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Title: Intra-household Allocation


1
Intra-household Allocation
  • Conflict and Cooperation in the Family

2
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.

3
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 js private
    consumption.

4
Voluntary 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.

5
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, (g1g2)/p), subject to gj?0.
  • Implies UjG(xj, G)/Ujx(xj, G) ? p , j1,2.

6
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) lt p for parent j
    (he/she is too poor), their contribution is
    zero.

7
Example 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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9
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
    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)

10
When 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)

11
Only 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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13
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.

14
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.

15
Separate 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.

16
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 goodlooks like
    separate spheres model.

17
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

18
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).

19
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.

20
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.

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22
Demand 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.

23
Cross-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.

24
Example 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).

25
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)
    (y1y2)(?1-?2)(??/?y2)/p
  • ?x2e/?y2??2 (y1y2)?2(??/?y2)

26
Two 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 .

27
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, persons share of household income.

28
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. 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.

29
Caring 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.

30
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31
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
    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.

32
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.

33
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34
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 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.

35
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. 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.

36
Equilibrium 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.

37
Three 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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40
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.

41
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 js time.
  • Have replaced purchases of G with home production
    of it.

42
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 js
    wage, yj is js non-labour income and s1 is
    transfers from the father to the mother.
  • Analogous for father.

43
Mothers 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.

44
Mothers 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.

45
Fathers 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

46
Non-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.

47
Voluntary 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.

48
When 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.

49
Transfers 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.

50
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 parents time input.

51
Effects 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.

52
Cooperative 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.
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