## THE between fertility and labour supply is crucial

THE EFFECT OF FERTILITY ON

LABOUR SUPPLY:

A REVISION OF ANGRIST &

EVANS’ FINDINGS

I. Introduction

The empirical study of the relationship

between fertility and labour supply is crucial for testing the existing

theories that link the family and the labour market. Up to now, most of the

evidence found points out a negative correlation between fertility and female

labour supply, but many of these results are esteemed to be blurry as far as

the problem of endogeneity of fertility is not solved:

“… it has proven difficult to find enough

well-measured exogenous variables to permit cause and effect relationships to

be extracted from correlations among factors such as the delay of marriage,

decline of childbearing, growth of divorce, and increased female labour force

participation” (Robert J. Willis, 1987 p. 74).

Indeed, there are good

reasons to believe that fertility and labour supply are jointly determined,

thus preventing from extracting any causal interpretation: The fact that

fertility is both used in the literature as an explanatory variable and as a

dependent variable of labour force is compelling. Fortunately, this problem can

be tackled through diverse ingenious instrumental variable (IV) strategies, such

as that of Lundborg, P., Plug, E., and Rasmussen, A. W. (2017), who analyse the

effect of childbearing on labour market outcomes among women with similar

working histories that become mothers for

the first time through in vitro fertilization (IVF). Because the success of

such treatment can be regarded as a random result from nature, it becomes

plausible to compare the labour-market outcomes of those women who effectively

gave 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, this

can also be regarded as a hypothetical experiment in which women seeking for

help are randomly assigned a baby by nature, allowing the author to compare the

labour-market outcomes of those women who were successful in getting pregnant to

those of who were not. In contrast with these two studies, however, in this

paper we will explore the causal relation from fertility to labour supply

following the alternative IV strategy based on the sibling sex mix in families with two or more children put forward by

Angrist, J. D. & Evans, W. N. (1998), with a subset of the same data base

used by the authors, to recreate some of their central estimates and

corroborate their findings.

Although standard household

theories predict that the labour-market consequences of having a first child

are stronger than those of having additional children, this difference in focus

allows us to ‘exploit’ the fact that parents prefer a mixed-sex composition of

their children, instead of having offspring of the same sex. More

interestingly, it has been observed that parents who have siblings of the same

sex are more likely to go on in having an additional child. And because the sex

mix is a randomly assigned factor, given by nature, a ‘dummy’ variable that

indicates whether or not the sex of the second child matches that of the first,

same sex, can be used as an

instrument for further gestation or ‘fertility’ among women with at least two

children (for the sake of simplicity and conciseness, we leave aside the use of

twinning as an alternative instrument

in this paper, which the authors find to deliver similar results). Thus, our

instrument captures the effect of moving from the second to the third child on

women’s labour supply.

II. The Data

In this study we use a

subset of the same Census Public Use Micro Samples (PUMS) of 1980 data set used

by Angrist, J. D. & Evans, W. N. (1998), but we leave aside the 1970 and

1990 samples originally considered by the authors, who detect a substantial

decline in fertility and an increase in women’s labour supply throughout the

period. By contrast, we focus exclusively in the causality effect from

fertility to labour supply in one particular period, which is the central

question 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 of

the 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 of

moving from the second to the third child), which consists of 355,356 observations.

The variables and their

descriptive statistics are provided in Table 1, where the covariate of our main

interest is the binary variable Morethan2

children (as indicator of ‘fertility’, the endogenous variable), and Samesex is its instrument. As we will

see ahead, the two components of the latter, 2Boys and 2Girls, are

also shown. Demographic and labour supply variables are also included in the

lower half. Notice that among all women of the sample, 40.19%

had a third child.

III. Sex-Mix and Fertility

We can model parents’

sex-mix preferences and utility in the following way: Suppose a couple has

already children, and they decide on the additional

number of children they want to have, . Because parents prefer a

mixed-sex composition of their offspring, having already a same-sex composition

reduces the utility from and increases, at turn, the marginal utility

of . Thus, under these

circumstances, parents are more likely to decide to have an additional child.

Accordingly, Table 2 reports

different estimations of the effect of ‘sex-mix’ on fertility that reveal this

phenomenon. Recall here, however, that we are only interested in women with two or

more children. Thus, Table 2 shows the relationship between the fraction of

women 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 their

offspring: two boys, two girls, one boy and one girl, and simply, two children

of the same sex. The last row displays the difference between the same-sex and

mixed-sex group averages.

