (2007), agents do not derive any urther utility rom home production, and this constraint will be binding, x = x(b). 3.2 Couple Household s Optimization The representative couple household maximizes the sum o male and emale utilities, by choosing male and emale hours o home production and o market labor supply and how many children to have, subject to a budget constraint, the home production requirement (1) and the home production technology (2). The household can earn a labor income w m n m + w n, where w m is the male wage and w the emale wage, which can be used to purchase consumption goods, c m and c, and time-saving inputs to home production, e. The representative couple s maximization problem is thereore given by (5) max c m,c,b,e n m,n,h m,h θu m (c m, n m, h m, b) + (1 θ)u (c, n, h, b) subject to the constraint set: (6) (7) (8) (9) c m + c + e = w m n m + w n x(b) = e γ ( z m h 1 m n m + h m 1 and n + h 1 + z h 1 n g 0 and h g 0 or g {m, } ) 1 γ 1 where (6) is the household s budget constraint and (7) the relationship between the required amount o the home good and the home production inputs. The inequalities in (8) constrain the male and emale total time working in the market and at home to be no larger than the time endowment o 1. As the marginal utility o leisure goes to ininity when leisure goes to zero, these constraint will be slack or both spouses. The inequalities in (9) are non-negativity conditions on the spouses time spent working in the market and at home. As the home production unction (3) eatures complementarity between male and emale home hours, optimality requires h m and h to be positive and the associated constraints never bind. The non-negativity constraints on market hours could in principle 16 In an earlier working paper version, Siegel (2012), I had proposed an alternative to generate imperect specialization within the household based on preerences eaturing imperect substitutability in the disutilities rom working at home or in the market. Both models generate qualitatively the same predictions or the eects o changing relative wages on the couple s time allocation and on the ertility choice. 12

13 bind, but since I ocus on a representative couple, I will choose parameter values or which both n m and n are positive (to capture the trends in igure 1). As a consequence all time constraint will be slack in equilibrium and I will ocus throughout on interior solutions. Also note that since I am ocusing on a representative couple, the household can choose any non-negative continuous quantity o children b. Combining (7) with (6), the eective budget constraint to the household s maximization problem can be written as (10) c m + c + ( zm h 1 m x(b) + z h 1 ) 1 γ 1 Combining these optimality conditions implies (11) c = 1 θ c m θ and or the optimal time allocations 1 γ = w m n m + w n (12) (13) (14) (15) h m = ( ) 1 zm w m ( ) 1 z ( 1 γ γ ( 1 γ h = w γ ( φl c m n m = 1 h m w m ( 1 θ n = 1 h θ ) γ [ z 1 ) γ [ z 1 ) 1 η mw mw ) 1 φ l c η m w 1 m 1 m + z 1 + z 1 1 w 1 w ] γ 1 1 x(b) ] γ 1 1 x(b) The optimal ertility choice satisies (16) φ b b σ b = θ ( ) γ ( ) 1 γ 1 1 [ 1 1 z mwm + z 1 1 c m γ 1 γ w ] (1 γ) 1 }{{} C x(w m,w ) x (b) Equations (10) to (16) ully characterize the couple s utility maximizing choices. When choosing a ertility plan, the couple is outweighing beneits and costs rom having children. The let-hand side in (16) is the marginal beneit o having an additional child, which at the optimum has to equal the marginal cost (in utility terms), the right-hand side. The term ( ) γ ( ) 1 γ [ C x (w m, w ) = γ 1 γ z mwm +z 1 1 w ] (1 γ) 1 is the minimal cost to produce one unit o the home good x. The marginal cost o having more children lies in the need or more home production. To increase home production, the couple devotes more time to home labor and uses more purchased inputs. Both adjustments reduce consumption o the parents. Notice that C x (w m, w ) is increasing in the male and emale wage rates as 13

