MODELLING CHILDREN IN A FAMILY BARGAINING MODEL
Elisabeth Gugl and Linda Welling
Department of Economics, University of Victoria
PO Box 1700, Victoria, BC Canada V8W 2Y2
FIRST DRAFT
Abstract:
We present a two period model, with the child as a public good
for the parents. Central to our model is the fact that children need
considerable time investment from parents. In the …rst period a
couple is married, with a young child. Parents decide on private
consumption, total e¤ort to devote to the child, and the division of
this e¤ort, given current and future expected wages and parental
leave policies. In the second period, parents can renegotiate the
marriage contract taking divorce as the threat point.
One innovation of this paper is that it incorporates more realistic
features of children: we model them as a mix of experience goods
and goods with an existence value, and capture their need for regular
time inputs to survive. Another innovation in the paper is that we
explicitly model the a¤ect e¤ect of …rst period childcare time on
the caregiver’s second period market productivity, and hence the
parents’relative opportunity costs in terms of second period market
wages.
Our paper builds on the two period bargaining model in Gugl
(2003). Parents can commit to a …rst period sharing rule, but at the
beginning of their marriage cannot commit to a sharing rule in the
second period. Their incentives to establish a secure threat point for
the second period induce an ine¢ cient allocation of household labor
in the …rst period. Since the quality of care in the …rst period has
repercussions for the child later in life - and hence for the utility of
altruistic parents - this ine¢ ciency has lasting e¤ects. The division
of parental time is generally ine¢ cient, since the parent who can
dictate the time arrangement focuses not only on what maximizes
marriage surplus, but also what maximizes his or her share of resources in marriage. This share depends on the threat point in the
second period and the threat point depends on …rst period division
of time thus causing ine¢ ciency in the model. We consider various
extensions on the basic model.
1
Introduction
The amount of time and energy devoted to children and their care in any economy is mind boggling, involving not only parents and more remote relatives but
1
also the large body of civil servants employed in analyzing and running various public support schemes. Crucial to the economic and social analysis of any
policy directed at the family is an understanding of the transmission mechanisms by which these policies a¤ect the interactions between and well-being of
the various members of a family. In this paper we review the various ways in
which children appear in theoretical economic models, and present a new model
which we argue is more comprehensive. Our model captures essential features
of nurturing children and potentially leads to di¤erent policy implications than
previous models.
Children frequently appear in economic models as household public goods,
and are arguably the most important example of these. (Becker 1984, and
Weiss and Willis (1985)). However, certain essential features which distinguish
children from other household public goods have been ignored/downplayed in
previous work. In this paper we capture these features, and illustrate their
impact on intrahousehold allocation.
First, children require time. Of course, we are not the …rst to incorporate this: various authors have highlighted parental investment in children, and
broadly interpreted this as time. For the most part, ”time is money” in these
models, and staying home to care for a child has little or no impact on future earnings. In this paper, we explicitly incorporate both current and future
implications of this time investment by using a dynamic framework. In the
…rst period children need parental time, and parents choose to allocate their
time between work and leisure/care for their child. The income implications of
parental childcare are re‡ected in the second period wage rates, which depend
on the amount of time spent in the labour force in the …rst period. In the second
period, both parental time and purchased goods are necessary inputs.
Second, children have experience good qualities: there is a learning curve
to parenting, and the pleasure one obtains from a child changes over time. We
capture this by assuming that, for each parent, own time spent caring for a child
in the …rst period a¤ects the utility parents receive from the child in the second.
Thus there are elements here of a ”warm glow” public good.
Third, children are not only a public good within marriage, but they remain
so if the parents divorce. In a number of papers Greenwood and various authors
have developed dynamic general equilibrium models with endogenous fertility
and divorce. In each of these, it is assumed that fathers care about children
only while they are married to the children’s mother, and never choose (in some
cases, are never given the choice) to engage in discretionary child care. (See,
for example, Greenwood et al (2000 ) We consider this as a special case; in our
basic model both parents derive utility from the existence of a child, regardless
of the marital state and the custodial arrangement. We allow the form of this
utility to vary across states.
Fourth, children are durable goods and once you have a child, you are responsible for raising it all your life. We incorporate this idea by …xing the total
investment in the child in the second period based on the amount of investment
in the …rst period. Although the “production function” for the child changes
from the …rst to the second period, capturing the changing needs of children as
2
they grow, having the child in the …rst period commits the parents to maintaining a particular level of child quality.
