Economic Theories of Fertility

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Economic Theories of Fertility

• Beyond Malthus

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Thomas Malthus (early 19C)

• fertility determined by the age at marriage and
  frequency of coition during marriage.
• an increase in peoples income would encourage
  them to marry earlier and have sexual intercourse
  more often.
• Gary Becker generalized and developed the
  Malthusian theory.

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Child Quality

• Gary Beckers seminal contribution pointed out
  that the psychic satisfaction parents receive
  from their children is likely to depend on the
  amount that parents spend on children as well as
  the number of children that they have.
• Children who have more spent on them are called
  higher quality children.
• Basic idea is that if parents voluntarily spend
  more on a child, it is because they obtain
  additional satisfaction from the additional
  expenditure.

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Current state of theory

• Now child quality is usually identified with the
  lifetime well-being of the child.
• Can be increased by investing more in the childs
  human capital or by the direct transfer of wealth
  to the child.
• Thus, we could think of child quality as the
  childs quality of life, as an adult as well as
  during his or her childhood.

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Home Production of Child Quality

• Define a production function for child quality
  (Q), or income as an adult, in terms of parents
  time and purchases of good and services.
• Assume that parents choose the same level of
  child quality for each.
• Q f(xc/N, tmc/N, tfc/N), where
• xc is the total amount of goods and services and
  tmc and tfc are the total amounts of mothers and
  fathers time devoted to the production of child
  quality.
• N is the number of children.
• Constant returns to scale production function.

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Expenditure on Children

• Total expenditure on children is, therefore,
  ?CNQ ?Cf(xc, tmc, tfc), where
• ?C is the marginal cost of children.
• Marginal cost of children depends on the prices
  of inputs into its production, namely
• dln(?C) (pcxc/?CNQ)dln(pc) (wmtmc/?CNQ)dln(wm)
  (wftfc/?CNQ)dln(wf)
• wj is the wage rate of parent j and pc is the
  price of purchased goods and services
• assumes that both parents work in the market
  sometime during the childrearing period.

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Parents standard of living (Z), or parental
consumption for short

• Produced by combining parents time and purchased
  goods and services.
• Zg(xz,tmz,tfz)
• where g(?) exhibits constant returns to scale.
• Consensus Preferences U(Z,N,Q).
• Ignore different preferences for simplicity.

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Parents Decision

• Choose N, Q and Z to maximize their utility
  subject to the lifetime budget constraint
• Y ?ZZ ?CNQ
• where ?Z is the marginal cost of parental
  consumption, which depends on input prices,
  analogously to ?C.
• Note product NQ in budget constraint.

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Characteristics of solution

• UN ??CQ ?pN
• UQ ??CN ?pQ
• UZ ??Z
• where pN and pQ are the marginal costs of the
  number and quality of children respectively.
• The marginal utility of income is ?.

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Implications

• Cost (or shadow price) of an additional child
  is proportional to the level of child quality
• Cost (shadow price) of raising child quality is
  proportional to the number of children the
  parents have.
• Important interaction between family size and
  child quality.

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Figure 6.1

• The optimal choice of N and Q is given at point
  A, at which the indifference curve U0 is tangent
  to the budget constraint C0NQY-?ZZ(Y,?Z,?C)/?C
  , where
• C0 is the parents real expenditure on children.
• Z(Y,?Z,?C) is the demand function for parents
  consumption.
• At the optimum UN/UQpN/pQQ/N.

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• Maximization of utility implies that the
  indifference curve must be more convex than the
  budget constraint, which means
• that child quality and quantity cannot be close
  substitutes for one another.
• Increase in income (Y) produces
• A pure income effect (A?B)
• An induced substitution effect (B?C)
• Latter can be large enough to produce a fall in
  fertility when income increases.

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A negative income elasticity of demand for
children?

• The income elasticity of fertility can be
  negative, even though children are normal
  goods, in the sense that parents want more of
  them when parental income increases.
• The reason is that the true income elasticity is
  defined with relative prices constant, but,
  because of the interaction, we cannot hold the
  ratio of the shadow price of an additional child
  to that of child quality (pN/pQ) constant when we
  measure the elasticity.

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The cost of children

• Determined by the cost of the inputs that
  determine the cost of child quality relative to
  the cost of the parents living standard.
• dln(?C/?Z) (qmC-qmZ)dln(wm)
• (qfC-qfZ)dln(wf) where pCpZ1 (numeraire)
• qmC(wmtmc/?CNQ) and qfC(wftfc/?CNQ)
• qmZ(wmtmz/?ZZ) and qfZ(wftfz/?ZZ)
• These are parents respective cost shares in
  producing Q and Z, respectively.

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Children time-intensive

• Rearing of children is assumed to be mothers
  time-intensive relative to other home production
  activities in the sense that qmCgtqmZ.
• Implies relative cost of children (?C/?Z) is
  directly related to the mothers wage.
• The relative cost of children also depends on the
  fathers wage as long as qfC?qfZ.

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Lifetime budget constraint

• In terms of full income
• Y (wmwf)T y (tmctmz)wm (tfctfz)wf
  xCxZ ?ZZ ?CNQ.
• Let U(Z,N,Q) U(Z,NQ) for simplicity.

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Demand function for children

• dln(NQ) ?CSyYdln(y)
• ?CSmY-?SZ(qmC-qmZ)dln(wm)
• ?CSfY-?SZ(qfC-qfZ)dln(wf)
• where ? is the elasticity of substitution in
  consumption between Z and NQ (?gt0)
• ?C is the elasticity of NQ with respect to full
  income
• SZ?ZZ/Y and SyYy/Y and
• SiYwi(T-tjc-tjz)/Y, jm,f, are shares of full
  income.

