Abstract
The Early Bird gets the Worm? Birth Order
E¤ects in a Dynamic Model of the Family
Elisabeth Gugl and Linda Welling
This version: May 2008
Abstract
Birth order e¤ects are found in empirical work, but lack theoretical foundations. Our new approach to modelling children provides this. Each child has
the same genetic make-up and parents do not favour a child based on its birth
order. Each child’s needs change as it grows, and births are sequential. At any
point in time siblings are at di¤erent developmental stages, and the bene…ts
of parental investment di¤er across these stages. Parental time investment in
children lowers current and future wages; this opportunity cost varies across
time. Birth order e¤ects emerge from the interaction of the changing bene…ts
and costs of parental investment.
JEL: D13, D91, J13.
This is a substantially revised version of a paper presented at the Western
Economic Association International 82nd annual conference, Seattle June 29July 3, 2007. We thank Joseph Price, our discussant, whose comments suggested
this re-orientation of the paper, and two anonymous referees.
Both authors’ address: Department of Economics, University of Victoria,
P.O. Box 1700 STN CSC, Victoria, BC, V8W 2Y2, fax: (250) 721 6214, Gugl:
phone: (250)721 8538, email: egugl@uvic.ca; Welling: phone: (250) 721 8546,
e-mail: lwelling@uvic.ca.
1
"The great advantage of living in a large family is that early lesson of life’s essential unfairness." ~Nancy Mitford (http://www.quotegarden.com/family.html)
1
Introduction
Recent empirical literature has found signi…cant di¤erences between …rst-born
and later-born children in both outcomes - such as educational attainment
and/or earnings (Ejrnæs and Pörtner 2004, Black et al. 2005, Conley and
Glauber 2006, Kantarevic and Mechoulan 2006) - and inputs (Price 2008).
Since family resources, both …nancial and time, are viewed as largely responsible for these e¤ects, much of this literature has attempted to separate the
e¤ects of birth order from family size. Hanushek (1992) suggests that these
di¤erences arise because children of di¤erent birth order within the same family
are born into di¤erent family con…gurations, with di¤erent intellectual environments. There are few theoretical economic models in which birth order e¤ects
emerge endogenously; this is not altogether surprising, since our models are typically atemporal when it comes to investment in children. The only empirical
paper investigating a possible cause for a birth order e¤ect is Price (2008). The
author looks at quality time that parents spend with each child in multiple child
families, and …nds that children of higher birth order receive less quality time
at any given age than did their earlier-born siblings at the same age. This does
not necessarily imply that younger siblings receive worse education within the
family than older children because siblings replace some of the quality time that
parents would otherwise spend with them; however, younger siblings lack the
opportunity to be teachers from an early age on (this was suggested by Zajonc
1976). Our paper …lls an important gap in the theoretical literature on families
with children by investigating how child quality is produced in the family and
how birth order may in‡uence the conditions of producing child quality for each
child.
We investigate the trade-o¤ parents make between spending time with di¤erent children and working for income (to purchase consumption goods for family
members). Parents care about both the contemporaneous utility of each child,
and their well-being as adults. We consider a three stage model of parenthood.
Two children are born and mature sequentially. Each child spends two periods
with their parents: the …rst born is a single child in its …rst period of life, and
an older sibling in its second; the second born is a younger sibling in its …rst
period, and the sole child in the household in its second. Thus each child spends
one period with parents and a sibling, and one period alone with its parents.
Older children have di¤erent needs than younger ones, so the sequential nature
of child rearing matters.
Our model is based on four assumptions about child development:
(1) Children develop in stages, and their needs change as they grow. While
this is clearly an oversimpli…cation, we distinguish two periods in a dependent
child’s life. In the …rst period a child’s dominant need is time spent with them;
2
in the second period, parental time remains an essential input, but the child
also needs consumption goods. These inputs a¤ect both contemporaneous
child utility and the child’s human capital when grown. This is consistent with
Waldfogel (2006), who summarizes evidence that suggests that interaction with
the mother (primarily) is of primary importance in the …rst three years, whereas
for the second half of the pre-school years good nutritional habits and socialization with peers becomes important as well.1 In a recent series of papers
Cunha &Heckman (2007), along with various co-authors, have emphasized that
the psychological literature tells us that human capital is multi-dimensional (at
least cognitive and non-cognitive), and investment at particular stages/ages of
childhood can be critical for adult ability in di¤erent dimensions. Cunha &
Heckman (2007) present a stylized theoretical model with two stages of childhood and two of adulthood. More than one stage of childhood is important
in their model because the production technology in the second stage may depend on inputs in the …rst stage, and …nal adult utility may be a¤ected by this
cumulative investment.
(2) Parental child care has an opportunity cost of foregone current and future earnings. The time each parent spends caring for a child has an impact
on parents’future wage rates and, therefore, on the potential earnings of each
parent once the child is grown. While this "family wage gap’is well-documented
empirically2 , most of the theoretical literature has overlooked this aspect; Willis
(1973) is an early exception. Cunha & Heckman (2007) follow tradition in this:
parents supply labour inelastically, so there are no labour market/family income
implications of early versus late investment for the family of origin. We explicitly incorporate both current and future implications of this time investment in
a dynamic framework.
(3) Child care can be provided by parents, or purchased, but parental time
is more valuable in producing child quality than is the same amount of time
provided by others. There is a large empirical literature emphasizing the importance of parental (or more often maternal) time spent with the very young
child.3 However, theoretical models have not yet captured this distinction.
(4) Child bearing is sequential. In most models parents choose quality and
quantity simultaneously. Sequential childbearing and di¤erent needs of children of di¤erent ages suggest that multiple children should be viewed as joint
products in household production, with the production mix varying over time.