We incorporate these features into a household allocation model, where
forward-looking parents recognize that they can renegotiate their marriage contract in the second period. The anticipation of future renegotiations, and altered
child care patterns, a¤ect their current decisions. The focus of this paper is how
parental time is determined when all of these elements are incorporated in the
decision.
Our model takes as given the existence of a household consisting of two adults
and one child; the household lives for two periods. The child requires one unit
of parental time in the …rst period, and a mix of time and consumption goods in
the second. Parents make all decisions, based on their endowments of one unit
of time per person per period, exogenously determined …rst period wage rates,
and perfectly anticipated income trajectories dependent on …rst period labour
force participation. The household allocation problem has three components:
who does the childcare in each period? how are goods distributed between the
adults in each period? how much time is devoted to child care /how is child
quality produced in the second period?
In the …rst period, the adults agree on the division of the surplus, but cannot make intertemporal commitments. In the second (and last) period, they
renegotiate their marriage contract and split the gains from marriage equally,
given the payo¤s available if they divorce and the …rst period child investment
choice.
We derive the optimal choices in each period, and investigate deviations from
e¢ ciency. We show that parents disagree over the optimal childcare, and examine conditions under which the wife/mother would prefer the husband/father to
spend more time with the child, so that she could spend more time in the labour
market. The e¢ cient time distribution lies between those allocations preferred
by the wife and the husband.
.
2
2.1
Related Literatures
Family Bargaining/Dynamic
Use of the Nash bargaining solution (NBS) to determine intra-household allocation is typically justi…ed on the basis that because partners in marriage are
involved in repeated interaction, it ”makes sense”to assume that they will manage to achieve an e¢ cient outcome. The (NBS) is chosen to pick the outcome
either because it is straightforward, or because it can be justi…ed as the outcome of an appropriately structured non-cooperative game, or both. A number
of recent papers have presented dynamic models of marital bargaining. A common thread is that although the NBS is e¢ cient in a static/one period model
or alternatively if spouses negotiate over intertemporal utility, with period-byperiod bargaining the outcome is not generally intertemporally e¢ cient. Current
3
choices a¤ect subsequent threat points, and rational agents consider this when
they make their moves. Typically, the ine¢ ciency shows up as an underinvestment in the public good in the initial period - thus reducing the gains from
marriage. This plays out in a variety of contexts: a number of references are
given and discussed in Gugl (2003). Gugl analyses labour supply and household
production, and …nds an oversupply of the former and undersupply of the latter,
relative to the choices which would be made if the spouses could write a binding
contract at the beginning of their marriage. In this paper we build on this by
carefully modelling childcare as the good produced in the household.
2.2
Divorce and Child Custody
From a social perspective, the nurture of children is one of the most important components of marriage, and conditions a¤ecting this nurture in the event
of divorce have long been a focus of economic analysis. The seminal paper of
Weiss and Willis (1985) identi…ed the ine¢ ciency arising from the agency problem created by sole custody, and subsequent papers have further examined the
interaction between custody arrangements and child support payments (ie, Del
Boca and Ribero (2003), Farmer and Tiefenthaler (2003)). Welling and Bearance (2002) extend the Weiss and Willis framework to consider goods shared
by the custodial parent and the child, and show how such goods a¤ect the
willingness to pay child support. Recent closely related papers by Francesconi
and Muthoo (2003) and Rasul (2006) compare existing custodial regimes with
those that would be chosen if parents could write binding contracts determining
custody prior to making investments in children
Here we consider custody as exogenous, and important to the allocation
achieved/ decisions made by a married couple because the custodial regime
determines the threat points should marriage break down in the second period.
Early investments of parental time a¤ect the threat points, and thus the second
period married outcomes through this indirect route, as well as directly through
the impact on second period wages. Other models that assume away utility
derived by a non-custodial parent from children eliminate this link. Here, the
impact of …rst period childcare on second period wages explicitly di¤erentiates
time and monetary investments.
