Intrahousehold Allocation

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Intra-household Allocation

• Conflict and Cooperation in the Family

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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.

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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.

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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.

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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.

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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.

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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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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)

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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)

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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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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.

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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.

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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.

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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.

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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

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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).

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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.

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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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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.

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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.

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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).

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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)

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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 .

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.