14 these represent the opportunity cost o male and emale home labor; devoting more time to home production lowers labor income and thereby reduces consumption. Since how costly it is to orgo consumption depends negatively on the original consumption level, the right-hand side o (16) is decreasing in c m. Ceteris paribus, higher wages imply an increase in the cost o having children, whereas higher consumption implies a lower cost. Also note that C x is independent o the bargaining weight θ. The reason or this is that the spouses relative time spent on home production is solely based on comparative advantage. From (12) and (13) it can be seen that the optimal ratio o male to emale home hours is hm h = ( ) 1 z m w z w m 3.3 Eect o Higher Female Wages. When emale relative wages improve, the couple inds it optimal to allocate a larger share o home production to the husband. Holding the number o children and the required amount o the home good constant, this leads to a reallocation o men s time rom market to home labor, and or women rom home to market labor. This impacts the marginal cost o having children in (16). Higher emale wages directly increase the opportunity cost o home production, but they also improve labor income and thereby consumption. While the ormer channel tends to reduce ertility as the couple substitutes rom having children towards consumption o market goods, the latter channel tends to increase it as the household experiences higher income and children are normal goods in the utility unction. The irst eect is the substitution eect and the second one the income eect. To derive a condition under which the substitution eect dominates and ertility alls, use the parametrization o home production requirement (1) to write the optimal number o children as (17) b = φ b c m κ 1 κ 2 θ γγ (1 γ) 1 γ[ z 1 mw 1 m 1 w ] (1 γ) 1 + z 1 }{{} =1/C x(w m,w ) 1 σ b +κ 2 1 Note that (17) implies that, holding consumption constant ertility alls in response to higher wages (due to the substitution eect) only i σ b + κ 2 1 > 0, which is a restriction on parameters I will assume throughout. Consumption itsel is pinned down by the budget constraint (10) given the optimal choices according to (11) to (15). In general this gives a system o two non-linear equations which has no closed orm solution. On the one hand, higher emale wages increase the cost o home production, which induces a substitution eect that implies a reduction in ertility. But on the other hand, they also have a positive income eect, which implies an increase in consumption and in ertility. When C x increases by more than c m, ertility alls. In the next subsection I 14

15 ocus on a special case, or which one can derive a closed-orm solution or the couple s optimal number o children. I show under what conditions ertility alls in response to higher emale wages, and when it does not. In the subsection thereater I show that the drawn conclusions generalize Special Case: σ b = 1 and φ l = 0 Consider the special case o σ b = 1 and φ l = 0. This means that agents have log utility rom children and do not value leisure time, such that n g + h g = 1 or g {m, }. These assumptions allow to derive a closed orm solution. The optimal ertility choice is b = ( ( φb 1 κ 1 κ 2 + φ b (w m + w )γ γ (1 γ) 1 γ[ z 1 m w 1 m + z 1 1 w ] (1 γ) 1 κ 0 )) 1 κ 2 b The sign o w, which captures how the optimal ertility choice responds to increasing emale wages, is given by (18) z 1 1 w = γz 1 + z 1 1 m wm 1 w (1 γ)z 1 + w m ( ( zm w m 1 w ) 1 (1 γ) ( z (1 γ)z 1 w mw 1 ) w ) 1 No marketization: First, consider a scenario in which there is no marketization o home production by setting γ = 0. Then b w < 0 i and only i < z z m. Hence, higher emale wages reduce ertility when women s relative wages are low compared to their relative productivities at home. w w m The reason is that when women s comparative advantage lies in home production, the emale share in the cost o children is larger than the emale share in total income. This implies that higher emale wages increase the cost o having children by more than household income. As a consequence, the substitution eect dominates the income eect, and ertility alls. No male home production: Next, consider a scenario with marketization but without men contributing to home production, which is nested as the optimal choice b when z m = 0. Here, w < 0 i and only i < 1 γ. Thus, when the degree o γ marketization, as measured by γ, is suiciently low, higher emale wages reduce ertility. When the usage o time-saving inputs is limited, the cost o home production increases by more than income. Intuitively this happens since there is a lack o possibilities to exploit the beneits rom higher emale wages. Since there is no substitution with men s time and substitution with purchased inputs is limited, women s home hours cannot all much; as a consequence her labor supply and the household s income do not increase much. As a result, the substitution eect is stronger than the income eect. I one assumes that in the long-run emale and male wages are equalized, a suicient condition or ertility to always all when emale wages go up is γ < 1. Hence, i there is no male home production 2 15 w w m

16 and the degree o marketization is suiciently low, ertility always alls when women s wages improve. With marketization and male home production: When γ > 0 and z m > 0, and the home production technology allows both or marketization and or substitutability between male and emale time, b w (19) γ w w m + < 0 i and only i ( zm z ) 1 ( w w m ) 1 < 1 γ Since the let-hand side is increasing in w w m, whereas the right-hand side is constant, this inequality is satisied i and only i w w m is suiciently low. This means that improvements in emale wages reduce the optimal number o children only i emale relative wages are suiciently low, echoing the conclusions drawn above. Notice that this inequality is more likely to hold when men s relative productivity in home production is high. To understand how the degree o substitutability between male and emale time aects this condition, consider irst a scenario where men and women have equal productivities at home (z m = z ). In this case, the let-hand side o (19) increases in or w w m < 1. This implies that or larger, i.e. when there are stronger complementarities between male and emale time in home production, the ertility decline ends at a lower emale relative wage threshold. When male and emale home productivities dier, there is an additional eect through the term ( ) 1 z m z. When z m < z, which is a reasonable assumption or some child care tasks, this term depends positively on, resulting in an additional channel through which the let-hand side o (19) increases in. 17 This implies that the larger the complementaries between male and emale time at home, the earlier the ertility decline ends in response to improvements in women s relative wages. With larger complementaries, the household gains more rom increasing the husband s home hours when the wie lowers hers; as a consequence the home production cost, C x, increases less when w rises. The condition or the ertility decline to end is also more likely to be satisied when the degree o marketization is higher. A larger γ means that the additional margin, the possibility to substitute between parental time and purchased inputs, is more important, which weakens the strength o the overall substitution eect on ertility relative to the income eect. From inequality (19) one can derive a suicient condition that ensures a bottoming out o ertility. Fertility stops alling, and potentially starts increasing, beore emale and male wages are equalized, only i (20) ( zm z ) 1 > 1 2γ 17 When z m > z and men are more productive than women in home production, the opposite result emerges and a larger reduces ( ) 1 z m z, making it less likely that inequality (19) is satisied. 16