3
The Model
We consider a married heterosexual couple, with partners called the husband (
h ) and the wife (w), over two periods, denoted by p = 1; 2: Partners care for
their minor child over both periods. In this section we …rst develop our model of
child quality, and then discuss the utility parents derive from the child. Finally,
we describe the family budget constraints
4
3.1
Child Quality
The basic resource available to parents is time. Each adult is endowed with one
unit of time in each period;and must choose how to allocate this time between
earning income and spending time with their child. Denote by tjp ; j = h; w;the
time parent j spends with their child in period p ; thus, for example, (1 tw2 )
is the time the wife spends earning income in period 2 : From the perspective
of the child, time with one parent is a perfect substitute for time spent with the
other, so tp = thp + twp . In the …rst period the infant child requires parental
time of t1 = 1: We abstract from any other input in the child ”production
function”, as an infant does not require signi…cant amounts of purchased food,
space etc. Child quality in this …rst period is de…ned to be equal to the inputs, so
q1 = t1 = 1. In the second period, the child needs private consumption goods,
xc > 0, as well as time investment from the parents; these are substitutes to a
limited extent. Child quality in the second period is determined by quality in
the …rst period of life. Second period child quality is therefore given by
q2 = xc t2 = xc (th2 + tw2 ) = q1 = 1;
0 <
; < 1;
+
< 1:
(1)
The assumption that child quality is …xed at unity in each period simpli…es
our analysis considerably.
3.2
Utility from children:
In each period, the utility parents receive from their child depends on both
the child’s quality and the amount of a¤ection they feel for the child. We
assume that a¤ection is increasing in the amount of time spent with the child
in the previous period.Since our model has only two periods, this means that
the …rst period choices drive most of the results. In the …rst period, the time
parent j spends with the child increases the parent’s enjoyment of the child, but
at a diminishing rate. In each period parents must determine the allocation
between themselves of the time spent with the child. While we …x the total
time at one for the …rst period, in the second period parents must also choose
how much of the desired child quality will be provided by time, and how much
by consumption goods. Further, we assume parents receive utility from the
existence of their child; this ”existence value”we denote by , and assume it to
be invariant across parents. Thus the utility derived from the child by parent
j in the two periods is given by the equations below.
cj 1
cj2
=
+ tj1 q1 = ( + tj1 ) (th1 + tw1 ) = + tj1
= ( + tj1 ) q2 = ( + tj1 ) cj1 as
1
5
(2)
Thus, a¤ection
does not enter
parents utility
in the …rst
period,
but
appears as a
predetermined
variable in the
second period.
Parents also derive utility from the consumption of private goods. The
private consumption of parent j in period p is denoted by xjp ; we assume that
these goods, like the private consumption of the child, have a unit price of 1.
The utility of parent j in period p is then
ujp = xjp + cjp
3.3
Budget constraints:
Each parent can earn a wage rate wjp in the labour market; this wage depends on
their education and other labor market factors. We assume that the wife’s …rst
period wage rate is no higher than the husband’s: ww1
wh1 ; an assumption
often made in the literature.1 The second period wage rate depends on …rst
period labour force participation: wj2 = (1 + (1 tj1 )) wj1
In the …rst period spouses commit to a sharing rule that awards of the total
utility to the wife and (1
) to the husband. The second period allocations
are determined by a bargaining process, the outcome of which depends in part
on the custodial regime if the parents divorce. For reasons that will become
clear later, we assume that the …rst period share of wife is bounded below by
child support payments, so 0 <
< 1; where denotes the fraction of the
husband’s earnings that will be paid in child support payments to the wife in
the case of divorce when the wife has sole custody.
4
Second Period
In the second period spouses renegotiate their marriage contract and split any
marriage surplus equally (given the linearity of the UPF, this division is consistent with a symmetric Nash bargaining game where the partners have equal
bargaining weights) Using the divorce outcomes as the threatpoint in the bargaining game, marriage surplus is determined by the di¤erence in the sum of the
parents’utilities in marriage and in divorce. After divorce, both parents receive
utility from having had a child with each other, but because they are apart and
a child is to some extent a relationship-speci…c investment, their utility from
the child is discounted (from the utility if married); denote this discount rate
by d < 1 Thus, if the parents divorce, the utility each derives from this child
is
cdj2 = d ( + tj1 ) < ( + tj1 )
We assume that if the couple divorces the wife will have sole custody of the
child and that support payments are a …xed fraction, , of the non-custodial
parent’s income. This implies that a divorced father simply maximizes his private consumption, since child quality is exogenously given and he cannot change
the support payment. In the divorce threatpoint, with the ex-husband paying
1 Alternatively,
we de…ne the wife as the partner with the lower …rst period wage rate.
