Intra-household Allocation
Conflict and Cooperation in the
Family
Non-cooperation as a starting point
• Assume that each person will act to
maximize their welfare as they evaluate it,
given the predicted behaviour of others.
• A “Nash equilibrium” is, in its essence, the
general formulation of this assumption.
• Provides a foundation for modelling
cooperative behaviour within a family.
Basic model
• Focuses on the behaviour of couples with
children.
• Expenditures on children (G) are assumed
to be a public good for the parents, but
each parent has their own preferences.
• The preferences of each parent j are
represented by the utility function:
• Uj = Uj(xj,G), where xj. is parent j’s private
consumption.
Voluntary contributions to expenditures
on children, G
•
•
•
•
gj (gj0), and so xj = yj - gj and
G= (g1+g2)/p,
where yj is the income of parent j and
p is the price of the child good relative to
that of the private good, xj.
Nash equilibrium
• Each parent chooses their contribution to
child expenditures to maximize their utility,
taking the contribution of their partner as
given; that is,
• parent j chooses gj to maximize
• Uj(yj-gj, (g1+g2)/p), subject to gj0.
• Implies UjG(xj*, G*)/Ujx(xj*, G*)  p , j=1,2.
Solution
• If UjG(xj*, G*)/Ujx(xj*, G*) = p for both
parents, it provides two equations in g1
and g2, which describe their strategies,
and these can be solved for the Nash
equilibrium contribution.
• If UjG(xj*, G*)/Ujx(xj*, G*) < p for parent j
(he/she is ‘too poor’), their contribution is
zero.
Example: Uj = jln(xj) + (1-j)ln(G),
When g1>0 and g2>0 and p=1,
• (2a) g1 = (1-1)y1 - (1g2)
• (2b) g2 = (1-2)y2 - (2g1)
• Nash equilibrium illustrated in Figure 2.1,
with 1>2
Figure 2.1
Reaction Functions
g2
1  α1  y1
α1
(1-α2)y2
(2a)
g2N
(2b)
g1N
(1-α1)y1
1  α2 

 y 2
 α2 
g1
Example solutions
• (3a)
g1N = [(1-1)y1 - 1(1-2)y2]/(1-12)
• (3b)
g2N = [(1-2)y2 - 2(1-1)y1]/(1-12)
• The contribution of parent j is increasing in their
income and decreasing in the other parent’s.
From equations (3),
• (4a)
GN = [(1-1)(1-2)(y1+y2)]/(1-12)
• (4b)
x1N = [1(1-2)(y1+y2)]/(1-12)
• (4c)
x2N = [2(1-1)(y1+y2)]/(1-12)
When both parents contribute
• Non-cooperative outcomes only depend
on the joint income of the parents, y1+y2.
• E.g. in example,
• (4a)
• (4b)
• (4c)
GN = [(1-1)(1-2)(y1+y2)]/(1-12)
x1N = [1(1-2)(y1+y2)]/(1-12)
x2N = [2(1-1)(y1+y2)]/(1-12)
Only one parent contributing
• Taking the father as parent 1, he will not
contribute when
• y1/(y1+y2) < 1(1-2)/(1-12)
• In this case,
• GN=(1-2)y2,
• x1N=y1 and x2N=2y2.
• Analogously, for the mother—see Fig. 2.2
• Individual incomes matter for the outcome.
Figure 2.2
Contributions to Child Expenditure
g1 = 0, g2>0
g1>0, g2>0
g1>0, g2 = 0
1 1   2 
1   2 
1  α1α 2
1  α1α 2
y1
y1  y 2
Non-Cooperative Equilibrium
The non-cooperative equilibrium can
indicate:
• what the “fallback position” would be if
communication and bargaining within the
family break down;
• how individual preferences and incomes
affect this fallback position.
‘Separate spheres’ non-cooperative
model (Lundberg and Pollack)
• Two household public goods.
• High costs of the coordination that would
be required for choices about voluntary
contributions to public goods based on
relative preferences and incomes.
• Traditional gender roles provide a focal
point that avoids coordination problems:
• Man decides about one, the woman about the
other; ‘social prescribed’ spheres of influence.
‘Separate spheres’ implications
• Reaction functions analogous to earlier, in
which the man’s demand functions depend
on the purchases of the ‘woman’s public
good’ and vice versa.
• Nash equilibrium: intersection of the two
public good demand functions.