Our model makes explicit the sequential nature of decisions. Ejrnæs and Pörtner (2004) consider sequential fertility, with parents deciding on future children
once they observe the nature of the current infant. Parents determine investments in these children once all are born. We abstract from the "nature" side,
and assume all children are identical at birth; parental investment begins im1 See
also Huang(2006) and Keane and Wolpin (2007) for related results.
references - waldfogel has papers in mid/late 90’s; since then, many papers on various (all?) western countries, some in Japan....."...women work shorter hours when they have
children and are, if anything, paid less per hour than comparable women without children"
Waldfogel (2006: 13)
3 See, for example, Ruhm (1998, 2000).
2 Need
3
mediately. Thus the …rst-born receives parental investment before the second
child arrives, and investment in the last-born continues after the …rst-born is
fully grown and self-su¢ cient.
We show that both the …nancial and time investments parents make in their
children typically di¤er with birth order. These di¤erences arise because the
opportunity cost of parental child care varies over time, depending on the current
wage rate, the anticipated growth in wages over time, and the particular inputs
required by children at each stage.
We focus on child-rearing, beginning with the birth of the …rst child. We
take as given that spouses achieve e¢ ciency in marriage - at least from this point
forward. Spouses could have entered a binding agreement at the beginning of
marriage or, starting with the arrival of the …rst child, bargained over life-time
utility. While our model can be extended to incorporate family bargaining, we
leave this to future research together with the analysis of changes in family
policies.
In the following section we …rst explain our basic three period model, and
specify our assumptions on child utility for respectively, young, older, and adult
children. We then set out our assumptions on parents. Parents receive utility
from their children through two mechanisms: …rst, they care about a child’s
current well-being, and second, they care about the adult well-being of their
children. Finally, we develop the contemporaneous budget constraints. Section
3 presents the results derived from solution of the parents’three-period decision
problem. Conclusions and extensions are in section 4.
2
The Basic Model
The family consists of a two adults and two children. Parents make decisions
over three periods. Births are deterministic and sequential: the …rst child is
born in the …rst period, and the second in the second period. A child spends
two periods with his or her parents, and then becomes a self-su¢ cient adult.4
The time path of family composition, then, is
Period 1(k = 1): wife (w), husband (h), young …rst-born
Period 2 (k = 2): wife, husband, older …rst-born, young second-born
Period 3 (k = 3): wife, husband, older second-born; independent older …rstborn.
Parents allocate their time between child care and work, and their earned
income between own and child consumption goods, in each of the three periods
to maximize the discounted present value of the sum of their individual utilities.
Let tjik denote the time parent i devotes to child j in period k = 1; 2; 3: Parents
care about their independent adult children as well as children living at home;
moreover, although they will not live to see it, they also care about the future
adult well-being of the second child. In this section we …rst describe the utility
4 In a fully dynamic model this third period would coincide with the …rst period, for the
next generation, of family life in our model.
4
functions of a child, then that of a parent. We then specify the household budget
constraints. In the next section we analyse the parents’decision problem.
2.1
Child’s utility
We di¤erentiate the …rst and second born children by superscripts: a denotes
the child born in the …rst period and b the child born in the second period. Let
ujk ; j = a; b denote the utility of child j in period k. We assume ub1 = 0 = ua3 :
in period k = 1 the unborn second child has utility of zero, and in period k = 3
the …rst born is an adult rather than a child, with a utility speci…ed in section
below.
Children live in the present: they have no memory of the past, and care
nothing about their future. Contemporaneous utility is produced di¤erently at
di¤erent ages of the child. While a dependent child j requires both time with
adults (tjk ) and consumption goods (xjk ), the relative importance of these two
broad categories changes as the child grows.
2.1.1
Young child
We take the extreme view that in the …rst period of life a child needs only
supervision equivalent to the time available to one adult; we normalize this
time to 1: Parents can provide child care themselves, or they can outsource by
hiring a nanny or enrolling the child in paid care. Let tjnk be the time a young
child spends in non-parental care in period k; k = 1; 2:5 Then the household’s
kth-period time constraint for a young child j is
tjwk + tjhk + tjnk = 1:
(1)
Utility depends on the time the parents spend with the child. Research on
early child development suggests that parental time is very important in terms
of the happiness of the child and we therefore assume that it produces higher
child utility than the same amount of time in other care.6 We assume parents
are perfect substitutes in child care, and focus on the sum tpk = twk + thk .
Using (1), assuming that parental time is more productive by a factor of p; the
utility in period k of a young child j is
ujk = Y (tjY k ); tjY k = (p
1)tjpk + 1; p > 1;
(2)
where Y ( ) is a strictly concave, at least twice di¤erentiable and increasing
function; in the appendix we make explicit further conditions su¢ cient to ensure
an interior solution for tY k :
5 Recall
there is no young child in the household in period 3.
more time with the child when young also increases a child’s skills and translates
into a higher earning potential when the child is grown. Parents care about their grown child’s
utility which is a function of the child’s earning potential. We discuss this point further below.
6 Spending
5
2.1.2
Older child
In the second period of life, a child no longer requires constant supervision; utility the child’s consumption of private goods as well as interaction with parents.7
As before, parents are perfect substitutes in child care. The utility in period k
of an older child j is
ujk = O tjpk ; xjck ;
(3)
where O ( ; ) is strictly concave, twice di¤erentiable and increasing in both
variables.
2.1.3
Adult child
After two periods as a dependant, the child becomes a self-su¢ cient adult who
requires no further input from the parents.8 Utility is a function of the child’s
earning potential, which is a function of the accumulated human capital of the
child. Human capital of the grown child depends on the parental time that
was spent with the child in each stage, as well as the amount of private goods
consumed by the older child. Given the sequential births, the …rst born becomes
an adult in period 3 while the second does not become self-su¢ cient until what
would be period 4. An adult child’s utility is a lump sum.
uaA
= A tap1 ; tap2 ; xa2
ubA
tbp2 ; tbp3 ; xb3
= A
(4)
where A ( ; ; ) is a strictly concave, at least twice di¤erentiable and increasing
function.
