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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.

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thomas malthus early 19c
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
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
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
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 timedevoted to the production of child quality.
    • N is the number of children.
    • Constant returns to scale production function.
expenditure on children
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
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
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
Characteristics of solution
  • UN = CQ = pN
  • UQ = CN = pQ
  • UZ = Z
  • wherepN and pQare the marginal costs of the number and quality of children respectively.
  • The marginal utility of income is.
  • 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
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.
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
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
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
Children time-intensive
  • Rearing of children is assumed to be mother’stime-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
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
Demand function for children

dln(NQ) = CSyYdln(y) +

[CSmY-SZ(qmC-qmZ)]dln(wm) +


  • 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
Effects of men’s and women’s wages
  • Income effectsrepresented 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
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 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
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.
different cases
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 andM=tmc-kN. Point B in Figure.
  • If the mother’s wage is low or the market price of child care is high, wM

demand function for children1
Demand function for children
  • dln(NQ) = CSyYdln(y) +


- 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.
  • 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
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
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
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.
  • 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
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
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
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.