This PDF is a selection from an out-of-print volume from the National
Bureau of Economic Research
Volume Title: Economics of the Family: Marriage, Children, and Human
Capital
Volume Author/Editor: Theodore W. Schultz, ed.
Volume Publisher: University of Chicago Press
Volume ISBN: 0-226-74085-4
Volume URL: http://www.nber.org/books/schu74-1
Publication Date: 1974
Chapter Title: Interaction between Quantity and Quality of Children
Chapter Author: Gary S. Becker, H. Gregg Lewis
Chapter URL: http://www.nber.org/chapters/c2963
Chapter pages in book: (p. 81 - 90)
NORMAN B. RYDER
ier kind of model altomay deserve increasing
been proscribed in the
n at all (and quite apart
rtion voluntarily infertile
roles are under
nd others from attempts
1 work in a theoretically
e that some redirection
of the economic theory
n, collected according to
with which their models
for centuries on official
the last few decades
create their own data to
mic theory of fertility is
for other purposes, from
y Study.
Interaction between Quantity and Quality of
Children
Gary S. Becker and H. Gregg Lewis
University of Chicago
Students of human fertility have been aware for a long time that there
may be some special relation between the number (quantity) of children
ever born to a family and the "quality" of their children as perceived by
others if not by the parents. One need only cite the negative correlation
between quantity and quality of children per family so often observed
in both cross-section and time-series data. One of us (Becker 1960) more
than a decade ago stressed the importance for understanding fertility
(quantity) of the interaction between quantity and quality, and we are
pleased to note that this interaction is emphasTized in this book, and especially by De Tray and Willis.
Some economists have argued that the negative relation between quan-
tity and quality often observed is a consequence of a low substitution
elasticity in a family's utility function between parents' consumption or
level of living and that of their children (see, e.g., Duesenberry 1960;
Willis 1969). The approach followed by De Tray in this volume is differ-
ent, but it makes equally special assumptions about the substitution
between quantity and quality in the utility function and in household
production.
We want to argue here that one can go a long way toward understanding
data on the interaction between quantity and quality as well as on
quantity or quality alone without assuming that, either in the utility
function or in household production, quantity and quality are more closely
related than any two commodities chosen at random. The analysis that
follows is sketchy and incomplete, mainly because we have only recently
developed this line of argument.
The key feature in our analysis is that the shadow price of children
with respect to their number (i.e., the cost of an additional child, holding
their quality constant) is greater the higher their quality is. Similarly,
the shadow price of children with respect to their quality (i.e., the cost
81
QUANTITY AND QUALITY i
GARY S. BECKER AND H. GREGG LEWIS
82
of a unit increase in quality, holding number constant) is greater, the
greater the number of children. Furthermore, with appropriate change of
language, the same may be said of the other commodities consumed by
the family. However, to simplify the analysis in this paper, we make the
quantity-quality distinction only for children. Thus, to illustrate our
If
whose ral
the marginal rates of subs
income concept is the
these shadow prices;
purpose is
P,4,
Pq, and
reasoning, we specify the following simple utility function:
U
U(n,q,y),
(1)
where n is the number of children, q their quality (assumed to be the same
for all of the children), and y the rate of consumption of all other commodities. We start out with a simple budget restraint:
I = nqx + Yitv,
(2)
is the price of y. We
where I is full income, it is the price of nq, and
make no special assumptions about the elasticities of substitution among
n, q, and y, either in the utility function or in the household production
functions that underlie the it's.
The first-order conditions for maximizing the utility function subject
to the budget restraint are:
MU,, = Aqir =
MUq
Xnit =
MU5
=
= Ap5,
(3)
where the MU's are the marginal utilities, the p's are marginal costs or
shadow prices, and X is the marginal utility of money income. The important point is that the shadow price of children with respect to number
(pa) is positively related to q, the level of quality, and the shadow price
with respect to quality (p5) is positively related to n, the number of
children. The economic interpretation is that an increase in quality is
more expensive if there are more children because the increase has to
apply to more units; similarly, an increase in quantity is more expensive
if the children are of higher quality, because higher-quality children
cost more.
These equilibrium conditions (3) together with the second-order condi-
tions can be found in several places in the literature on quantity and
quality (see, e.g., Houthakker 1952; Theil 1952; Becker 1960; and
Willis's paper in this volume), but a number of their important implications for income and price effects apparently have not been explored.
1.