The presence in the second period of two children, of di¤erent ages and
with di¤erent needs, both requiring some supervision time, raises the issue of
economies of scope in child care: is a given amount of parental child care as productive if it is shared by two children, rather than devoted to one? Can parents
purchase child care for two children at the same rate as for one? Economies of
scope in supervision time implies that any detrimental e¤ects of sharing a caregiver with a sibling are compensated for by the bene…ts from sibling interaction.9
We assume economies of scope for the parents only, hence tap2 = tbp2 = tp2 :
7 While human capital as built up by the parents’ time spent with the child in the …rst
period feeds into the child’s earning potential and therefore third period utility of the child,
we assume that the happiness of a child in the second period is una¤ected by the time parents
have spent with it in the previous period. Of course this is an extreme assumption and past
parental time is very likely to in‡uence both human capital and child happinessin the current
period, but we …nd it reasonable to make a distinction between a child’s utility when it is still
a child and has a high discount factor and a child’s human capital that translates into utility
most signi…cantly in later periods.
8 Wishful thinking on our parts, no doubt. More seriously, we do not consider bequests.
9 If the children were twins, the appropriate concept would be economies of scale.
6
2.2
Parent’s utility
Parents obtain utility from their individual private consumption in period k
(denoted by xik ) as well as from the contemporaneous utility of any dependent
children. Thus the utility of parent i = w; h in period k = 1; 2; 3 is given by
uik = Uik (uak ; ubk ; xik )
(5)
In their planning parents also consider the well-being of their adult children,
so i’s intertemporal utility is given by
Ui = ui1 + ui2 + ui3 + uaA + ubA :
(6)
Parents have an intertemporal discount factor of 1, so their utility in future
periods weighs as heavily as the current period in their decisions. This implies
that they care about child utility tomorrow as much as they care about child
utility today. Moreover, parents treat each child’s utility over the child’s life
span equally.10 These assumptions would be natural in the case of twin births,
and are also appealing in the context of sequential births, because they guarantee
that parents will not favour their …rst-born because they value current utility
more than future utility. Of course, that does not guarantee identical treatment,
since the costs of producing child utility depend on the changing opportunity
cost of parents’time as well as the children’s developmental stages.
2.3
Full budget constraints
There are two uses for parental time: child care, and labour force participation.
In the …rst period, parent i is employed, for the amount of time lik = 1 tik , at
wage rate wi1 . The husband has an initial absolute advantage in earnings, so
ww1 < wh1 :The wage rate in each subsequent period is increasing in the time
spent in the labour force in the previous period:
wi2
wi3
= wi1 (1 + li1 ); > 0
= wi2 (1 + li2 ) = wi1 (1 + li1 )(1 + li2 )
(7)
Earnings in k = 2; 3 are therefore a function of past as well as current choices.
While the parents’wage rates grow, alternative child care can be purchased for
nk = n dollars per unit of time, and the private consumption good for both
parents and children has a unit price of 1 in each period. Thus the household’s
1 0 There are two obvious ways in which intertemporal discounting might enter here: parents
may value current utility more highly than future utility, at any point in time, and they might
weight more heavily the adult utilty of one or the other of their children. Introducing either,
or both, of these factors would alter our results in predictible ways.
7
budget constraints for each of the three periods are
h
a
xw
1 + x1 + ntn
a
b
xh2 + xw
2 + x2 + ntn
= ww1 lw1 + wh1 lh1 =
=
X
X
wj1 lj1
(8)
j=w;h
wj1 (1 + lj1 ) lj2
(9)
j=w;h
b
xh3 + xw
3 + x3
=
X
wj1 (1 + lj1 ) (1 + lj2 ) lj3
(10)
j=w;h
3
Parental Investment
We assume that parents’utility functions yield transferable utility, and therefore allocative e¢ ciency is independent of distribution. Maximizing the sum
of the parents’intertemporal utilities allows us to …nd the unique point in the
parents’production possibility set that yields Pareto e¢ ciency (Bergstrom and
Cornes 1981 and 1983). Maximizing this sum also uniquely determines the sum
h
of parental consumption, xpk = xw
k + xk ; k = 1; 2; 3. As parents are perfect
substitutes in the production of child utility, e¢ ciency dictates that the spouse
with the lower wage rate in period k is the primary caregiver, while the other
parent is the primary wage earner in that period. We assume parameters such
that the optimal time allocation has the latter working full time, and the former part-time, for pay; by construction the wife always provides parental child
care. Thus we have lhk = 1 and tpk = twk 2 (0; 1) ; k = 1; 2; 3 and therefore
tY k = (p 1)twk + 1.
The e¢ cient outcome is determined by the choices of time ftw1 ; tw2; tw3 g and
P3
private goods fxp1 ; xp2 ; xp3 ; xa2 ; xb3 g which maximize [ k=1 (uwk +uhk )+2(uaA +ubA )]
subject to the time and budget constraints above, given assumptions on the
nature of economies of scope. We consider two cases of transferable utility: i) quasi-linear utility, where parent i’s utility is additive in own consumption and child utility and there are no income e¤ects on spending on
the children, and ii) more general transferable utility which incorporates income e¤ects. In either case, the decision problem can be solved recursively:
third-period choices are functions of tw2 ; second-period choices are functions
of tw1 and …rst-period values can be found once the second-period value function is derived. These derivations are in Appendix A, as are all proofs. Assuming interior solutions in the third and second periods for twk , all third
and second-period choices are functions of …rst-period maternal child care,
tbw2 (tw1 ) ; taw2 (tw1 ) ; xa2 (tw1 ); xp2 (tw1 ); tw3 (tw1 ) ; xb3 (tw1 ); xp3 (tw1 ).