Table 2 allows to infer that

women with two children of the same sex are noticeably more likely to go on in

having a third child than women with one boy and one girl. Concretely, 43.18% of the mothers with same sex children go on

in 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 Supply

A. Wald Estimator

Because our Samesex instrumental variable is

essentially randomly assigned by nature, we can safely extract a causal

interpretation from the regression of fertility on labour supply. Consider the

following bivariate regression model:

Where is labour supply (Workedforpay) or any of our other measures of labour-market

outcomes described in Table 1, and is our endogenous fertility measure, Morethan2. As usual, we denote as our Samesex

binary instrumental variable, and we define the estimator for binary

instrumental variables (), also called the Wald

Estimator, as:

In which is the mean of for the observations where is equal to one, and the other terms are

defined in an analogue manner. Here, the numerator captures the relationship

between and , while the denominator

captures 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 the

sex-mix () of their offspring.

This can be easily obtained by running a simple regression

of the endogenous variable with the instrument (as we will see, similar to a

first-stage estimation without covariates in the two-stage least-square, 2SLS,

estimation framework), to obtain the denominator, and a regression of the

outcome variable and the instrument to obtain the numerator (notice that both

the numerator and denominator are ‘scalars’), and then dividing one scalar over

the other (also, a procedure equivalent to running the second-stage estimation

without covariates in the 2SLS method). Effectively, the first column of Table

3 reports the components of separately, showing in the first row the

denominator of the Wald estimate, , where it can be seen that

the effect of Samesex on Morethan2 is equal to 0.0605 (which is the same as the difference between the same-sex and

mixed-sex group averages reported in the last row of Table 2); and in the

remaining rows, different estimations of the numerator, (one for each labour-market outcome),

suggesting that indeed, additionally to having more children than women with

one boy and one girl, women with two children of the same sex present a lower

labour supply. Specifically, the Wald estimates reported in the second column,

obtained from dividing the numerator by the denominator, indicate that having

more 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-Squares

Estimation

We now try a different approach

to the problem by using the two-stage least-squares (2SLS) estimator. While the

Wald 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, which

include Age, Age at first birth, Familyincomelog,

and Education. However, at this point

we deviate somewhat from the original authors’ estimation in that we treat Familyincomelog as a covariate instead

of a dependent variable, for it is more likely that the mothers’ family wealth

determines how prone they are to participate in the labour market, and not the

other way around (i.e. if a mother’s family is relatively rich, she might be

less urged to work while childbearing than a mother from a poor family). This

is made evident by the insignificant effects found by the authors when treating

this variable as an outcome variable. At the same time, we also make emphasis

on the role of mothers’ education because several theories put forward the idea

that the impact of fertility on labour supply varies with the years of

schooling, and there is some empirical evidence in the same sense, showing that

the more educated women’s labour supply is more sensitive to fertility than the

labour supply of the less educated women (Gronau, R. 1986).

Another advantage of using

2SLS is that it also permits us to control for any ‘secular’ additive effects

of childbearing as we use the Samesex

instrument. Indeed, because Samesex

is an interaction term comprising the sex of the first two children, it is

potentially correlated with the sex of either child, which can ultimately be a

problem if the sex of offspring affects in some way parents’ attitude towards

the labour market (see Angrist, J. D. & Evans, W. N. (1998) for a

proof). Thus, we can add the Boy1st (S1)

and Boy2nd (S2) regressors

described in Table 1 to eliminate the possibility of an omitted-variables bias

arising from these sources. We can then write the following regression model

linking the labour supply and labour-market outcomes with the endogenous fertility

variable, , the vector of other

exogenous variables, , and the additive effects

for the sex of each child, as:

Now, the first-stage

equation relating the endogenous Morethan2

variable to the sex-mix is:

Where is the first-stage effect of the instrument on

. A variant of this approach

also used by the authors exploits the possibility of formulating an over-identified

model by decomposing the Samesex

instrument into two separate indicators: 2Boys

and 2Girls. To see this more clearly,

realize how we can express our instrument as:

Where S1 and S2

are, 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 same

sex), from where we can extrapolate the two separate instruments: 2Boys, S1S2, and 2Girls,

(1-S1)(1-S2), also

reported in Table 1. This over-identification strategy is advantageous because

we might expect any bias arising from the so called ‘secular’ effects of child

sex on labour supply to be different for each of these two instruments, while

the 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, we

must drop one of this variables to avoid perfect multi-collinearity problems, in

which case we choose to withdraw S2i.