It is well known that
is unity; that is:
Consider, however, the "o
flq, and ij,,, derived b
follows directly from the
p's in (3) that the simil
ticities is 1/R = I/(1 +
I
R
I
That is, on the average
true elasticities in the
downward bias is simple
the it's but not the p's
However, increases in n
Thus, the percentage inc
an index of the
is less
This price effect of a
the price effect resulting
wage rates. The increas
money income, because
hold are increased by th
We think that it is p1
with respect to quality (
to
quantity (ia).
Becaus
and the effect on prices,
negative even though th
Income Effects
Let the "true" income elasticities of demand for the number (n) and
quality (q) of children and for all other commodities (y) be
iq, and
respectively.
These elasticities are derived in the usual way by chang-
ing "income" while holding constant the "prices" of n, q, and y. The
appropriate prices for this purpose are the shadow prices (marginal costs)
1 This price effect, howev
Becker (1960), and followe
for the rich as for the poor
rich choose more expensive
their number is higher for
children. Similarly, the relev
rich because they choose in
R AND H. GREGG LEWIS
)nstant) is greater, the
appropriate change of
nmodities consumed by
his paper, we make the
Fhus, to illustrate our
I
83
QUANTITY AND QUALITY OF CHILDREN
and
whose ratios in equilibrium (see eq. {3 1) are equal to
the marginal rates of substitution in the utility function. The appropriate
income concept is the total "expenditure" on n, q, and y calculated at
these shadow prices; that is, the correct measure of income for this
p,,,
purpose is
function:
(4)
(1)
assumed to be the same
It is well known that the mean value of the true income elasticities
is unity; that is:
ption of all other corn-
npn
Lint:
(2)
is the price of y. We
of substitution among
re household production
utility function subject
are marginal costs or
income. The imwith respect to number
and the shadow price
to n, the number of
increase in quality is
se the increase has to
ntity is more expensive
higher-quality children
the second-order condi-
on quantity and
2; Becker 1960; and
important implicanot
been explored.
the number (n) and
ties (y) be 'in,
and
e usual way by chang-
of n, q, and y. The
prices (marginal costs)
(5)
Consider, however, the "observed" income elasticities, which we denote by
and
derived by changing I while holding it and its, constant. It
follows directly from the budget restraint (2) and the definitions of the
p's in (3) that the similarly weighted mean of the observed income elasticities is I/R = I/(I + nqit), which is less than unity; that is:
I
1>—=
(3)
yP1,
R
I
I±nqit =
R
+
qpq_
R
YPv_
'iq + —Tjy.
R
(6)
That is, on the average, the observed elasticities are smaller than the
true elasticities in the ratio I/R. The economic explanation for this
downward bias is simple. The direct effect of the increase in I, holding
the it's but not the p's constant, in general is to increase n, q, and y.
to rise.
However, increases in n and q cause the shadow prices p,, and
Thus, the percentage increase in real income in the sense of R deflated by
an index of the
is less than the percentage increase in money income I.
This price effect of an increase in money income resembles somewhat
the price effect resulting from a rise in money income caused by a rise in
wage rates. The increase, in ratio terms, is less in real income than in
money income, because the costs of producing commodities in the household are increased by the rise in the price of time (see Becker 1965).'
We think that it is plausible to assume that the true income elasticity
with respect to quality (ii,) is substantially larger than that with respect
to quantity
Because of the downward bias in the observed elasticities
and the effect on prices, the observed elasticity for quantity
may be
negative even though the true elasticity
is not. Assume for simplicity
I This price effect, however, does offer a correction to the argument advanced by
Becker (1960), and followed by many others, that the price of children is the same
for the rich as for the poor (aside from the cost-of-time argument), even though the
rich choose more expensive children. The relevant price of children with respect to
their number is higher for the rich precisely because they choose more expensive
children. Similarly; the relevant price of cars, houses, or other goods is higher for the
rich because they choose more expensive varieties.
84
GARY S. BECKER AND H. GREGG LEWIS
that
= 0. Let income 1 increase while holding x and constant. The
direct effect of the increase in I is to increase q (and y) while leaving n
unchanged. But then the shadow price with respect to quantity (Pu = qx)
= nit) and that of
will rise while the shadow price of quality
=
are unchanged, causing q and y to be substituted for n, and therefore
n will decline.