3.1
Quasi-linear utility
Let uik = xik + uak + ubk , i = w; h; k = 1; 2; 3:Then the household chooses private
consumption goods and the time allocation in each period to maximize:
8
3
X
(uwk + uhk ) + 2(uaA + ubB )
k=1
=
=
3
X
(xpk + 2(uak + ubk ) + 2(uaA + ubB )
k=1
[xp1 + 2Y (taY 1 )]
+[xp2 + 2 O(taw2 ; xa2 ) + Y tbY 2 ]
+[xp3 + 2O(tbw3 ; xb3 ) + 2(A(taw1 ; taw2 ; xa2 )
+ A(tbw2 ; tbw3 ; xb3 ))
subject to constraints (2) ; (3) ; and (4) :
We …rst state two results which clarify the source of the birth order e¤ects
in our model.
Lemma 1 With full economies of scope in maternal child care, so tw2 = taw2 =
tbw2 , and all else equal, the intertemporal opportunity cost of additional parental
child care is greater in the …rst period than in the second if and only if parental
child care time is greater in the …rst period than in the second.
Proof: see appendix
This relationship follows from the assumed intertemporal discount factor (of
unity), and the constant growth rate of parent i’s wages, given the previous
period’s labour force participation. In each of the …rst two periods, a young
child is present in the household, and the requirement of full-time care means
that parental child care time for this child has a within-period opportunity cost
of (wik n); k = 1; 2. The third period di¤ers because only an older child
is present, so the contemporaneous opportunity cost of parental child care is
higher in this …nal period than in the previous two.
Time spent in child care in the …rst period, out of the paid labour force,
lowers the wage rate in the second period, and hence lowers the opportunity
cost of parental time in that period. Similarly, when parents determine the …rst
period time allocation, they recognize that the greater the amount of time spent
at home in the second period, the less will be the income loss from lower second
period wages, and so the lower will be the opportunity cost of …rst period child
care time. This intertemporal feedback e¤ect will induce di¤erences in optimal
choices in the two periods, as is demonstrated by Lemma 2: .
Lemma 2 Suppose i) = 0, so future wage rates are independent of current
labour force participation; ii) maternal child care is a public input; and iii) the
utility of adult children is independent of their upbringing. Then i) taw1 < tw2 ;
ii) tbw3 < tw2 and iii) xa2 < xb3 if the cross partial for time and goods is negative
2
O
in the production of utility for the older child (so @x@j @t
0).
j
k
Proof: see appendix
9
wk
Thus the …rst born receives more attention when older than younger, while
this is reversed for the second-born. Moreover, the parents substitute goods
for time for the second born, while the reverse is true for the …rst born.
When there are no future labour market costs to parental child care, time
spent on child care in any period is indistinguishable from money. In this
case, only current bene…ts matter. If parents’spending on dependent children
a¤ects the latter’s adult utility, and parents consider this, then the complementarity of time and goods in this function also enters into the time allocation
determination.
Given the possibility of intertemporal substitution on the …nancial side, any
di¤erences in parents’choice of time in the …rst two periods will be determined
largely by the bene…ts accruing from child care time. The presence of two
children in the second period, but only one in the …rst, increases the bene…t of
second period time spent at home. Thus we have Proposition 1:
Proposition 1 With quasi-linear utility for parents, and full economies of
scope for the mother, the …rst-born and the second-born receive unequal treatment.
Proof: see appendix
Lemma 3 Suppose parental time is rival, and taw2 = tbw3 = 0. Then the
…rst order conditions for the parents’ decision problem can be satis…ed by equal
treatment for the two children: (taw1 ; xa2 ) = (tbw2 ; xb3 ):
Proof: see appendix
Thus, if parental time is rival, second period utility for children is such that
either they require no parental time in that period, and parents optimally choose
to devote no time to their children in this stage, then sequential child-bearing
does not necessarily give rise to birth order e¤ects. However, when parents
do spend time with both children at each stage, siblings receive di¤erent treatment even if parents care only about the contemporaneous utility of dependent
children.
Proposition 2 Suppose maternal time is rival, and parents choose to spend
time with each child in each stage. Then the …rst born receives more of both
maternal time and consumption goods than the second born, at each developmental stage.
Proof: (See Appendix)
Overall, this implies that the …rst born receives higher utility in each period
of her life and hence also enjoys a higher earning potential when grown-up. Note
that it is still possible to …nd tbw2
taw2 as Price does in his empirical work,
but his …nding that a signi…cant number of families spend equal time on both
children seems to be less plausible than when we assume maternal time is a
public input.
Given the same model except for maternal time being a public input in the
second period, we get tw2 > taw1 : This is not at odds with Price’s …ndings as
10
he restricts the children in the sample to be at least 3 years of age (check age
limit again). We argue that the second period starts right around this age as
pre-school now provides bene…ts that are complements to parental time and
so would …t better with our time-plus-consumption speci…cation of the second
period utility function of the child than the …rst period utility function.
3.2
Transferable utility with income e¤ects
Proposition 3 Suppose uaA = ubA = 0, so parents care about their children’s’
utility only while they are co-resident. Let uwk + uhk =
uak + ubk xpk +
a
b
2 uk + uk , so that utility is still transferable within the household but we now
have both an income e¤ ect and a substitution e¤ ect on child care time. Then
the …rst-born and the second-born child are treated di¤ erently.
Proof (see appendix)
Notice that "di¤erent treatment" is not necessarily "unequal treatment".