6
of his income in support, his utility is
dh
d
= (1
< 1
) wh2 + d ( + th1 ) :
(3)
The divorced mother, who has custody of the child, must determine her time
allocation as well as the division of consumption goods between herself and the
child. Thus her problem is
max dw
xc ;tw2
s:t: xc [(th2 + tw2 )]
=
=
=
max f(ww2 (1
tw2 ) + wh2
xc ;tw2
xc ) + d ( + tw1 )g
(4)
1
n
max ww2 + wh2 + d ( + tw1 )
tw2
ww2 tw2
(th2 + tw2 )
Notice that many of the terms in the above problem are functions of tw1 , and
so are predetermined; therefore solving the mother’s second period problem is
equivalent to minimizing the cost of producing child of unit quality (of course
quasi-linear utility leads to no income e¤ect!) where the inputs of time and
child’s private goods have unit prices of ww2 and 1, respectively: The …rst order
condition for the time spent with her child is
d(dw )
dtw2
( + )
=
ww2 +
tw2
=0
Solving this for tw2 , and substituting this value into the production constraint
to solve for the optimal amount of the child’s private goods, yields
+
tw2
=
xc
=
ww2
ww2
+
Observe that as ww2 increases, the mother chooses less childcare time in the
second period, substituting more consumption goods to maintain child quality.
Overall, given the assumption that a unit of the child’s consumption goods has
a price of unity, the minimized cost of maintaining a child of unit quality is
(ww2 )
= tw2 ww2 + xc
(5)
1
+
= ww2+
0
+
1
+
= ww2+ K where K =
> 0> "
7
+
o
The minimum cost of second period child quality is increasing (at a decreasing
rate) in the contemporaneous wage rate of the caretaker, Since second period
wages are increasing in …rst period labour force participation, the …rst period
time allocation decision of the couple will a¤ect the second period cost of childcare. 2
= ww2+ K = (1 + (1
tw1 ))ww1+ K;
@
<0
@tw1
(6)
Having solved for the optimal choices of the divorced mother, the mother’s
utility in divorce is
dw
ww2+ K + d ( + tw1 )
+ d( + tw1 )
= ww2 + wh2
= ww2 + wh2
(7)
The parents’utilities when divorced provide the threat points for the cooperative solution in the second period. Notice that if parents stay married (as
they do in this model), with quasi-linear utilities the second period utility possibility frontier is linear over a certain range of utilities. We assume that negative
transfers cannot be made - that is, a transfer cannot exceed the combined labor
income of the ex-spouses.
In any case, the sum of second period utility is maximized when the values
of private goods for the child, and the time allocations of the parents, satisfy
max
xc ;th2 ;tw2
[wh2 (1
th2 ) + ww2 (1
tw2 )
xc ] + 2 + tw1 + th1
(8)
s:t: 1 = xc (tw2 + th2 )
or equivalently
h
max wh2 (1
th2 ;tw2
th2 ) + ww2 (1
tw2 )
(tw2 + th2 )
+ 2 + tw1 + th1
i
2 Given our assumptions about the parameters
and
an interior solution is indeed a
maximum but we need to make sure that the solution is feasible, given the time and …nancial
constraints of the wife. First, child care time cannot exceed the wife’s total available time in
the second period, so tw2
1. This will always be true if
:
ww2
Second, the total cost of the child cannot exceed the wife’s full income, so
ww2+
1
+
ww2 + wh2 ;
+
8
The …rst order conditions are
@U2
@th2
@U2
@tw2
( + )
=
wh2 +
(tw2 + th2 )
=
ww2 +
(tw2 + th2 )
=0
( + )
=0
We have a corner solution whenever wh2 6= ww2 : so long as the parents remain
married, the partner with the lower opportunity cost of time does all the child
care. Suppose ww2 < wh2 : Then the …rst order condition for tw2 binds, and
reduces to
@U2
@tw2
( + )
=
ww2 +
(tw2 )
=0
Notice that this equation is the same as that determining the mother’s optimal
time allocation when the parents are divorced, and she is the custodial parent.
