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THE EFFECT OF FERTILITY ONLABOUR SUPPLY:A REVISION OF ANGRIST &EVANS’ FINDINGS I. IntroductionThe empirical study of the relationshipbetween fertility and labour supply is crucial for testing the existingtheories that link the family and the labour market. Up to now, most of theevidence found points out a negative correlation between fertility and femalelabour supply, but many of these results are esteemed to be blurry as far asthe problem of endogeneity of fertility is not solved:”… it has proven difficult to find enoughwell-measured exogenous variables to permit cause and effect relationships tobe extracted from correlations among factors such as the delay of marriage,decline of childbearing, growth of divorce, and increased female labour forceparticipation” (Robert J. Willis, 1987 p. 74).

Indeed, there are goodreasons to believe that fertility and labour supply are jointly determined,thus preventing from extracting any causal interpretation: The fact thatfertility is both used in the literature as an explanatory variable and as adependent variable of labour force is compelling. Fortunately, this problem canbe tackled through diverse ingenious instrumental variable (IV) strategies, suchas that of Lundborg, P., Plug, E., and Rasmussen, A. W. (2017), who analyse theeffect of childbearing on labour market outcomes among women with similarworking histories that become mothers forthe first time through in vitro fertilization (IVF).

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Because the success ofsuch treatment can be regarded as a random result from nature, it becomesplausible to compare the labour-market outcomes of those women who effectivelygave birth after the treatment to those of who did not.Similarly, Cristia J. P. (2008)focuses on women who sought for medical ‘advice and testing’ to get pregnant for the first time as well. Indeed, thiscan also be regarded as a hypothetical experiment in which women seeking forhelp are randomly assigned a baby by nature, allowing the author to compare thelabour-market outcomes of those women who were successful in getting pregnant tothose of who were not. In contrast with these two studies, however, in thispaper we will explore the causal relation from fertility to labour supplyfollowing the alternative IV strategy based on the sibling sex mix in families with two or more children put forward byAngrist, J.

D. & Evans, W. N. (1998), with a subset of the same data baseused by the authors, to recreate some of their central estimates andcorroborate their findings.

Although standard householdtheories predict that the labour-market consequences of having a first childare stronger than those of having additional children, this difference in focusallows us to ‘exploit’ the fact that parents prefer a mixed-sex composition oftheir children, instead of having offspring of the same sex. Moreinterestingly, it has been observed that parents who have siblings of the samesex are more likely to go on in having an additional child. And because the sexmix is a randomly assigned factor, given by nature, a ‘dummy’ variable thatindicates whether or not the sex of the second child matches that of the first,same sex, can be used as aninstrument for further gestation or ‘fertility’ among women with at least twochildren (for the sake of simplicity and conciseness, we leave aside the use oftwinning as an alternative instrumentin this paper, which the authors find to deliver similar results). Thus, ourinstrument captures the effect of moving from the second to the third child onwomen’s labour supply.

II. The DataIn this study we use asubset of the same Census Public Use Micro Samples (PUMS) of 1980 data set usedby Angrist, J. D.

& Evans, W. N. (1998), but we leave aside the 1970 and1990 samples originally considered by the authors, who detect a substantialdecline in fertility and an increase in women’s labour supply throughout theperiod. By contrast, we focus exclusively in the causality effect fromfertility to labour supply in one particular period, which is the centralquestion of this work. Additionally, we also ignore the subsample of ‘married’women considered by them; which leaves us only with the analysis of a subset ofthe subsample of all women with two or more children contained in the 1980 PUMS(recall that we are interested only in the marginal effect on labour supply ofmoving from the second to the third child), which consists of 355,356 observations.The variables and theirdescriptive statistics are provided in Table 1, where the covariate of our maininterest is the binary variable Morethan2children (as indicator of ‘fertility’, the endogenous variable), and Samesex is its instrument.