• Non-cooperative equilibrium allocation
depends on individual incomes, y1 and y2.
Low coordination costs
• Outcomes are purely determined by preferences
and relative incomes.
• There is at most one public good to which both
will contribute (Browning et al).
• When the intra-family income distribution is such
that there is such a public good, individual
incomes do not matter for outcomes.
• When parents’ incomes are ‘very similar’, each
contributes to a different public good—looks like
‘separate spheres’ model.
Cooperative Equilibrium
• Cooperation between parents to achieve
an allocation between parents’ private
consumption and child expenditure such
that one parent cannot be made better off
without making the other worse off;
• i.e. a Pareto-efficient allocation.
• must maximize U1(x1, G) subject to:
• (a) U2(x2, G)  U2* and (b) y1 + y2 = x1 + x2
+ pG
Equivalent Formulation
•
•
•
•
Equivalently, it must maximize
U1(x1, G) + U2(x2, G)
subject to constraint (b),
where  is the Lagrange multiplier
associated with the “efficiency constraint”
(a).
• This is what Chiappori (1992) calls the
“collective” approach (model).
Cooperative solution
• Maximisation implies that:
• U1x(x1e, Ge)= U2x(x2e, Ge)
• p = U1G(x1e, Ge)/U1x(x1e, Ge)
+ U2G(x2e, Ge)/U2x(x2e, Ge)
i.e. the Samuelson (1954) condition for the
efficient provision of public goods.
• Cf. UjG(xj*, G*)/Ujx(xj*, G*) = p in Nash
equilibrium→ inefficiency.
Utility possibility frontier
• The locus of Pareto optimal utility levels
for the two parents corresponding to given
values of y1, y2, p and the parameters of
their utility functions.
• Different  imply different positions on
frontier.
Figure 2.3
Utility Possibility Frontier
U2
A
U2N
N
B
U1N
U1
Demand functions
• G=Ge(y1+y2, p,)
• xj=xje(y1+y2, p,), j=1,2
• In general,  is a function of individual
incomes and the price of the public good;
i.e. =(y1,y2,p)
• Cooperation and efficiency are indicated
by the presence of this common
(unknown) function in all the demand
functions.
Cross-equation restrictions
• Because G/yj=(Ge/)(/yj), j=1,2,
and similarly for xj ,
• (G/y1)/(G/y2)=(/y1)/(/y2)=
(x1/y1)/(x1/y2)=(x2/y1)/(x2/y2)
• i.e. the marginal propensities to consume
out of different sources of income must be
proportional to each other across all of the
goods.
• Provides a test of intra-family efficiency.
Example: utility functions used earlier
pGe = [(1-1)+ (1-2)](y1+y2)/(1+)
x1e = 1(y1+y2)/(1+)
x2e = 2(y1+y2)/(1+).
Equivalent to giving each parent a share of
joint income, 1/(1+) and /(1+)
respectively, and letting each choose
according to their own preferences.
• An income “sharing rule” (Chiappori 1992).
•
•
•
•
Income effects
• Possible interpretation of (y1,y2,p) is that
it reflects bargaining in the family, with 
increasing in y2 and decreasing in y1.
• Define =/(1+); then in example,
• Ge/y2={[(1-)(1-1)+ (1-2)] +
(y1+y2)(1-2)(/y2)}/p
• x2e/y2=2 + (y1+y2)2(/y2)
Two effects of mother’s income (y2) on child
expenditure (G):
1. It increases family income (y1+y2);
•
(1-)(1-1)+ (1-2) in example.
2. It may increase mother’s bargaining
power (/y2>0);
•
•
(y1+y2)(1-2)(/y2) in example
could reinforce (1>2) or offset (1<2) the
income effect .
Distribution factors
• Variables that affect the intra-family
decision process (i.e.  ) without affecting
individual preferences or resources.
• These may include marriage market
attributes and divorce laws that, in some
circumstances, affect bargaining between
spouses within marriage.
• Also, person’s share of household income.
Inferences about individual welfare
• Suppose we can observe x1 and x2
separately (often can only observe x1 +x2 );
• i.e. man and woman consume some different
goods (e.g. men’s and women’s clothing).
• Let  be dependent on mother’s income
share, s2 =y2/(y1+y2).
• From above, holding y1+y2 constant,
• x2e/s2=(y1+y2)2(/y2)
• E.g. if /y2>0, higher s2 increases x2 and,
conditional on G, the mother’s welfare.