This occurs because we have assumed that child quality is predetermined; in a
more general model where quality is a choice variable we would expect action on
this margin as well. Thus in this model, the sole source of surplus from marriage
in the second period is that d < 1! If, on the other hand, wh2 < ww2 ;then the
couple also bene…ts from having the lower waged spouse taking care of the child
in marriage, while custody is awarded to the higher-waged spouse in divorce.3
Result 1: In the Nash bargaining solution in the second period, the decomposition of childcare into parental time and child consumption goods is the same in
marriage as when the mother has sole custody in the divorce threatpoint. The
husband works full-time,while the wife works part-time. The minimized cost of
childcare is increasing in the …rst period wage of the wife and the second period
increment to the wage ( );and decreasing in the …rst period childcare time of
the wife.
For the moment we assume that ww2 is always lower than wh2 ; even if the
wife works full-time and the husband takes care of the child in the …rst period.
For the NBS, each spouse receives half the marriage surplus in addition to their
respective threatpoint utility, so
3 The assumption that the wage rate of the wife does not exceed the wage rate of the
husband in the …rst period does not rule out the possibility that the wife’s second period wage
rate is higher than her husband’s.
9
uj2
=
uj2
=
U2
dh dw
+ dj
2
dk + dj
2
U2
The parents will split the second period utility equally only if they have equal
utilities if divorced. Since divorce utilities depend on …rst period decisions
as well as the initial wage rates and the child support transfers, equal utilities
within marriage is unlikely in general. To verify that marriage and not divorce
is the outcome in the second period, consider …rst the wife’s utility
2
3
wh2 + ww2
+ (2 + tw1 + th1 )
4
5
(1
) wh2 d ( + th1 )
+ wh2 + ww2
+ d ( + tw1 )
uw2
=
1
2
uw2
=
[ wh2 + ww2
]+
+
1
(tw1 (1 + d) + th1 (1
2
(9)
d))
The …rst term in square brackets is the private consumption of the mother
when divorced, and is positive (non-negative?) by assumption. The last term in
square brackets is always strictly positive as d<1. We therefore conclude that
the wife is better o¤ in marriage than divorced. The husband’s second period
utility in marriage
2
3
w + ww2
+ (2 + tw1 + th1 )
1 4 h2
5
+ (1
) wh2 + d ( + th1 )
uh2 =
(10)
2
wh2 ww2 +
d ( + tw1 )
uh2
=
(1
) wh2 +
+
1
(tw1 (1
2
d) + th1 (1 + d))
also exceeds his utility in divorce.
Result 2: The couple stays married in the second period, since both are
strictly better o¤ married than divorced.
Consider comparing the NBS utilities with an outcome in which each spouse
obtains utility equal to their own full income plus the utility derived from the
child in marriage, minus half the cost of the child. For this latter case, spouse
j would receive utility of
uj2
= wj2 + ( + tj1 )
10
ww2+ K
Although we have assumed that parents take the child support to be a …xed
fraction of the non-custodial parent’s income, independent of the income of
the custodial parent, for purposes of comparison we consider a "representative"
divorced couple, for whom the child support payments are such that each parent
bears half the opportunity cost of raising the child, so wh2 = ww2+ K. In this
case under NBS we have
BS
uN
= ww2
w2
ww2+ K +
+
1
[tw1 + th1
2
ww2+ K +
+
1
[tw1 (1
2
d (th1
tw1 )]
for the wife and
BS
uN
= wh2
h2
d) + th1 (1 + d)]
for the husband. Only if
tw1 + th1
d (th1
2
tw1 + th1 + d (th1
2
tw1 )
tw1 )
= tw1
= th1
that is, will each spouse obtain utility equal to their own full income plus the
utility derived from the child in marriage, minus half the cost of the child in the
NBS, only if th1 = tw1 . This also means that both parents have invested equally
in learning to love their child. If the father spends less time than the mother in
the …rst period then, given the mother has sole custody in divorce, the father
receives less than his own full income minus half the child cost plus child utility
in marriage. Why? Since the child quality is …xed at one in each period, a
parent’s own …rst period time investment determines the second period utility
each parent derives from the child. Unilateral decreases in the amount of time
the father spends with the child in the …rst period decrease his utility when
divorced, while increasing the …rst period commitment - and hence the second
period utility - of the mother.