The …rst-order conditions for the private consumption goods of the two children
require that private goods are chosen to equalize the full impact of these goods
across the children. This suggests scope for di¤erent choices for the two children,
if parents choose di¤erent amounts of child care. First, recall that uaA and ua2
are both functions of both xa2 and tw2 , while ubA depends on the latter but not
the former, and ub3 depends only on third period choices. Thus, changes in time
and goods allocation in the second period which leave the FOC for xa2 satis…ed
will change third period choices.
3.3
Discussion of results
Consider the constellation of conditions which must hold if the two children
are treated identically. In our model, this means that each child receives the
same amount of parental child care as their sibling did in the same developmental stage, and the same amount of the private consumption good as their
sibling in the second stage: that is, (taw1 ; taw2 ; xa2 ) = (tbw2 ; tbw3 ; xb3 ). If maternal
time is a pure public input, so tbw2 = taw2 = tw2 , identical treatment implies
taw1 = tw2 = tbw3 : the mother spends the same amount of time in the labour
force in each of the three periods. Given positive returns to human capital acquisition in the labour market ( > 0) and constant prices for outsourced child
care and private consumption goods, this implies that full family income, and
the quantity of private consumption goods purchased, increases each period.
Identical treatment of the two children further implies that, since spending on
the older child’s private consumption is the same in the second and third periods, the parents consume more private goods in the third period than in the
second. Thus parents allocate all of the increase in full family income over time
to increased private consumption.
Why is identical treatment unlikely? In our setup, unless maternal child
care is purely rival, the contemporaneous bene…t of spending additional time
11
at home in the second period, caring for two children, is higher than the corresponding bene…t in the …rst period, when there is only one child. This suggests
that more time would be devoted to parental child care in the second period.
However, given the positive impact of current labour force participation on future market wages, there is also an intertemporal link between the periods, as
the opportunity cost of second period maternal care is decreasing in …rst period
maternal care. With no discounting, this increases the return to parental time
devoted to the young …rst born above that devoted to the young second born:
since there is no third child, no similar future bene…t increases the bene…t to
maternal time in the second period.
Clearly, these considerations change if the spacing between the births of
the two children is greater. Our model is presented in terms of two stages
of child development; nothing except more complex notation prevents us from
incorporating more stages. The changing opportunity cost of parental time and
the di¤erent needs of children at di¤erent stages will introduce birth order e¤ects
which depend on the alternative demands on parental resources at di¤erent
stages. The greater the spacing between children, the less likely it is that
demands will overlap, and the more siblings will resemble only children.
Using data from the American Time Use survey and restricting the age range
of children from 4 to 13, Price (2008) …nds that many parents spend equal time
with their children at any given point in time and that time spent with children
typically decreases from one period to the other (as the children grow older).
This is consistent with an extension of our model in which what is now called
the second period of a child’s life is split up into more periods in which children’s
needs for parental time play a lesser and lesser role as they grow. One could
also test our assumption of economies of scope by checking how often parents
spend time with one child while the other sibling is present.
4
Extensions and conclusions
This paper examines birth order e¤ects. Parents do not favour one child over the
other: each child in each of its developmental stages enters their intertemporal
utility function with the same utility weight. However, we do not require that
parents devote the same resources to each child in each developmental stage.
Central to our …ndings of birth order e¤ects is the assumption that investments
in children must begin at birth. Moreover children’s needs change as they grow,
so that parents deal with di¤erent production functions for child quality in any
period where children of di¤erent ages are present in the household.
4.1
Parents pay for early parenting "mistakes"
As emphasized before, our model does not explicitly incorporate the feedback
of earlier child quality on current child quality. Thus our model does not address the "critical" or "sensitive" periods for investment in certain dimensions
12
of human capital as in Cunha and Heckman (2007). Some aspects of this can
be incorporated by restricting the signs of the cross-partials of the utility function for the adult children, A(taw1 ; tw2 ; xa2 ) and A(tw2 ; tbw3 ; xb3 ) : the cumulative
e¤ects of investment appear if the marginal productivities of second period expenditures are increasing in the time spent with the young child. With the same
investment into the child in its second period of life, having spent less time in
the …rst period will lead to a lower marginal productivity of second-period investment. This reduces the negative impact of …rst-period parental child care on
the sum of second-period utility of the parents. Explicit use of CES functional
forms in our model would allow us to investigate the interaction of a deeper
model of child development with birth order e¤ects.
4.2
Adapting the CH model
Suppose now that the full extent of parental concern for children is captured
in a simpli…ed version of the CH human development model. In this case we
subsume the three functions for each child Y( ); O( ; ), and A( ; ; ) into a single
function:
ha
h
b
=
=
[ (taw1 ) + (1
[
(tbw2 )
+ (1
)(taw2 xa2 ) ]
1
)(tbw3 xb3 )
1
]
(11)
(12)
Retaining the quasi-linear form for parents’utility, the intertemporal decision problem becomes
P3 one of choosing the time and consumption goods allocations to maximize k=1 xpk + 2(ha + hb ), subject to the same budget constraints
as before.
The …rst order conditions for this problem are given in the appendix; the
next two Lemmas follow easily:
Lemma 4 When parental time is rival, the …rst and second born do not receive
identical treatment.
Proof: see appendix.
This result is driven by the sequential nature of child rearing, and the assumptions that the investment a child needs di¤ers at each stage.
Lemma 5 Suppose parental time is a public input, so taw2 = tbw2 = tw2 : Then
the …rst and second born do not receive identical treatment.
Proof: see appendix.
The argument above applies here, as well: so long as children require di¤erent
inputs at di¤erent stages, and there is at least one period in which siblings in
di¤erent stages share an input and one in which they do not, then identical
treatment will not be optimal.