‘Caring’ preferences
• Preferences take the form
V1 = V1[U1(x1,G), U2(x2,G)],
• and similarly for parent 2, where the Uj()
are “private” utility indices for each parent
and Vj[] is “social utility” to parent j
• A natural way to represent parents caring
for each other (i.e. Vj/Uk>0 for jk).
• If Vj=Uj(xj,G), then these preferences
collapse to egoistic ones.
Figure 2.3
Utility Possibility Frontier
U2
A
U2N
N
B
U1N
U1
Implications
• Demand functions of the same general form as
above.
– Any outcome that is efficient in the context of caring
preferences would also be efficient if the parents were
egoistic.
• Points A and B in Figure are best choices under
caring—the indifference curves associated with
V2(U1,U2) and V1(U1,U2) respectively are tangent
to the utility possibility frontier.
• Caring preferences eliminate two segments at
the extremes of the frontier because parents
who care for one another do not want their
partner’s ‘private’ utility to fall below some
minimum level.
– above A and below B, only joint income matters.
Bargaining within Families
• Noted that  may reflect bargaining, but a
bargaining theory was not advanced.
• Each partner has the alternative of not
cooperating, providing an alternative level
of utility, which we call their threat points.
• Possible cooperative solutions lie on UPF,
between the two threat points, T1 and T2.
Figure 2.4
Two Bargaining Rules:
Nash Bargaining and Dominant Partner
U2
Ni = [U1 (·) - Ti1][U2 (·) - Ti2].
NBi = ‘Nash-bargained’ solutions.
Di = ‘dominant partner’ solutions.
NBB
NB = constant
NBA
TB2
DB
DA
TA2
T1
NA =constant
U1
Bargaining rules
• Dominant partner--couple maximizes his
or her utility.
– E.g. father dominant, solution is DA in figure
– he would offer his wife just enough to accept
this arrangement—her threat point.
• Nash bargaining: maximizes the product of
the gains from cooperation, where these
gains are U1-T1 and U2-T2—NBA in Figure.
• Effect of change in threat point—Figure.
What should be the threat point?
• Rubinstein-Binmore multi-period bargaining
game.
• Partners alternate in proposing how to “divide
the cake”: utility from cooperation in the family
which we normalise to be 1.
• i.e. u1+u2=1, where uj is the proposed utility of
partner j in marriage .
• In any period in which they remain married but
do not reach an agreement, partner j receives
utility bj.
Equilibrium of bargaining game
• b1+b2<1 due to inefficiency of non-cooperation.
• If either partner asks for a divorce, they will get
m1 and m2 respectively, where m1+m2<1.
• If the time between offers is “small”, the unique
equilibrium of the bargaining process is
• uje = bj + (1-b1-b2)/2, j=1,2;
• that is, the gains from cooperative relative to
non-cooperative marriage are shared equally.
• Three cases.
Three cases
a) bj + (1-b1-b2)/2 > mj, j=1,2
•
Divorce threat not credible for either party
b) b1 + (1-b1-b2)/2 < m1,
•
Divorce threat is credible for the husband
and u1e=m1 and u2e=1-m1>m2
c) b2 + (1-b1-b2)/2 < m2
•
Divorce threat is credible for the wife and
u1e=1-m2>m1 and u2e=m2
Figure 2.5 Bargaining
U2
Case (a)
Divorce threat
not credible
*
U i  bi 
1  b1  b2 
2
U(U,U,U )
e e* e e*
1
22
1
1
 (m1,m2)
(b1,b2)
U1
1
U2
Case (b)
Divorce threat credible
for person one
*
U i  bi 

1  b1  b2 
2

e*
e*
(U
U 1 ,1U,U1 2)
(m1,1-m1)
(b1,b2)
 (m1,m2)
U1
Threat points and non-cooperative
marriage
• Suppose divorce threat is not credible.
• Then Non-cooperative marriage provides
threat point (i.e. b1 and b2 above).
• Does individual income affect threat point?
• No, when both contribute to the public good in
‘voluntary contributions’ formulation above
(i.e. when their incomes are ‘similar’).
• Yes, in ‘voluntary contributions’ formulation
when incomes are sufficiently ‘dissimilar’.
• Yes, in ‘separate spheres’ formulation.
Home production
• Explicit treatment of household production
is standard in family economics.