5
First Period
Given the renegotiation in the second period, using divorce as the threatpoint,
we can now solve for the household’s labour allocation decisions in the …rst period. Once we have these, we can track the consequences of this …rst period
decision for second period wages, labour force participation, the trade-o¤ between goods and time in maintaining child quality, and the endogenous a¤ection
for the child. While the couple never divorces in this model, it is nonetheless
true that the divorce regime a¤ects the marriage allocations.
Although our functional forms are simple, the model is su¢ ciently complex
that explicit solutions are not possible. Moreover, the constraint that the child
11
requires full-time care in the …rst period (with no third party available to provide
it) means that we cannot always appeal to standard solution concepts. In this
section we …rst consider two ad hoc rules for determining childcare, and ask if
the parents would agree on one or both of these rules. We then solve for the
e¢ cient labour allocation in the …rst period, and then compare other solutions
to this benchmarks. To make the comparisons somewhat easier, in all cases
we solve for the childcare time of the wife,tw1 , and invoke the constraint that
th1 = 1 tw1 .
5.1
Exogenously determined allocations:
We consider two allocations which might arise as rules of thumb, and ask if
there would be agreement between the parents as to which rule was better.
5.1.1
Gender role specialization:
Suppose that, because of enforced gender roles, the wife does all the child care,
so tw1 = 1;and th1 = 0. In this case, the intertemporal utilities of the husband
and wife are
Uh
= uh1 + uh2 = (1
) (wh1 + 2 + 1)
[1 d]
) (1 + ) wh1 + +
2
+ (1
Uw
= uw1 + uw2 = (wh1 + 2 + 1)
+ww1 + (1 + ) wh1
5.1.2
(11)
(12)
(ww1 ) +
[1 + d]
+
2
Equal sharing
Suppose parents split …rst period child care duty equally. Then the intertemporal utilities are
Uh
=
(1
)
+ (1
Uw
ww1 + wh1
+ 2 + 2 :5
2
1
) 1+
wh1 + [2 + 1]
2
2
ww1 + wh1
+2 +2
2
=
+
1+
2
wh1
:5
1+
12
2
(13)
+ 1+
ww1
ww1
2
1
+ [2 + 1]
2
(14)
Would the partners agree or disagree on which rule is better? That is, who
would prefer (tw1 ; th1 ) = (1; 0); and who would prefer (tw1 ; th1 ) = (:5; :5)? Note
that if the husband could choose between these two child care arrangements, he
would prefer that his wife were the sole care taker if
(1
) (2
1
1) + d
2
:5
<
(1
)
wh1
ww1
2
+ (1
)
3
wh1
4
To evaluate this inequality, recall that d 2 (0; 1) and < 1. Then the
LHS of the above inequality is more likely to be negative the smaller are both
these parameters: that is, the greater the reduction in utility from a child upon
divorce and the less important …rst period childcare time is in determining the
parent’s a¤ection for the child. The RHS of this inequality is unambiguously
positive, given our assumptions. Moreover, this latter expression will be larger
the greater is the di¤erence between the spouses’…rst period market wages, the
smaller the fraction of the husband’s income paid as child support in the event
of divorce, and the larger is the wage increase in the second period from …rst
period labour force participation.
The wife would rather do all the child care than share it equally with her
husband if
wh1
ww1
2
>
(2
:5
1)
+
4
( wh1
ww1 ) +
1+
+
4
1
(ww1 )
d
2
the LHS is positive. Thus the wife would always rather do all the child care than
share the care equally with her husband, if these are the only choices available.
Result 3: it is possible that the couple would agree on standard gender roles,
but not on equal sharing, if these were the only two possibilities.
Again, notice that in this model divorce serves as the threat point in the
second period, but never occurs.