13
4.3
Probabilistic Second Birth and Spacing of Children
We assumed above that fertility was deterministic. Suppose instead that the
birth of the second child is probabilistic rather than certain. The household’s
intertemporal objective function is now the expected sum of utilities of the
parents. That is, parents maximize intertemporal utility knowing that they
have one child in the …rst period, but are uncertain whether they will have a
second child or not. Thus parents who, in the second period, have one child, and
those with two children, will have made identical investments in the …rst period.
In the second period, the uncertainty is resolved, and second-period maternal
care for the …rst-born will depend on the existence of the second child: the …rst
child will enjoy both more maternal care and child quality if the second child
is present. This is consistent with Zajonc (1976), who notes that only children
tend to perform less well than …rst borns.
Our model also predicts that if parents can keep trying to have children in
later periods, since the older child moves more and more towards self-su¢ ciency,
a child born in the third period would receive less attention than if the child
were born in the second period. Price (2008) …nds that the longer the spacing
between siblings the bigger the gap in quality time reported for each child at the
same age. Price speculates that parents divide family resources equally across
children at each point in time out of a sense of fairness. Our model o¤ers a
di¤erent explanation for the same observed phenomenon: Parents spend equal
time with their children due to the public nature of maternal child care; as the
spacing of child births increases the bene…t of maternal child care for the older
child decreases and hence less time is spent with both children.
5
References
Bergstrom, Th. C. and R. C. Cornes (1983) "Independence of Allocative Ef…ciency from Distribution in the Theory of Public Goods," Econometrica 51,
1753-66.
Bergstrom, Th. C. and R. C. Cornes (1981) "Gorman and Musgrave are
Dual: An Antipodean Theorem on Public Goods," Economic Letters 7, 371378.
Black, Sandra E., Paul J. Devereux and Kjell G. Salvanes (2005) "The More
the Merrier? The E¤ect of Family Size and Birth Order on Children’s Education," Quarterly Journal of Economics, vol. 120, no. 2, 669-700.
Conley, Dalton and Rebecca Glauber (2006) "Parental Educational Investment and Children’s Academic Risk - Estimates of the Impact of Sibship Size
and Birth Order from Exogenous Variation in Fertility," Journal of Human Resources, vol. 41, no. 4, 722-737.
Cunha, F. and J. Heckman (2007) "The Technology of Skill Formation",
NBER working paper 12840, http://www.nber.org/papers/w12840.
Ejrnæs, Mette and Claus C. Pörtner (2004) "Birth Order and the Intrahousehold Allocation of Time and Education," Review of Economics and Statistics,
14
vol 86, no. 4, 1008-1019.
Hanushek, Eric A (1992) "The Trade-o¤ between Child Quantity and Quality", Journal of Political Economy, Vol 100, No. 1, pp. 84-117.
Heckman, James J. (2007) "The economics, technology and neuroscience of
human capability formation", NBER working paper 13195.
Huang, F. (2006) "Child Development Production Function", working paper
24-2006, Singapore Management University, School of Economics, https://mercury.smu.edu.sg/rsrchpubupload/
Kantarevic, Jasmin and Stéphane Mechoulan (2006) "Birth Order, Educational Attainment, and Earnings - An Investigation Using the PSID," Journal
of Human Resources, vol. 41, no. 4, 755-777.
Price, Joseph (2008) "Parent-Child Quality Time: Does Birth order Matter?
Journal of Human Resources, vol. 43, no. 1, 240-265.
Ruhm, Christopher J. (1998) “The Economic Consequences of Parental
Leave Mandates: Lessons from Europe,”Quarterly Journal of Economics, vol.112,
no.1, 285-317.
Ruhm, Christopher J. (2000) “Parental Leave and Child Health,” Journal
of Health Economics, vol. 19, 931-960.
Todd, P.E and K.I. Wolpin (2007) "The Production of Cognitive Achievement in Children: Home, School, and Racial Gap Test Scores", Journal of
Human Capital, vol. 1no.1, 91-136.
Waldfogel, Jane (2006) what children need, Harvard University Press.
Willis, Robert J. (1973) “A New Approach to the Economic Theory of Fertility Behavior,” Journal of Political Economy, vol. 81, no. 2, part 2: New
Economic Approaches to Fertility, S14-S64.
Zajonc, R.B. (1976) "Family Con…guration and Intelligence", Science, New
Series, Vol 192, No 4236, pp. 227-236.
6
Appendix A
Unless speci…cally mentioned, the husband spends all his time in employment
in each period, that is thk = 0; k = 1; 2; 3:
With quasi-linear utility, the utility of parents is
15
6.1
3
X
k=1
uwk + uhk
!
+ 2(uaA + ubB )
= [(ww1 n) (1 taw1 ) + wh1 + 2Y (taY 1 )]
+[(ww1 (1 + (1 taw1 )) n) 1 taw2
tbw2 + (1 + )wh1
xa2
+2 O(taw2 ; xa2 ) + v tbY 2 ]
+[ ww1 (1 + (1
taw1 )) 1 +
1
taw2
tbw2
(1
tw3 ) + (1 + )2 wh1
+2 A(taw1 ; taw2 ; xa2 ) + O(tbw3 ; xb3 ) ]
+2A(tbw2 ; tbw3 ; xb3 )
The household’s problem reduces to one of choosing
the sequences of maternal child care and private goods
for each child: {ftawk ; tbwk gk=1;2 ; tbw3 xa2 ; xb3 g The FOCs are:
9
8
@ua
@ua
>
>
2 @ta1 (p 1) + 2 @taA
=
<
@
w1
Y1
=0
=
(ww1 n) + ww1 (1 (taw2 + tbw2 )
>
>
@taw1
;
:
b
b
a
+ ww1 (1 + (1 (tw2 + tw2 ))(1 tw3 )
9
8
@ua
@ua
2 @ta2 + 2 @taA
=
<
@
w2
w2
a
=0
=
(ww1 n) + ww1 (1 tw1 )+
;
:
@taw2
b
a
ww1 (1 + (1 tw1 ))(1 tw3 )
9
8
@ubA
@ub2
>
>
2 @tb (p 1) + 2 @tb
=
<
@
w2
Y2
a
=0
=
(ww1 n) + ww1 (1 tw1 )+
>
>
@tbw2
;
:
a
b
ww1 (1 + (1 tw1 ))(1 tw3 )
(
)
@ub
@ub
@
2 @tb 3 + 2 @tbA
=
=0
w3
w3
@tbw3
ww1 (1 + (1 taw1 )) (1 + (1 (taw2 + tbw2 )))
@
@ua2
@uaA
=
2
1
+
2
=0
@xa2
@xa2
@xa2
@
@ub3
@ubA
=
2
1
+
2
=0
@xb3
@xb3
@xb3
Proof of Lemma 1:
by comparison of …rst and third …rst order conditions above.