• Consider a very simple home production
technology in which each parent may
contribute time (tj) to the raising of their
children:
G= h1t1 + h2t2
• where hj is the productivity of j’s time.
• Have replaced purchases of G with home
production of it.
Non-cooperative equilibrium
• Even if parents do not cooperate, it may
be in the interest of one parent to make
financial transfers to the other.
• Private consumption of the mother is given
by x2 = (T-t2)w2 + y2 + s1,
• where T is total time available, wj is parent
j’s wage, yj is j’s non-labour income and s1
is transfers from the father to the mother.
• Analogous for father.
Mother’s decision
• Assume the mother chooses her time
allocation, t2, to maximize her utility, taking
the time allocation of her husband and the
financial transfer from him as given.
• She chooses t2 to maximize U2((T-t2)w2 +
y2 + s1, h1t1 + h2t2), which implies
• U2G(x2*,G*)/U2x(x2*,G*) = w2/h2
• LHS is MRS and RHS is MC.
Mother’s reaction function
•
•
•
With the Cobb-Douglas utility function
assumed earlier, this condition, the
budget constraint and the home
production technology implies that her
reaction function is
t2 = (1-2)[(w2T+y2+s1)/w2)] - 2h1t1/h2
Best strategy: reduce her home
production time when father increases
his.
Father’s decision
• He chooses his time allocation and monetary
transfers to his wife, s1, so as to maximize his
utility, U1((T-t1)w1 + y1-s1, h1t1+h2t2), subject to:
• Mother’s reaction function
• t10, s10
• Implies two conditions:
w1/(1-2)h1  U1G(x1*,G*)/U1x(x1*,G*) = (11)x1/1G
w2/(1-2)h2  U1G(x1*,G*)/U1x(x1*,G*) = (11)x1/1G
Non-cooperative Equilibrium
• Both cannot hold with equality if w2/h2 
w1/h1.
• If, for example, w2/h2 < w1/h1, only the
second can hold with equality, and if it
does so, t1=0 and s1>0.
• i.e. full specialisation in market work by the
father (note: mother may also work in
market).
• He effectively buys the time of the mother
through voluntary transfers.
Voluntary transfers and ‘income
pooling’ equilibrium
• Transfer from the father to the mother is
• s1 = (1-1)(w1T+y1)- 1(w2T+y2)
• i.e. transfer rises with his ‘full income
(w1T+y1) and declines with hers (w2T+y2).
• Family full income: YF=(w1+w2)T+y1+y2
• GN = (1-1)(1-2)YF/(w2/h2)
• x1 = 1YF and x2 = 2(1-1)YF.
When intra-family income distribution
matters
• If neither condition above holds with
equality, s1=0 and t1=0.
• Father finds the child good too expensive.
• Occurs when (w2T+y2)/(w1T+y1)>(1-1)/1;
• Father is ‘too poor’ relative to the mother.
• GN= (1-2)(w2T+y2)/(w2/h2).
• Redistribution of income from father to
mother raises GN and x2 and lowers x1.
Transfers and income pooling
• When w2/h2< w1/h1, the mother will never
make financial transfers to the father.
• She has the comparative advantage in
child rearing.
• Who, if anyone, makes transfers depends
on the relative cost of the child good as
well as relative full incomes.
• Non-cooperative outcome may provide the
threat points for cooperative bargaining.
Specialisation
• The tendency for one or both parents to
specialise fully is a reflection of the
particular production technology assumed.
• It may not hold with diminishing marginal
productivity of each parent’s time input.
Effects of parents’ wages when w2/h2<w1/h1,
• Increases in the mother’s wage (w2) give rise to
both a substitution effect and an income effect
on provision of the child good.
• Increases in the father’s wage (w1) only affect
the provision of the child good through an
income effect; and there would be no effect of w1
(or y1) if the father does not make transfers to
his partner.
• Higher productivity in child rearing for the mother
(i.e. higher h2) raises GN.
Cooperative equilbrium
• Full specialisation in market work by one
parent if w2/h2  w1/h1.
• If w2/h2 < w1/h1, then t1=0 and
• w2/h2 = U1G(x1*,G*)/U1x(x1*,G*) +
U2G(x2*,G*)/U2x(x2*,G*) (Samuelson
cond.)
• Analogous to earlier formulation with
p=w2/h2 and family full income YF
replacing “y1+y2”.
• Mother’s wage has a substitution effect, as
well as income and bargaining effects.