Hence, the following alternative regression model using the two separate

instruments can be specified:

Where the first-stage

regression is now:

Table 4 reports the results

of the first-stage estimations for both the just-identified and the

over-identified regression models. We can see, on the one hand, that mothers

with two children of the same sex are 6.2% more

likely to have a third one; and on the other hand, that the effect of 2Girls on fertility is higher than that

of 2Boys, suggesting that parents are

more willing to persist in having children until they can have a boy. As for

our covariates of particular interest here, we can see that having a higher

level of education reduces the likelihood of having a third child by 2%, and that greater family wealth slightly

increments this probability (this can interpreted as mothers from rich families

being less worried about the economic difficulties of childbearing).

At this point, however, we

would 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 in

each first-stage regression (in both the just-identified model, where Samesex instrument was used, and in the

overidentified model, where we used the 2Boys

and 2Girls instruments). When testing

for the ‘strength’, we are actually interested in the correlation between the

endogenous variable Morethan2 and

each of our instruments (indeed, the ‘first-stage condition’ means that the

instruments(s) considered should bring some knowledge to the endogenous

variable). Effectively, this correlation is measured by the first-stage Partial R2, where in the case of the

just-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 the

first-stage F-statistic in each of

the models is sufficiently large to reject the null-hypothesis that the

instruments are ‘weak’ (see

appendix 2).

Having checked this aspect of our instruments’

validity, we can now run the regression of the effect of on the

different labour-market outcomes using both the just-identified and the

over-identified models. Simultaneously, we conduct simple ordinary

least-squares (OLS) regression to compare the results and have an insight of

the magnitude of the bias arising from our endogenous variable. The results are

presented in Tables 5 through 7.

As can be seen, when using

the single Samesex instrument (the

just-identified model), having a third child reduces the probability of

participating in the labour market by around 11.5 percentage

points, the number of hours per week worked by 8-9

per year, and the amount of earnings by more than $1,745.

Likewise, when using the over-identified model, the probability of

participating 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 estimations

suggested that mothers of two boys are less likely than mothers of two girls to

have a third child, the 2SLS estimates in Table 5 allow us to infer that

separating our Samesex instrument

into its two components doesn’t change the magnitude of the coefficients very much

and, in the end, the same conclusions are reached. Finally, the bias arising

from the endogeneity of our fertility variable (Morethan2) clearly overestimates the

effects of fertility on the labour-market outcomes, as can be appreciated when

comparing the OLS with the 2SLS results described above. As for our other

covariates of interest, we see that both education and family income yield the

expected results: The former has a positive effect in all of the three

labour-market outcomes, incrementing the labour supply (Workedforpay) by 3.1%, hours

worked by 5-6, and labour income by $795-$798. Conversely, family income reduces

labour supply by 2.7%, hours worked by 7-8, and the mothers’ labour income by $757-$758 (recall these are all mothers of at

least two children, and they are more likely to abandon the labour-market if

they have the economic support from wealthy relatives).

V. Concluding Remarks

We have seen that both the

Wald and the 2SLS estimates consistently confirm the thesis that increasing

fertility (moving from the second to the third child) reduces women’s

participation in the labour market. Effectively, on the one hand, these results

are in line with those found in the cited literature, but they appear to be

less ‘harsh’. For example, while Lundborg et. al. (2017) calculate a reduction

of 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 26

percentage points’, our calculations only account for a 10.6%-11.5% decrease. This is due to the fact that our IV

strategy focuses on the effect of the third-born child instead of the

first-born child, which is expected by the standard household theories to have

a lower impact on the labour supply of women.

On the other hand, however, the effects

calculated in this paper (big or small) shouldn’t be over-dimensioned. In

effect, 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 children

decreased by around 15.7 percentage points between 1970 and 1980; while, at the

same time, the participation of women in the labour market rose by about 13.2

percentage points in the same period. If we then use the 2SLS estimation of the

impact of fertility on labour supply (Workedforpay)

using the Samesex instrument reported

in the upper part of the central column of Table 5 (-0.115),

we can deduce that declining fertility accounted for an increase in

labour-market participation of roughly 1.8 percentage

points (0.157×0.115). Thus, our study also lets

us conclude that, although fertility has a significant negative impact on

labour supply, the increase in the labour-market participation rate has been so

substantial that declining fertility only accounts for a small fraction of the

whole change.