16
xb3
6.2
Proof of Lemma 2:
If there are full economies of scope for maternal child care, then the decision
problem above is modi…ed as follows:
i) the sum (taw2 + tbw2 ) is replaced everywhere in the decision problem, and
its solution, with the single term tw2 ;
ii) the super-script j = a; b is dropped from the second period maternal child
care terms; and
iii) the second and third …rst order conditions above are replaced by the
single condition
8
>
>
<
@
=
>
@tw2
>
:
@ub
@ua
(ww1
2
+ @tb 2 (p 1)
2 @tw2
Y2
n) + ww1 (1 taw1 ) + ww1 (1 + (1
+2
@ua
A
@tw2
+
@ubA
@tw2
taw1 ))(1
tbw3 )
9
>
>
=
>
>
;
Inspection of the FOC’s shows that
(i) the opportunity cost of parental time is equal to ww1 n in periods 1
and 2. Since in the second period the older child also bene…ts from this care,
the marginal bene…t of maternal child care for the younger child must be lower
in the second period than in the …rst. Given concavity, this means higher
maternal child care in period 2 than in period 1.
(ii) In the …nal two periods parents choose both time and consumption goods
for the older child in each period. Since the younger child needs full time
supervision in the second period, and there is no younger child in the third
period, the opportunity cost of parental time is higher in the third period than
in the second. This implies that the marginal bene…t of parental time in the
third period is higher than that in the second. If tbw3 < tw2 , parents have more
disposable income to devote to consumption goods in the …nal period than in
the middle period, so can divert more private goods towards the child in the
2
O
@O
@O
former period. With @x@j @t
0 and tbw3 < tw2 , if xa2 = xb3 , then @x
j
b
@ta :
k
3
wk
2
Then satisfaction of the …rst order conditions for the older child’s consumption
goods requires xb3 xa2 :
6.3
Proof of Proposition 1: Economies of scope for mother
Suppose taw1 = tw2 = tbw3 and xa2 = xb3 are equilibrium choices. Then all FOCs
must hold with equality. However, comparing @t@a with @t@w2 ; the opportunity
w1
cost of maternal time would be the same, but there are more bene…ts of maternal
time in period 2. Hence we have a contradiction.
Now consider the following experiment. Retain xa2 = xb3 , and suppose
a
tw1 = tw2 at the value which satis…es the FOC for tw2 . Then the partial
wrt taw1 <0 (since the marginal bene…t in the …rst period is less than in the
second).. Notice that lowering taw1 will raise the positive terms in the partial
for that variable, without a¤ecting the negative terms, moving that closer to 0.
On the other hand, this change will decrease the absolute value of the negative
17
terms in the partial with respect to tw2 , reducing that partial, while having an
@ua
A
ambiguous e¤ect on the @tw2
term. This partial can still be equal to zero if
this latter term in increasing in taw1 :
6.4
Proof of Proposition 2: mother’s time is rival
Consider the original problem, and the original six FOC’s.
i) Suppose children receive identical treatment, so taw1 = tbw2 ; taw2 = tbw3 and
a
x2 = xb3 are equilibrium choices, satisfying all FOC’s above. Then the bene…ts
of maternal time are equal in the partials @t@a with @t@b ; while the opportunity
w1
w2
cost of maternal time is lower for taw1 than it is for tbw2 : Hence taw1 = tbw2 cannot
be part of the solution.
ii) Suppose the grown child’s earning potential depends only on …rst period
maternal time, so
=
@ubA
@xb3
=
@ua
A
@ta
w2
=
@ubA
@tbw3
= 0:
a) Is
tbw2 and taw1 are symmetric and
a
> t w1
the opportunity cost of maternal
time in period 1 must be less than that in period 2. Since a young child’s
utility is increasing in parental time, tbw2 > taw1 implies that the marginal
bene…t of maternal time for the …rst born must also be less than that of the
second born if the FOC’s are to hold. However, this requires tbw2 < taw1 so we
have a contradiction.
Thus we have taw1 > tbw2
b) Is tbw3 > taw2 possible? Comparing the FOC’s for these two variables, For
maternal time in the second stage of each child, taw2 is likely to be greater than
tbw3 ; because the cost of alternative child care in the second period is likely to
cause a lower opportunity cost of second period maternal time than the impact
of second-period maternal time has on the opportunity cost of third-period
maternal time. Thus bene…ts are the same but opportunity cost is higher in the
third period and so tbw3 < taw2 : This would also imply a lower child consumption
of the second born than the …rst born and hence a lower utility of the second
born child than the …rst born in his second period of life.