17 To summarize, ertility alls in response to higher emale wages only i the gender wage gap is large. Moreover, both the substitution o emale time spent in home production with male time as well as the substitution with inputs purchased in the market are important. The ertility decline ends beore the gender wage gap is ully closed, when men s inputs to home production suiciently well complement emale time and when there is a suicient degree o marketization. Obviously, once ertility stops alling, it in general increases. Whether the model implies that overall ertility looks rather stable or more like a pronounced U-shape, is a quantitative question whose answer depends on other parameters, and on which I return in section Log-Linearization o the General Case (σ b 1) To study the eect o higher emale wages on the ertility decision in the more general case with σ b 1, I rely on a irst-order approximation. For tractability I continue to assume φ l = 0, implying n g + h g = 1 or g {m, }. Given the optimal choices or emale consumption and or the time allocations as a unction o male consumption, according to (11) to (15), the budget constraint (10) implies (21) ( ) γ ( c m 1 1 θ + γ 1 γ ) 1 γ [ z 1 1 m 1 w ] (1 γ) 1 mw + z 1 } {{ } C x(w m,w ) x(b) = w m + w which pins down c m or a given number o children b. The household chooses b optimally according to (17). These two equations ully characterize the optimal choices. As there is no closed orm solution, I log-linearize the two equations. In particular, I consider the log deviations o a variable between this period and the previous one. To keep the notation simple, I denote previous period s values by a tilde and log deviations between two periods by a hat; or a generic variable Z, the previous period s values is Z = Z t 1, and the log deviations between the two periods is Ẑ = log(z) log( Z) Zt Z t 1 Z t 1. The log-linearization o the optimality condition or the ertility choice (17) gives 1 ( ) b = ĉ m σ b κ 2 1 Ĉx and o the reduced-orm budget constraint (21) Ĉ x + x ( b) b w m ŵ m + w ŵ ˆb cm = θ ĉm b w m + w cm θ where Ĉ x = z 1 1 m w m 1 γ + z 1 w 1 ( 17 ) z 1 1 m w m ŵm + z 1 1 w ŵ

18 Solving these equations or ŵ m = 0, gives or b (the log change in ertility) as a unction o ŵ (the log change in women s wages), b = 1 cm θ (σ b κ 2 1) w m+ w cm θ + x ( b) 1 w m + w cm θ w 1 (1 γ)z w 1 z 1 1 m w m + z 1 w 1 ( w m + w ) ŵ The sign o b ŵ is thereore given by the term in parentheses, which can re-written as 1 1 w z 1 1 m w m + z 1 w 1 ( z 1 1 m w m + z 1 w 1 ( w m + w )(1 γ)z 1 w As the terms outside o these parentheses are necessarily positive, the crucial term is the one inside, which is the very same as in condition (18). Thereore, the conclusions drawn in subsection above generalize to cases with σ b 1, as long as σ b > 1 κ 2 (which is the parameter restriction ensuring a normal substitution eect). Higher emale relative wages lower ertility when the gender wage gap is large, but the ertility decline might end beore the gap is closed, depending on the properties o the home production technology. In the quantitative model calibrated in the next section, I ind that these results also go through when allowing or φ l 0. ) 1 4 Calibration To quantitatively assess the eect o the narrowing o the gender wage gap on ertility, I calibrate the model. I choose parameters such that the model replicates in 1965 eatures o the data or married couples. I target married men s and married women s hours worked, both at home and in the market, and an indicator or married ertility. All parameters are time-invariant, and the only exogenous change over time is in the gender wage gap. As the Aguiar and Hurst (2007) data is available once per decade, I study the choices o representative couples born ten years apart. Below I irst discuss what data is targeted and then the choice o parameters. I calibrate the model in the ollowing order: the home production unction, the required amount o the home good, and the preerence parameters. 4.1 Targets To construct the targets, I need to map the hours worked data o igures 1 and 4 into the model. Since the model is about the decisions o prime-aged couples, I restrict the sample to married individuals o age 20 to 60. In the model, agents have a time endowment o one, which they can split between market work, home work, and leisure time. Assuming that 18