5.2
E¢ cient time allocation:
Consider …rst the choice which maximizes the intertemporal utility of the (always married) household. Using the …rst period time constraint, the problem
is to
max[(1
tw1
tw1 )ww1 + tw1 wh1 + 2 + tw1 + (1
+(1 + tw1 )ww1 + (1 + (1
13
tw1 )
tw1 ))wh1 + 2 + (1
(15)
(tw1 )]
The …rst order condition for the e¢ cient amount of …rst period childcare
undertaken by the wife, denoted by tew1 , satis…es
(1 + )(wh1
e
ww1 ) + [tw1
1
(1
tew1 )
1
]
@ =@tew1 = 0
(16)
Notice that if …rst period utility is either linear in own child care time
( = 1) or independent of own child care time ( = 0), the left hand side of this
equation is always strictly positive, so that it is always e¢ cient for the wife, the
lower-waged spouse, to do all the child care in both periods.
Since an increase in the time the wife spends in childcare in the …rst period
decreases her second period wage rate and hence the cost of child care in the
second period, @ =@tew1 < 0: Thus at any solution for which the parents split
e 1
…rst period childcare, so tew1 (; 1);it must be true that [tw1
(1 tew1 ) 1 ] < 0;
e
implying that tw1 > 0:5:
Result 4: if the parents do not completely specialize in the …rst period, it is
e¢ cient for the wife to do more of the childcare in the …rst period and all of
the childcare in the second. This means that in the second period, the wife will
care more for the child than will the husband.
5.3
First mover outcomes:
If the spouses were to choose their own childcare time simultaneously, there
is no guarantee that in the Nash Equilibrium time spent with the child will
sum to the required full time child care. To circumvent this problem, we now
investigate the outcome when one spouse can determine the division of child
care by him or herself (i.e. has all the bargaining power in the …rst period over
t1 ). That is, we suppose that one of the partners can choose their childcare
time, knowing that the other partner will cover the remaining time. For the
moment we assume that no matter how much time the mother spends in the
labour force in the …rst period, her second period wage rate can never exceed
that of her husband:
max
ww2
5.3.1
=
min
(1 + ) ww1 < wh2
= wh1
Wife chooses childcare
Consider …rst the outcome when the wife chooses the time allocation. Using the
constraint that tw1 + th1 = 1;given no discounting, her problem is to maximize
her intertemporal utility; her choice solves
maxtw1 [ ((1 tw1 ) ww1 + tw1 wh1 + 2 + tw1 + (1 tw1 ) )
+ (1 + tw1 )wh1 + (1 + (1 tw1 )ww1
+ + 21 [1 d (1 2tw1 )]
14
(17)
w
The FOC for her choice, ww1
;satis…es
[(wh1
ww1 )+ [tw
w1
1
(1 tw
w1 )
1
]]+
wh1
ww1
@ =@tw1 +d = 0 (18)
If the couple split the total marriage utility equally in the …rst period, so =
0:5, then the wife would choose her childcare time (and, hence, her husband’s)
e¢ ciently if [ ( wh1 ww1 ) @ =@tw1 + d]= = (wh1 ww1 ) @ =@tw1 , where
the partial for childcare is evaluated at the e¢ cient level. This will hold if
@ =@tw1 + (d
(1
)wh1 ) = 0: The …rst term is the decrease in second period
childcare cost of an increase in the wife’s …rst period childcare time. The second
term is the increase in the divorced husband’s share of an increase in his second
period wage due to a increase in his …rst period childcare time, and represents
the change in the second period utility of the husband as the wife’s …rst period
childcare time decreases. ???
5.3.2
Husband chooses childcare
Suppose now that the husband dictates how much time he and the wife should
spend taking care of the child (again, taking into account both that the sum
of the time devoted to child care in the …rst period must be one, and that the
second period child quality is unity). Then his problem is
max (1
) [(1
tw1
+ (1
tw1 )ww1 + tw1 wh1 +
) (1 + tw1 )wh1 +
+ tw1 +
+ [1 + d (2(1
tw1 )
+ (1
tw1 )] (19)
1)] =2
The FOC for thw1 is
(1
)[(wh1
h
ww1 ) + [tw1
1
(1
thw1 )
1
]] + wh1
d=0
(20)
) 2 wh1 + (1
) (wh1 ww1 ) = d;
Notice that thh1 = 1 tw1 = 21 if (1
that is, if the positive e¤ects on father’s share of …rst and second periods’family
full income are equal to the negative e¤ect of decreasing the father’s love for the
child in divorce.