@ 2 ua
iii) Suppose @ta @tAa > 0; this would not change the prediction so long as
w2
w1
child care has a bigger impact on the opportunity cost than wage growth through
maternal human capital investment in the earlier periods.
..
tbw2
6.5
tbw2
@ua
A
@xa
2
> taw1 possible? The FOCs for
implies tbw2 + taw2 > taw1 : This implies
Proof of Proposition 3
Introduce income e¤ect, switch o¤ human capital investment character of parenting
18
@
@taw1
=
8
>
<
@ua
(Y (taY 1 )) xp1 + 2) @ta1 (p 1)
Y1
(ww1 n) + ww1 (1 tw2 )
a
(Y (tY 1 ))
+ ww1 (1 + (1 tw2 ))(1 tbw3 )
(
>
:
8
>
<
0
@ub
@ua
9
>
=
>
;
=0
2
+ @tb 1 (p 1)
O(tw2 ; xa2 ) + Y (tbY 2 ) xp2 + 2 @tw2
Y2
(ww1 n) + ww1 (1 tw1 )
a
b
O(tw2 ; x2 ) + Y (tY 2 )
+ ww1 (1 + (1 tw1 ))(1 tbw3 )
)
@ub2
0
O(tbw3 ; xb3 ) xp3 + 2 @tw3
=0
O(tbw3 ; xb3 ) [ww1 (1 + (1 tw1 )) (1 + (1 tw2 ))]
0
@
@tw2
=
@
@tbw3
=
@
@xa2
=
0
O(tw2 ; xa2 ) + Y (tbY 2 ) xp2 + 2
@
@xb3
=
0
O(tbw3 ; xb3 ) xp3 + 2
>
:
(
@ua2
@xa2
@ub2
@xb3
O(tbw3 ; xb3 ) = 0
2
3
O(tbw3 ; xb3 ) : In order for both FOCs to hold, we
O(tw2 ; xa2 ) + Y (tbY 2 ) >
0
0
must have
O(tw2 ; xa2 ) + Y (tbY 2 ) xp2 + 2 >
O(tbw3 ; xb3 ) xp3 + 2 ; but
since () is strictly concave 0 O(tw2 ; xa2 ) + Y (tbY 2 ) < 0 O(tbw3 ; xb3 ) and
through human capital accumulation of the parents and the fact that there are
no child care costs in period 3, xp2 < xp3 : So we have a contradiction.
6.5.1
Cunha-Heckman variation:
First order conditions are
@
@taw1
@
@taw2
@
@tbw2
@
@tbw3
=
=
=
=
8
<
:
8
<
:
8
<
:
(
2
2
9
1
[ (taw1 ) + (1
)(taw2 xa2 ) ] 1 (taw1 ) 1 =
=0
(ww1 n) + ww1 (1 (taw2 + tbw2 )
;
b
b
a
+ ww1 (1 + (1 (tw2 + tw2 ))(1 tw3 )
1
)(taw2 xa2 ) ] 1 (1
) xa2 (taw2 xa2 )
(ww1 n) + ww1 (1 taw1 )
+ ww1 (1 + (1 taw1 ))(1 tbw3 )
9
1
2
[ (tbw2 ) + (1
)(tbw3 xb3 ) ] 1 (tbw2 ) 1 =
=0
(ww1 n) + ww1 (1 taw1 )
;
+ ww1 (1 + (1 taw1 ))(1 tbw3 )
2
[ (taw1 ) + (1
1
[ (tbw2 ) + (1
)(tbw3 xb3 ) ] 1 (1
ww1 (1 + (1 tw1 )) (1 + (1
19
>
;
O(tw2 ; xa2 ) + Y (tbY 2 ) = 0
Suppose taw1 = tw2 = tbw3 and xa2 = xb3 are equilibrium choices, then all FOCs
@
must hold with equality. However comparing @x@ a with @x
b we can see that
6.6
9
>
=
) xb3 (tbw3 xb3 )
(taw2 + tbw2 )))
1
9
=
;
1
)
=0
=0
@
@xa2
@
@xb3
6.7
=
=
2
[ (taw1 ) + (1
)(taw2 xa2 ) ]
1
1
(1
) taw2 (taw2 xa2 2 )
2
[ (tbw2 ) + (1
)(tbw3 xb3 ) ]
1
1
(1
) tbw3 (tbw3 xb3 )
1
1
1=0
1=0
Proof of Lemma 4: Use the …nal FOC (for xb3 ) to simplify the third (tbw3 ), yielding
tbw3
=
xb3
ww1 (1 + (1
taw1 ))(1 + (1
(taw2 + tbw2 ))
(13)
Then do the same for the fourth and second:
taw2
=
xa2
ww1 (1 + (1
taw1 ))(1 + (1 + (1
tbw3 )))
n
(14)
By inspection, the RHS of the two equations are not (in general) equal - will
only be equal for particular combinations of ( ; n).
6.8
Proof of Lemma 5:
If there are full economies of scope for maternal child care, then the decision
problem above is modi…ed as follows:
i) the sum (taw2 + tbw2 ) is replaced everywhere in the decision problem, and
its solution, with the single term tw2 ;
ii) the super-script j = a; b is dropped from the second period maternal child
care terms; and
iii) the second and third …rst order conditions above are replaced by the
single condition
8
>
>
>
<
@
=
>
@tw2
>
>
:
2
f[ (taw1 ) + (1
)(tw2 xa2 ) ]
1
1
(1
1
) xa2 (tw2 xa2 )
+[ (tw2 ) + (1
)(tw3 xb3 ) ] 1 (tw2 )
(ww1 n) + ww1 (1 taw1 )
+ ww1 (1 + (1 taw1 ))(1 tbw3 )
Given this substitution, use the technique above.
20
1
g
1
9
>
>
>
=
>
>
>
;
=0