19 people need 8 hours o rest a day, 2/3 o time is discretionary. Hence to map the weekly data rom Aguiar and Hurst (2007) into my model, I divide all hours by (2/3 24 7). For market hours worked, I use data rom the Panel Study o Income Dynamics (PSID) rom 1968 to 2007, which I average over 10 year periods. 18 Since in the PSID market hours worked are given per year, I divided these by (2/ ). Not to conound trends in market and home hours with potential composition eects that could be due to demographic changes, I irst compute average hours or individuals at dierent ages in 5 year bins, and then take the simple average over the bins, which gives every age group the same weight. As the theory I propose applies to couples, the model s implications are or married ertility. Since the total ertility rate is constructed by age-speciic birthrates, it does not allow or a break-down into unmarried and married ertility. I thereore use children-ever born to women at age 40 44, or which there is data by marital history (shown in the right panel o igure 2), published by the United States Census Bureau in the Fertility Data Historical Time Series 19. This is an indicator o completed ertility. However, most couples have children at an earlier point in their lives than at age (and thereby take a decision under earlier wages). To construct an indicator or married ertility that corresponds to the model, I shit the series o children-ever born. I take the average number o children an ever-married women at age ever had, and shit it back by 16 years, since the mean age o a mother when giving birth is in the data. 20 The drawback o this indicator or the married ertility choice is that the constructed series ends early; the model s predictions cannot be evaluated against constructed data or 2005 (yet). However, one advantage o using this measure o completed ertility over other ertility indicators, such as the total ertility rate, is that it is not conounded by changes in the timing o births. For the gender wage gap I take the ratio o median emale to median male wages in the PSID, shown in igure 3. To eed it into the model, where a period corresponds to 10 years, I take simple averages. 4.2 Home Production Technology I set the elasticity o substitution between male and emale time spent in home production to replicate the rise in the ratio o married men s home hours to married women s observed in the data. In the model, equations (12) and (13) imply or relative male home hours hm,t h,t = ( z m w,t ) 1 z w m,t ; their ratio betweens periods T and 0 is thereore h m,t /h,t h m,0 /h,0 = 18 I discuss the advantages o using data rom the PSID, rather than rom the CPS, and compare the data rom these two surveys in section A o the online appendix The average age o a mother (total over all birth orders) was 24.9 in 1968 and 27.4 in 2003, according to Vital Statistics o the United States, 2003, Volume I, Natality. The average over is

20 ( w,t /w m,t ) 1 w,t /w m,t. To match the change in married men s relative home hours over 1965 to 2003 rom 0.25 to 4, given the change in emale relative wages rom 7 to 0.78, this requires or the inverse elasticity o substitution = In turn this implies or the relative home productivities zm z = ( h m,1965 h,1965 ) w m,1965 w,1965. Restricting z m + z = 1, men s productivity in home production ollows as z m = and women s as z = 04. In the literature the range o estimates or the labor share in home production is very wide. Studies that include housing as capital or equipment used or home production typically ind a relatively low value, close to the one o market production, e.g. Greenwood, Rogerson, and Wright (1995), while Benhabib, Rogerson, and Wright (1991), who exclude housing, estimate a very high value o In my model, the need or home production arises at the margin only rom having children (in the household) and does not correspond closely to either study. Since parents can acquire home production inputs in the market, such as hiring nannies or paid domestic help, the share o time-saving inputs acquired in the market, γ, should be higher than the Benhabib, Rogerson, and Wright (1991) value. As a benchmark I consider an intermediate value o γ = 0.29, but I conduct a series o robustness checks in section Home Production Requirement The required amount o home production (1), x(b) = κ 0 + κ 1 b κ 2 is calibrated in two steps, conditional on the home technology parameters pinned down earlier. First I choose κ 0 to ensure that the level o home hours are matched by the model. Second, I choose the additional home production required when having children, κ 1 b κ 2, to replicate some cross-sectional variation in home hours. While the parameterization o the home production technology ensures that relative male home hours are replicated by the model, I choose κ 0 to match in 1965 the level o male time spent on home production (as a raction o the time endowment o one). This yields κ 0 = To calibrate by how much the need or home production increases due to children, I make use o the observed variation o married men s and women s home hours against the number o children in the household. In the model, equations (12) and (13) im- ) + z h 1 1 ( ) γ [ = z mwm + ply or aggregated parental time input (3) ( z m h 1 m z 1 1 w ] γ 1 x(b), which can be rearranged to give 1 γ γ (22) κ 1 b κ 2 = ( 1 γ γ ( zm h 1 m + z h 1 ) γ [ 1 1 z mw m + z 1 ) w ] γ 1 κ 0 This relationship can be used to iner the home production requirement rom the Aguiar and Hurst (2007) data, which includes not only time use inormation, but or some 20