Title: Intra-household Allocation
1Intra-household Allocation
- Conflict and Cooperation in the Family
2Non-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.
3Basic 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 js private
consumption.
4Voluntary contributions to expenditures on
children, G
- gj (gj?0), and so xj yj - gj and
- G (g1g2)/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.
5Nash 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, (g1g2)/p), subject to gj?0.
- Implies UjG(xj, G)/Ujx(xj, G) ? p , j1,2.
6Solution
- 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) lt p for parent j
(he/she is too poor), their contribution is
zero.
7Example Uj ?jln(xj) (1-?j)ln(G),
- When g1gt0 and g2gt0 and p1,
- (2a) g1 (1-?1)y1 - (?1g2)
- (2b) g2 (1-?2)y2 - (?2g1)
- Nash equilibrium illustrated in Figure 2.1, with
?1gt?2
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9Example 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
parents. From equations (3), - (4a) GN (1-?1)(1-?2)(y1y2)/(1-?1?2)
- (4b) x1N ?1(1-?2)(y1y2)/(1-?1?2)
- (4c) x2N ?2(1-?1)(y1y2)/(1-?1?2)
10When both parents contribute
- Non-cooperative outcomes only depend on the joint
income of the parents, y1y2. - E.g. in example,
- (4a) GN (1-?1)(1-?2)(y1y2)/(1-?1?2)
- (4b) x1N ?1(1-?2)(y1y2)/(1-?1?2)
- (4c) x2N ?2(1-?1)(y1y2)/(1-?1?2)
11Only one parent contributing
- Taking the father as parent 1, he will not
contribute when - y1/(y1y2) lt ?1(1-?2)/(1-?1?2)
- In this case,
- GN(1-?2)y2,
- x1Ny1 and x2N?2y2.
- Analogously, for the mothersee Fig. 2.2
- Individual incomes matter for the outcome.
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13Non-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.
14Separate 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.
15Separate spheres implications
- Reaction functions analogous to earlier, in which
the mans demand functions depend on the
purchases of the womans 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.
16Low 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 goodlooks like
separate spheres model.
17Cooperative 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
18Equivalent 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).
19Cooperative 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.
20Utility 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.
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22Demand functions
- GGe(y1y2, p,?)
- xjxje(y1y2, p,?), j1,2
- 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.
23Cross-equation restrictions
- Because ?G/?yj(?Ge/??)(??/?yj), j1,2, 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.
24Example utility functions used earlier
- pGe (1-?1) ?(1-?2)(y1y2)/(1?)
- x1e ?1(y1y2)/(1?)
- x2e ??2(y1y2)/(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).
25Income 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)
(y1y2)(?1-?2)(??/?y2)/p - ?x2e/?y2??2 (y1y2)?2(??/?y2)
26Two effects of mothers income (y2) on child
expenditure (G)
- It increases family income (y1y2)
- (1-?)(1-?1) ?(1-?2) in example.
- It may increase mothers bargaining power
(??/?y2gt0) - (y1y2)(?1-?2)(??/?y2) in example
- could reinforce (?1gt?2) or offset (?1lt?2) the
income effect .
27Distribution 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, persons share of household income.
28Inferences 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. mens and womens clothing). - Let ? be dependent on mothers income share, s2
y2/(y1y2). - From above, holding y1y2 constant,
- ?x2e/?s2(y1y2)?2(??/?y2)
- E.g. if ??/?y2gt0, higher s2 increases x2 and,
conditional on G, the mothers welfare.
29Caring preferences
- Preferences take the form
- V1 V1U1(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/?Ukgt0 for j?k). - If VjUj(xj,G), then these preferences collapse
to egoistic ones.
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31Implications
- 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
caringthe 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 partners
private utility to fall below some minimum
level. - above A and below B, only joint income matters.
32Bargaining 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.
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34Bargaining 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 arrangementher threat point. - Nash bargaining maximizes the product of the
gains from cooperation, where these gains are
U1-T1 and U2-T2NBA in Figure. - Effect of change in threat pointFigure.
35What 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. u1u21, 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.
36Equilibrium of bargaining game
- b1b2lt1 due to inefficiency of non-cooperation.
- If either partner asks for a divorce, they will
get m1 and m2 respectively, where m1m2lt1. - If the time between offers is small, the unique
equilibrium of the bargaining process is - uje bj (1-b1-b2)/2, j1,2
- that is, the gains from cooperative relative to
non-cooperative marriage are shared equally. - Three cases.
37Three cases
- bj (1-b1-b2)/2 gt mj, j1,2
- Divorce threat not credible for either party
- b1 (1-b1-b2)/2 lt m1,
- Divorce threat is credible for the husband and
u1em1 and u2e1-m1gtm2 - b2 (1-b1-b2)/2 lt m2
- Divorce threat is credible for the wife and
u1e1-m2gtm1 and u2em2
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40Threat 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.
41Home 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 js time.
- Have replaced purchases of G with home production
of it.
42Non-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 js
wage, yj is js non-labour income and s1 is
transfers from the father to the mother. - Analogous for father.
43Mothers 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.
44Mothers 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)(w2Ty2s1)/w2) - ?2h1t1/h2
- Best strategy reduce her home production time
when father increases his.
45Fathers 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, h1t1h2t2), subject
to - Mothers 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
46Non-cooperative Equilibrium
- Both cannot hold with equality if w2/h2 ? w1/h1.
- If, for example, w2/h2 lt w1/h1, only the second
can hold with equality, and if it does so, t10
and s1gt0. - 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.
47Voluntary transfers and income pooling
equilibrium
- Transfer from the father to the mother is
- s1 (1-?1)(w1Ty1)- ?1(w2Ty2)
- i.e. transfer rises with his full income
(w1Ty1) and declines with hers (w2Ty2). - Family full income YF(w1w2)Ty1y2
- GN (1-?1)(1-?2)YF/(w2/h2)
- x1 ?1YF and x2 ?2(1-?1)YF.
48When intra-family income distribution matters
- If neither condition above holds with equality,
s10 and t10. - Father finds the child good too expensive.
- Occurs when (w2Ty2)/(w1Ty1)gt(1-?1)/?1
- Father is too poor relative to the mother.
- GN (1-?2)(w2Ty2)/(w2/h2).
- Redistribution of income from father to mother
raises GN and x2 and lowers x1.
49Transfers and income pooling
- When w2/h2lt 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.
50Specialisation
- 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 parents time input.
51Effects of parents wages when w2/h2ltw1/h1,
- Increases in the mothers wage (w2) give rise to
both a substitution effect and an income effect
on provision of the child good. - Increases in the fathers 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.
52Cooperative equilbrium
- Full specialisation in market work by one parent
if w2/h2 ? w1/h1. - If w2/h2 lt w1/h1, then t10 and
- w2/h2 U1G(x1,G)/U1x(x1,G)
U2G(x2,G)/U2x(x2,G) (Samuelson cond.) - Analogous to earlier formulation with pw2/h2 and
family full income YF replacing y1y2. - Mothers wage has a substitution effect, as well
as income and bargaining effects.