Economic Theories of Fertility Beyond Malthus. Thomas Malthus (early 19C) fertility determined by the age at marriage and frequency of coition during.

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

A

D

B C

Q0

C0

N1

U1

N0 N

Q

Q1

C1

U0

• 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 qfCqfZ.

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: HkN as well as M0.

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

M tmc-kN M* C

h(M) 1

mw

p

2

mw

p

B

A

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.