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 (gj0), 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 gj0.

• 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

22

21y

αα

1

111α

yα

(1-α2)y2

(1-α1)y1

g2

g1 g1N

g2N

(2a)

(2b)

Example solutions

• (3a) g1N = [(1-1)y1 - 1(1-2)y2]/(1-12)

• (3b) g2N = [(1-2)y2 - 2(1-1)y1]/(1-12)• 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)]/(1-12)

• (4b) x1N = [1(1-2)(y1+y2)]/(1-12)

• (4c) x2N = [2(1-1)(y1+y2)]/(1-12)

When both parents contribute

• Non-cooperative outcomes only depend on the joint income of the parents, y1+y2.

• E.g. in example,• (4a) GN = [(1-1)(1-2)(y1+y2)]/(1-12)

• (4b) x1N = [1(1-2)(y1+y2)]/(1-12)

• (4c) x2N = [2(1-1)(y1+y2)]/(1-12)

Only one parent contributing• Taking the father as parent 1, he will not

contribute when • y1/(y1+y2) < 1(1-2)/(1-12)• 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

21

21

11

αα

21

2

11

αα

21

1

yyy

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(x1

e, Ge)= U2x(x2e, Ge)

• p = U1G(x1e, Ge)/U1x(x1

e, Ge)

+ U2G(x2e, Ge)/U2x(x2

e, 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

N

U1N U1

U2

U2N

A

B

Demand functions

• G=Ge(y1+y2, p,)• xj=xj

e(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 x2

e/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, x2

e/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 jk).

• If Vj=Uj(xj,G), then these preferences collapse to egoistic ones.

Figure 2.3 Utility Possibility Frontier

N

U1N U1

U2

U2N

A

B

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

NB = constant

NA =constant

NBA

NBB

DA

DB

TA2

T1 U1

U2

TB2

Ni = [U1 (·) - Ti1][U2 (·) - Ti

2].

NBi = ‘Nash-bargained’ solutions.

Di = ‘dominant partner’ solutions.

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 • uj

e = 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 u2

e=1-m1>m2

c) b2 + (1-b1-b2)/2 < m2 • Divorce threat is credible for the wife and

u1e=1-m2>m1 and u2

e=m2

Figure 2.5 Bargaining

(m1,m2)

U1

U2

(b1,b2)

(m1,m2)

eeUU 11 ,

Case (a) Divorce threat not credible

2

1 21* bbbU ii

U1

U2

(b1,b2)

Case (b) Divorce threat credible for person one

2

1 21* bbbU ii

eeUU 11 ,

(Ue1,Ue

2) (U*1,U*

2)

(m1,1-m1)

(U*1,U*

2)

Figure 2.5 Bargaining

(m1,m2)

U1

U2

(b1,b2)

(m1,m2)

eeUU 11 ,

Case (a) Divorce threat not credible

2

1 21* bbbU ii

U1

U2

(b1,b2)

Case (b) Divorce threat credible for person one

2

1 21* bbbU ii

eeUU 11 ,

(Ue1,Ue

2) (U*1,U*

2)

(m1,1-m1)

(U*1,U*

2)

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• t10, s10

• Implies two conditions:w1/(1-2)h1 U1G(x1*,G*)/U1x(x1*,G*) = (1-1)x1/1Gw2/(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.