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Effects of mens and womens wages

• Income effects represented by the terms ?CSmY and
  ?CSfYare proportional to that parents earnings
  share of full income.
• Substitution effect of -?SZ(qjC-qjZ)
• Negative for mothers if qmC-qmZgt0
• Could be near zero for fathers if qfC-qfZ is
  small.
• Could be positive for fathers , because of his
  wifes comparative advantage in child-rearing
  i.e. qfC-qfZlt0

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Purchased child care and fertility

• Assume that fathers are not involved in home
  production (tfctfz0).
• Use tmc in the child quality production function
  to denote total time for child care (rather than
  just mothers time).
• tmc H h(M), 0lth?(M) lt1, h??(M)lt0, h(0)0
  (h?(M)dh/dM etc.).
• H is the amount of the mothers time devoted to
  children, M is the amount of time purchased in
  the market at price p.

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Choice of purchased child care

• Because each child may require a minimum amount
  of mothers time, k, there are constraints H?kN
  as well as M?0.
• At optimum,
• h?(M) (p-?M/?)/wm - ?H/?N
• ?H and ?M are the Lagrange multipliers (shadow
  prices) associated with these two inequality
  constraints.
• These are zero when the constraint is satisfied
  with an inequality and positive if satisfied with
  an equality.

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Different cases

• Because mothers time and purchased care are not
  perfect substitutes, it is likely that the
  parents use both sources of care. That is, HgtkN
  and Mgt0 then h?(M) p/wm. Tangency point A in
  Figure.
• If the price of market child care is sufficiently
  low, or mothers wage high, wm gt p/h?(M) for all
  values of M and Mtmc-kN. Point B in Figure.
• If the mothers wage is low or the market price
  of child care is high, wMlt p/h?(0), and no child
  care would be purchased. Point C in Figure.

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Demand function for children

• dln(NQ) ?CSyYdln(y)
• ?CSmY-?SZ(qHC-qmZ)dln(wm)
• - qMC?CSC?SZdln(p)
• qHCwmH/?CNQ, qMCwmh(M)/?CNQ, SC?CNQ/Y and now
  SmYwm(T-H-tmz)/Y.
• Higher price of child care has a negative effect,
  unless M0.

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Implications

• Even though children may be more time intensive
  than the production of Z in the sense that
  wmtmc/?CNQ gt qmZ, the substitution effect of a
  higher mothers wage could be positive if
  purchased child care time is a large enough
  proportion of child care time so as to make qHClt
  qmZ.
• A tendency for the impact of the mothers wage on
  fertility to vary with the level of wages and the
  price of market child care.

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• There is a tendency for qHC-qmZ to fall as the
  mothers wage increases or the price of child
  falls
• This reduces the size of the (negative)
  substitution effect.
• Women with very high wage levels find that
  wmgtp/h?(M) for all values of HgtkN, so that HkN.
  In this situation, the marginal cost of children,
  ?C, is not affected by changes in the wage, only
  the price of child care.

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Child mortality risk and fertility

• Failure to survive is ultimate manifestation of
  low quality.
• Does lower child mortality risk help account for
  the demographic transition from high
  fertility-high mortality environment to a low
  fertility-low mortality one?

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Simple model

• Parents utility function, Uu(z) v(n),
• where z denotes parental consumption, n is the
  number of children who survive to become adults.
• That is, children who die in childhood are not a
  source of utility to their parents.
• Each birth has survival chances, which can be
  represented by a probability distribution with
  mean equal to the survival probability s.

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• Surviving children n is the outcome from
  subjecting the number of births, b, to this
  random survival process.
• Denote the probability density function of n,
  conditional on b and s, as f(n,b,s).
• Then the expected utility of parents is given by
  E(U)u(z) g(b,s)
• where g(b,s) is the expected utility from having
  b births when on average sb survive
• (i.e comes from integrating v(n)f(n,b,s) over n
  from 0 to b).

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Parents optimisation

• Assume that each birth has a fixed cost c.
• Parents choose b to maximize
• E(U)u(y-cb) g(b,s).
• Implies cgb/u'
• gb?g/?b is the marginal expected utility from an
  additional birth and u' is the marginal utility
  of parents consumption.

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Implications

• db/ds -gbs/D 0
• Where D c2u''gbb lt0 and gbs0
• A higher probability of child survival reduces
  the price of a surviving birth, thereby
  encouraging higher fertility.
• Thus, lower child mortality does not lower
  fertility in this model.
• There must be some other consideration.

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Richer model

• Cigno suggests that parents can influence the
  chances that their own children survive to become
  adults (an element of child quality) by spending
  more on each child.
• That is, c is now chosen by the parents and it
  affects the survival distribution, f(n,b,s,c).
• It is now possible that db/dslt0
• if exogenous factors affecting s substitute for
  parents expenditure to improve child survival.

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Effects of contraceptive costs

• When family size and child quality are net
  substitutes
• Lower cost of averting births
• Reduces fertility (if income effect is small)
• Raises human capital investment (child quality)
• A higher return to human capital investment in
  children
• Raises human capital investment.
• Reduces fertility.

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Impacts of technical change

• Contraceptive costs/rate of return effects work
  through quantity-quality interaction, tending to
  magnify initial impacts because of effects on
  pN/pQ.
• Technical change (e.g. green revolution) has
  affected rate of return to human capital
  investment and contraception.
• Can account for important stylised facts of
  economic development.