Transcript Economic Theories of Fertility

Economic Theories of Fertility
Beyond Malthus
Thomas Malthus (early 19C)
• fertility determined by the age at marriage
and frequency of coition during marriage.
• an increase in people’s income would
encourage them to marry earlier and have
sexual intercourse more often.
• Gary Becker generalized and developed
the Malthusian theory.
Child ‘Quality’
• Gary Becker’s 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.
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
child’s human capital or by the direct
transfer of wealth to the child.
• Thus, we could think of child quality as the
child’s “quality of life”, as an adult as well
as during his or her childhood.
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 mother’s and father’s
time devoted to the production of child quality.
• N is the number of children.
• Constant returns to scale production function.
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.
Parents’ standard of living (Z),
or “parental consumption” for short
• Produced by combining parents’ time and
purchased goods and services.
• Z=g(xz,tmz,tfz)
• where g() exhibits constant returns to
scale.
• Consensus Preferences: U(Z,N,Q).
• Ignore different preferences for simplicity.
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.
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 .
•
•
•
•
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.
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
C0=NQ=[Y-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/UQ=pN/pQ=Q/N.
Figure 6.1
Quantity – Quality Interaction
Q
C1
C0
Q1
C
B
U1
D
Q0
A
U0
N0
N1
N
• 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.
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.
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 pC=pZ=1 (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.
Children time-intensive
• Rearing of children is assumed to be
mother’s time-intensive relative to other
home production activities in the sense
that qmC>qmZ.
• Implies relative cost of children (C/Z) is
directly related to the mother’s wage.
• The relative cost of children also depends
on the father’s wage as long as qfCqfZ.
Lifetime budget constraint
• In terms of “full income”
• Y= (wm+wf)T + y = (tmc+tmz)wm + (tfc+tfz)wf
+ xC+xZ = ZZ + CNQ.
• Let U(Z,N,Q) = U(Z,NQ) for simplicity.
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 (>0);
• C is the elasticity of NQ with respect to
full income;
• SZ=ZZ/Y; and SyY=y/Y and
SiY=wi(T-tjc-tjz)/Y, j=m,f, are shares of full
income.
Effects of men’s and women’s wages
• Income effects represented by the terms
CSmY and CSfY—are proportional to that
parent’s earnings share of full income.
• Substitution effect of -SZ(qjC-qjZ)
• Negative for mothers if qmC-qmZ>0
• Could be near zero for fathers if qfC-qfZ is
small.
• Could be positive for fathers , because of his
wife’s comparative advantage in child-rearing;
i.e. qfC-qfZ<0
Purchased child care and fertility
• Assume that fathers are not involved in
home production (tfc=tfz=0).
• Use tmc in the child quality production
function to denote total time for child care
(rather than just mother’s time).
• tmc = H + h(M), 0<h(M) <1, h(M)<0,
h(0)=0 (h(M)=dh/dM etc.).
• H is the amount of the mother’s time
devoted to children, M is the amount of
time purchased in the market at price p.
Choice of purchased child care
• Because each child may require a
minimum amount of mother’s 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.
Figure 6.2
Purchased Child Care
h(M)
 p 


w
 m
2
 p

 wm



1
B
A
C
M*
tmc-kN
M
Different cases
• Because mother’s time and purchased care are
not perfect substitutes, it is likely that the parents
use both sources of care. That is, H>kN and
M>0; then h(M) = p/wm. Tangency point A in
Figure.
• If the price of market child care is sufficiently low,
or mother’s wage high, wm > p/h(M) for all
values of M and M=tmc-kN. Point B in Figure.
• If the mother’s wage is low or the market price of
child care is high, wM< p/h(0), and no child care
would be purchased. Point C in Figure.
Demand function for children
• dln(NQ) = CSyYdln(y) +
[CSmY-SZ(qHC-qmZ)]dln(wm)
- qMC[CSC+SZ]dln(p)
qHC=wmH/CNQ, qMC=wmh(M)/CNQ,
SC=CNQ/Y and now SmY=wm(T-H-tmz)/Y.
• Higher price of child care has a negative
effect, unless M=0.
Implications
• Even though children may be more time
intensive than the production of Z in the
sense that wmtmc/CNQ > qmZ, the
substitution effect of a higher mother’s
wage could be positive if purchased child
care time is a large enough proportion of
child care time so as to make qHC< qmZ.
• A tendency for the impact of the mother’s
wage on fertility to vary with the level of
wages and the price of market child care.
• There is a tendency for qHC-qmZ to fall as
the mother’s 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 wm>p/h(M) for all values of H>kN, so
that H=kN. In this situation, the marginal
cost of children, C, is not affected by
changes in the wage, only the price of
child care.
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?
Simple model
• Parents’ utility function, U=u(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.
• 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).
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 c=gb/u′
• gb=g/b is the marginal expected utility
from an additional birth and u′ is the
marginal utility of parents’ consumption.
Implications
• db/ds = -gbs/D ≥0
• Where D= c2u′′+gbb <0 and gbs≥0
• 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.
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/ds<0
– if exogenous factors affecting s substitute for
parents expenditure to improve child survival.
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.
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.