More generally, when the utility function and budget restraints are
those given above in equations (1) and (2), the observed income elasticities for quantity and quality are related to the corresponding true elasticities as follows:2
=
Even if
—
were consi
only on q,, but also on
in money income I
income I rises—a plausi
with income, even with
income, contributing to
at lower levels of income
in some fertility data.3
2.
kGnq)flu
(1
QUANTITY AND QUALIT'
Price Effects
Before discussing price
slightly as follows:
(1
1—k —
(1—
—
I
(1—
(7)
so that the shadow pric
where
These shadow prices for
in p5. The
p,, and
depend on quantity but
= kanq + (1 — 2k)aqy;
(1
kOnq)2
—
(8)
(1—
The a's are the familiar Allen partial elasticities of substitution in the
utility function; the
are averages of the a's, and they must be positive;
D and (1 —
are positive by the second-order conditions. Equations
(7) verify that the observed quantity elasticity (ij,,) may be negative
is positive. Furthermore, if
even when the true quantity elasticity
lq> n,,, as we assume, ijq > unless q is a much better substitute than
n for y, for it follows from (7) and (8) that
1—k
(9)
Moreover,
may exceed
by more than
exceeds n,,; that is, the
downward bias in
may be less than that in ii,,. This is easily seen for
the case in which anq =
= = a. Then D, which is positive, is
equal to (1 —a)[1 +a(l —2k)] and
(1
may even
(1 — a), so that
if
a
>
k.
Indeed,
—
—
a,3,,
exceed
as may be seen from (7) and (8) by assuming
anq=an,,=aq,,=a.
a(l — 2k)],
2
•1
Then
which will exceed
(1 — k)(l —
if,
for example, k
the Mathematical Appendix to this paper.
=
—
=0
and
a)[1 +
and a =
3's.
child costs (such as
Similarly, the componei
thus has the attributes
"family good."
"handing down" of
that the fixed compone
that is,
>
a) First consider ±
induced, say, by an e
Since this increases the
shadow price of qualit
But the fall in numbe
nx), which induces su
be not only a fall in q
relative, that is, to ot
and quality are better
random. Exactly the
in the educati
in quality, which in
quantity (p,, = + q
increase
jr8
Consequently, both
3 See the discussion and
paper.
AND H. GREGG LEWIS
constant. The
x and
(and y) while leaving n
:t to quantity (Pn = qx)
nit) and that of y(p,, =
uted for n, and therefore
budget restraints are
income elasticicorresponding true elas-
85
QUANTITY AND QUALITY OF CHILDREN
Even if ii,, were constant, ij,, need not be, since the latter depends not
but also on the substitution elasticities and the share of nqx
declines as
in money income I (nqx/I = k/( 1 — k)). For example, if
income I rises—a plausible assumption, we think—ii,, would tend to rise
may rise With
with income, even with constant n,,, and, of course,
Indeed, ij,, could be negative
income, contributing to the increase in
only on
at lower levels of income and positive at higher levels, the pattern observed
in some fertility data.3
2.
Price Effects
Before
discussing price effects, we generalize the budget constraint (2)
slightly as follows:
(10)
nq)"lq
(7)
so that the shadow prices or marginal costs are now
(11)
(1 —
(8)
i
of substitution in the
d they must be positive;
er conditions. Equations
may be negative
)Ositive. Furthermore, if
:h better substitute than
(anvile —
(9)
that is, the
This is easily seen for
which is positive, is
exceeds n,,;
= (1
k)(riqrin)/
Indeed,
assuming
e,
k=
may even
0 and
+
— a)[1
and a = 3/'4.
These shadow prices for n and q each contain a "fixed" component: it,, in
p,, and itq in
The component nit, in child costs consists of costs that
depend on quantity but not on quality. Contraception costs and prenatal
child costs (such as maternity care) are moderately good examples.
Similarly, the component qitq depends on quality but not quantity, and
thus has the attributes of a "public good," or a better expression is a
"family good." Perhaps some aspects of training in the home and the
"handing down" of some clothing are reasonable examples. We assume
that the fixed component is more important for quantity than for quality,
that is, nit,,> qitq.
a) First consider the pure substitution effects of an increase in it,,
induced, say, by an exogenous improvement in contraceptive technique.