As we willsee ahead, the two components of the latter, 2Boys and 2Girls, arealso shown. Demographic and labour supply variables are also included in thelower half. Notice that among all women of the sample, 40.

19%had a third child.III. Sex-Mix and FertilityWe can model parents’sex-mix preferences and utility in the following way: Suppose a couple hasalready  children, and they decide on the additionalnumber of children they want to have, . Because parents prefer amixed-sex composition of their offspring, having already a same-sex compositionreduces the utility from  and increases, at turn, the marginal utilityof . Thus, under thesecircumstances, parents are more likely to decide to have an additional child. Accordingly, Table 2 reportsdifferent estimations of the effect of ‘sex-mix’ on fertility that reveal thisphenomenon. Recall here, however, that we are only interested in women with two ormore children. Thus, Table 2 shows the relationship between the fraction ofwomen who have a third child and the sex-mix of the first two children.

Specifically,women are divided into four groups according to the sex composition of theiroffspring: two boys, two girls, one boy and one girl, and simply, two childrenof the same sex. The last row displays the difference between the same-sex andmixed-sex group averages.Table 2 allows to infer thatwomen with two children of the same sex are noticeably more likely to go on inhaving a third child than women with one boy and one girl.

Concretely, 43.18% of the mothers with same sex children go onin having a third child, while a markedly lower 37.13%of mothers with mixed-sex children decide to have another one.IV.

Fertility and Labour SupplyA.    Wald EstimatorBecause our Samesex instrumental variable isessentially randomly assigned by nature, we can safely extract a causalinterpretation from the regression of fertility on labour supply. Consider thefollowing bivariate regression model:Where  is labour supply (Workedforpay) or any of our other measures of labour-marketoutcomes described in Table 1, and  is our endogenous fertility measure, Morethan2. As usual, we denote as  our Samesexbinary instrumental variable, and we define the estimator for binaryinstrumental variables (), also called the WaldEstimator, as:In which  is the mean of  for the observations where  is equal to one, and the other terms aredefined in an analogue manner. Here, the numerator captures the relationshipbetween  and , while the denominatorcaptures that of  and . Thus, any effect of  on  is attributable to the effect of  on . This is, the  estimates the average effect of  on  for those women whose fertility () has been affected by thesex-mix () of their offspring.This  can be easily obtained by running a simple regressionof the endogenous variable with the instrument (as we will see, similar to afirst-stage estimation without covariates in the two-stage least-square, 2SLS,estimation framework), to obtain the denominator, and a regression of theoutcome variable and the instrument to obtain the numerator (notice that boththe numerator and denominator are ‘scalars’), and then dividing one scalar overthe other (also, a procedure equivalent to running the second-stage estimationwithout covariates in the 2SLS method).

Effectively, the first column of Table3 reports the components of  separately, showing in the first row thedenominator of the Wald estimate, , where it can be seen thatthe effect of Samesex on Morethan2 is equal to 0.0605 (which is the same as the difference between the same-sex andmixed-sex group averages reported in the last row of Table 2); and in theremaining rows, different estimations of the numerator,  (one for each labour-market outcome),suggesting that indeed, additionally to having more children than women withone boy and one girl, women with two children of the same sex present a lowerlabour supply. Specifically, the Wald estimates reported in the second column,obtained from dividing the numerator by the denominator, indicate that havingmore than two children decreased the supply of labour (the Workedforpay variable) by 13.89 (-0.0084/0.0605)percentage points, weeks worked by 6.

456,hours worked by XX, and labour income by $2,273.666 per year.B.     Two-Stage Least-SquaresEstimationWe now try a different approachto the problem by using the two-stage least-squares (2SLS) estimator.