If (1
) wh1 +(1
) (wh1 ww1 ) d > 0; for an interior solution we must
h 1
have [tw1
(1 thw1 ) 1 ] < 0; which implies thw1 > 12 > thh1 : if the ”income
e¤ect” dominates the ”love e¤ect”, the father will choose to devote less than
half the time to child care.
Result: comparing individual choices to e¢ cient outcome: (pending; both
would choose wife doing more than half of …rst period care, as in e¢ cient outcome)
15
6
Discussion and implications:Why don’t you
take care more?
In the equilibrium of our model, the lower-waged spouse, the wife, works parttime in the second period while the higher-waged spouse works full time. For
a wide range of parameter values, in the …rst period all the child care is done
by the wife. By construction, total parental time is lower in the second period
than in the …rst; thus our model re‡ects the pattern of a mother re-entering the
labour force as the child grows. When maternal-custody-upon-divorce serves
as the threatpoint for the second period bargaining game, even when initial
wage rates are su¢ ciently similar that parents are inclined to share childcare in
the …rst period, maintaining child quality in the second period becomes more
expensive the more the wife works in the …rst period, reinforcing the incentive
for the lower-paid spouse to undertake more of the …rst period care. Changing
the child custody default in the divorce threatpoint would alter the results of our
model, as joint custody should increase the incentive for shared …rst period child
care. An alternative interesting change would be to consider the implications
of child support being …xed as equal to half of the second period costs ,rather
than a …xed proportion of the husband’s income. When child support is a
…xed proportion of the husband’s income, the wife incurs all the bene…ts of
lower second period costs, increasing her incentive to undertake the …rst period
childcare. If actual childcare costs were split, the wife would be more likely to
work part-time in the …rst period, since her incentive to minimize these costs
declines. In a model where divorce is only a threat, but never occurs, this is not
e¢ cient; however, if divorce could arise - as, perhaps, the result of a bad draw
on the realization of marital happiness independent of the child - this might
change.
Our model endogenizes parental altruism, rather than taking caring as completely exogenous. The exogenous component of caring, , is supplemented
by "learned" caring, as a parent spends time with the child in the early stages
of the child’s life. We have modelled these components as strictly positive and
common to both parents; clearly we could make these parent-speci…c, and trace
the results of these di¤erences through. The Greenwood et al papers referenced
earlier explicitly set both the existence value (for divorced fathers) and the husband’s …rst period child care time, th1 , equal to zero; we extend the choice set
of the couple in the …rst period to allow the husband to spend time with the
child. This choice undoubtedly complicates the problem facing the household,
but does take a step towards reality.
Of the simplifying assumptions in our model, the most serious is that of
quasi-linear utility for the parents. With quasi-linear utility thesecond period
utility possibility frontier has a linear segment, which simpli…es the solution
of second period problem and hence makes the …rst period problem somewhat
tractable. This simplicity comes at the cost of income e¤ects: all of the results
16
we have on the trade-o¤s involved in childcare derive from substitution e¤ects.
It is possible to introduce income e¤ects and public goods, and still retain transferable utility, but this becomes very complicated. One implication of this is
that perhaps the Nash bargaining solution is not the best concept to use for
examining the intertemporal choice problems which are endemic in family life.
7
REFERENCES (incomplete)
Becker, G (19...), A Treatise on the Family
Del Boca, D. and R. Ribero (2003), ”Visitations and Transfers after Divorce”, Review of Economics of the Household, 1, pp.187-204
Farmer, A. and J. Tiefenthaler (2003), ”Strategic Bargaining over Child
Support and Visitation”, Review of Economics of the Household, 1,
205218
Francesconi, M. and A. Muthoo (2003), ”An economic model of child custody”, IZA dp 857
Greenwood J., N. Guner and J. Knowles (2002?), ”More on Marriage, Fertility, and the Distribution of Income”, International Economic Review
Gugl, Elisabeth (2003), ”Changes in personal income tax and alimony regulations in a two-period bargaining model of the family”, mimeo, University of
Victoria
Rasul, I. (2006), ”The economics of child custody”, Economica, vol __, pp.
1Weiss, Y. and R. Willis (1985), ”Children as collective goods in divorce
settlements”, Journal of Labor Economics, 3, pp. 268-292
Welling, L. and M. Bearance (2002), ”Who’s minding the kids? An economic
comparison of sole and joint custody”, Journal of Socio-Economics, 31, pp. 1529
17