While theWald Estimator allowed us to identify the effect of fertility on labour supply,the 2SLS estimator allows us to relate our labour-market outcomes (Workedforpay, Hoursweek, and Labourincome)to fertility controlling for a list of other exogenous covariates, whichinclude Age, Age at first birth, Familyincomelog,and Education. However, at this pointwe deviate somewhat from the original authors’ estimation in that we treat Familyincomelog as a covariate insteadof a dependent variable, for it is more likely that the mothers’ family wealthdetermines how prone they are to participate in the labour market, and not theother way around (i.e. if a mother’s family is relatively rich, she might beless urged to work while childbearing than a mother from a poor family). Thisis made evident by the insignificant effects found by the authors when treatingthis variable as an outcome variable. At the same time, we also make emphasison the role of mothers’ education because several theories put forward the ideathat the impact of fertility on labour supply varies with the years ofschooling, and there is some empirical evidence in the same sense, showing thatthe more educated women’s labour supply is more sensitive to fertility than thelabour supply of the less educated women (Gronau, R. 1986).Another advantage of using2SLS is that it also permits us to control for any ‘secular’ additive effectsof childbearing as we use the Samesexinstrument.

Indeed, because Samesexis an interaction term comprising the sex of the first two children, it ispotentially correlated with the sex of either child, which can ultimately be aproblem if the sex of offspring affects in some way parents’ attitude towardsthe labour market (see Angrist, J. D. & Evans, W. N.

(1998) for aproof). Thus, we can add the Boy1st (S1)and Boy2nd (S2) regressorsdescribed in Table 1 to eliminate the possibility of an omitted-variables biasarising from these sources. We can then write the following regression modellinking the labour supply and labour-market outcomes with the endogenous fertilityvariable, , the vector of otherexogenous variables, , and the additive effectsfor the sex of each child, as:Now, the first-stageequation relating the endogenous Morethan2variable to the sex-mix is:Where  is the first-stage effect of the instrument on.

A variant of this approachalso used by the authors exploits the possibility of formulating an over-identifiedmodel by decomposing the Samesexinstrument into two separate indicators: 2Boysand 2Girls. To see this more clearly,realize how we can express our instrument as:Where S1 and S2are, as we know, our indicators for male firstborn and second-born children, Boy1st and Boy2nd (notice how the instrument renders cero if both S1 and S2 are of different sex, and one if they are of the samesex), from where we can extrapolate the two separate instruments: 2Boys, S1S2, and 2Girls,(1-S1)(1-S2), alsoreported in Table 1. This over-identification strategy is advantageous becausewe might expect any bias arising from the so called ‘secular’ effects of childsex on labour supply to be different for each of these two instruments, whilethe effect of childbearing can be expected to be independent of whether Samesex equals to 2Boys or 2Girls. In this formulation,however, since S1i, S2i, S1iS2i, and (1-S1i)(1-S2i) are linearly dependent, wemust drop one of this variables to avoid perfect multi-collinearity problems, inwhich case we choose to withdraw S2i.Hence, the following alternative regression model using the two separateinstruments can be specified: Where the first-stageregression is now:Table 4 reports the resultsof the first-stage estimations for both the just-identified and theover-identified regression models. We can see, on the one hand, that motherswith two children of the same sex are 6.

2% morelikely to have a third one; and on the other hand, that the effect of 2Girls on fertility is higher than thatof 2Boys, suggesting that parents aremore willing to persist in having children until they can have a boy. As forour covariates of particular interest here, we can see that having a higherlevel of education reduces the likelihood of having a third child by 2%, and that greater family wealth slightlyincrements this probability (this can interpreted as mothers from rich familiesbeing less worried about the economic difficulties of childbearing). At this point, however, wewould like to know something about the validity of our instrumental variables.For this purpose, we can perform a test for the strength of our instruments ineach first-stage regression (in both the just-identified model, where Samesex instrument was used, and in theoveridentified model, where we used the 2Boysand 2Girls instruments). When testingfor the ‘strength’, we are actually interested in the correlation between theendogenous variable Morethan2 andeach of our instruments (indeed, the ‘first-stage condition’ means that theinstruments(s) considered should bring some knowledge to the endogenousvariable). Effectively, this correlation is measured by the first-stage Partial R2, where in the case of thejust-identified model is equal to 0.0043,and in the overidentified model is 0.0044.

This result is interesting because it proves a small correlation, although thefirst-stage F-statistic in each ofthe models is sufficiently large to reject the null-hypothesis that theinstruments are ‘weak’ (seeappendix 2).Having checked this aspect of our instruments’validity, we can now run the regression of the effect of  on thedifferent labour-market outcomes using both the just-identified and theover-identified models. Simultaneously, we conduct simple ordinaryleast-squares (OLS) regression to compare the results and have an insight ofthe magnitude of the bias arising from our endogenous variable. The results arepresented in Tables 5 through 7.As can be seen, when usingthe single Samesex instrument (thejust-identified model), having a third child reduces the probability ofparticipating in the labour market by around 11.

5 percentagepoints, the number of hours per week worked by 8-9per year, and the amount of earnings by more than $1,745.Likewise, when using the over-identified model, the probability ofparticipating in the labour market falls by around 10.6 percentage points, the number of hours worked by 7-8, and the earnings by almost $1,610.

Thus, although the first-stage estimationssuggested that mothers of two boys are less likely than mothers of two girls tohave a third child, the 2SLS estimates in Table 5 allow us to infer thatseparating our Samesex instrumentinto its two components doesn’t change the magnitude of the coefficients very muchand, in the end, the same conclusions are reached. Finally, the bias arisingfrom the endogeneity of our fertility variable (Morethan2) clearly overestimates theeffects of fertility on the labour-market outcomes, as can be appreciated whencomparing the OLS with the 2SLS results described above. As for our othercovariates of interest, we see that both education and family income yield theexpected results: The former has a positive effect in all of the threelabour-market outcomes, incrementing the labour supply (Workedforpay) by 3.1%, hoursworked by 5-6, and labour income by $795-$798. Conversely, family income reduceslabour supply by 2.

7%, hours worked by 7-8, and the mothers’ labour income by $757-$758 (recall these are all mothers of atleast two children, and they are more likely to abandon the labour-market ifthey have the economic support from wealthy relatives).V. Concluding RemarksWe have seen that both theWald and the 2SLS estimates consistently confirm the thesis that increasingfertility (moving from the second to the third child) reduces women’sparticipation in the labour market. Effectively, on the one hand, these resultsare in line with those found in the cited literature, but they appear to beless ‘harsh’.

For example, while Lundborg et. al. (2017) calculate a reductionof working hours per week of 5.9, our estimates show a reduction of only 4.5 working hours, and as Cristia J.

P. (2008) finds that’having a first child younger than one year reduces female employment by 26percentage points’, our calculations only account for a 10.6%-11.5% decrease. This is due to the fact that our IVstrategy focuses on the effect of the third-born child instead of thefirst-born child, which is expected by the standard household theories to havea lower impact on the labour supply of women.

On the other hand, however, the effectscalculated in this paper (big or small) shouldn’t be over-dimensioned. Ineffect, referring to the calculations made by Angrist, J. D. & Evans,W. N.

(1998) in their ‘Table 1’, we can observe that the probability of having more than two childrendecreased by around 15.7 percentage points between 1970 and 1980; while, at thesame time, the participation of women in the labour market rose by about 13.2percentage points in the same period. If we then use the 2SLS estimation of theimpact of fertility on labour supply (Workedforpay)using the Samesex instrument reportedin the upper part of the central column of Table 5 (-0.115),we can deduce that declining fertility accounted for an increase inlabour-market participation of roughly 1.8 percentagepoints (0.157×0.115).

Thus, our study also letsus conclude that, although fertility has a significant negative impact onlabour supply, the increase in the labour-market participation rate has been sosubstantial that declining fertility only accounts for a small fraction of thewhole change. 

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