Since this increases the shadow price of quantity (p,,) relative to both the
shadow price of quality
n would fall.
and the shadow price of
= itq +
But the fall in numbers reduces the shadow price of quality
nit), which induces substitution in favor of quality. The outcome would
be not only a fall in quantity but also a relatively large rise in quality—
relative, that is, to other commodities—without assuming that quantity
and quality are better substitutes than any two commodities chosen at
random. Exactly the same result holds if ltq falls, say, because of an
increase in the education of parents. The fall in
induces an increase
in quality, which in turn induces an increase in the shadow price of
quantity (p, = it, + qit) and thus a relatively large decrease in quantity.
Consequently, both De
(in his paper which follows) that an
3 See the discussion and alternative explanation of this finding earlier in Willis's
paper.
86
GARY S. BECKER AND H. GREGG LEWIS
increase in the education of mothers has a strong positive effect on the
quality and a strong negative effect on the number of their children, and
the common belief that important advances in birth control knowledge
not only significantly reduce the number of children but also significantly
increase their quality, are consistent with the preceding analysis. Quantity
and quality are closely related, because the shadow price of quality depends on quantity and the shadow price of quantity depends on quality.
We repeat that no special assumptions about substitution in household
production or consumption are required to derive a special relation between
quantity and quality.
b) Now consider the pure substitution effects of equal percentage in-
creases in it,,, ltq, and it due, say, to increases in wage rates. To put the
argument in extreme form, assume ltq = 0 and it,, > 0. The equal increases
in it,,, itq, and it relative to
= can be treated simply as a relative fall
in it,, = p,,. A fall in p,, initially would induce equal percentage declines in
QUANTITY AND QUALITY
tity and quality of all goo
quite rewarding.
Mathematical Appendix
The budget restraints speci
and q. It is precisely this n
between quantity and qualit
The derivation of the el
and child quali
children
entiating the budget restrai
nonlinearity of the budget r
is followed, it is all too easy
tion elasticities in the utilit
that makes use of quite ía
First, we replace the cu
plane surface by adding nqit
I + nqit =
n and q if they were equally good substitutes for y. However, since the
n would
equal percentage declines in n and q would lower p,, more than
fall relative to q. Thus, the income-compensated elasticity of quantity with
respect to equal percentage changes in it,,., itq and it tends to be greater
numerically than the corresponding elasticity for quality. De Tray finds
that an increase in women's wage rates reduces the number of children by
a much bigger percentage than the quality of children.
This difference is, of course, accentuated if quantity is a better substitute
than quality for other commodities, which we think is a plausible, though
special, assumption. For then a fall in
directly induces a fall in n relative
to q, which accentuates the decline in p,, relative to Pn.
We conclude, therefore, that the observed price elasticity of quantity
exceeds that of quality, just the opposite of our conclusion for observed
income elasticities.4 This reversal of the quantity-quality ordering for price
and income elasticities is not only a somewhat unexpected implication of
the analysis, but also gives a consistent interpretation to the findings of
De Tray and others.
Of course, most of our discussion applies not only to the interaction
between the quantity and quality of children, but also to the quantity and
quality of cars, houses, food, tea, education, publications, and large numbers of other goods. The observed price and income elasticities of quantity
and quality will differ in predictable directions from the "true" elasticities.
A systematic analysis and reconsideration of the interaction between quan-
This conclusion about income elasticities, derived from the budget restraint
(2), is modified somewhat when the budget restraint is of the more general form
(10), since the shadow price of quality is less sensitive to any given percentage change
in quality than the price of quality is to a change in quantity. Conceivably then,
ii,, could be greater than
but we consider this
at the same time that ii,, <
unlikely, since
is probably much greater than ui,,.
where
R =1+
The two income concepts I
restraint.
Define
It is well known that the tr
the relation
The observed income
I while holding the it's cons
1—k———
may be
Equation
logarithmically with respect
tions (A3) and (A4).
than the true elasticities in
We now define two
follows:
=
Elty
= i—k
KER AND H. GREGG LEWIS
ong positive effect on the
iber of their children, and
birth control knowledge
dren but also significantly
eceding analysis. Quantity
adow price of quality deintity depends on quality.
substitution in household
a special relation between
:s of equal percentage inn wage rates. To put the
0. The equal increases
simply as a relative fall
percentage declines in
r
tity and quality of all goods from the viewpoint of this paper should be
quite rewarding.
Mathematical Appendix
The budget restraints specified in equations (2) and (10) are not linear in n
and q. It is precisely this nonlinearity, of course, that leads to the "interaction
between quantity and quality" that we discuss in this paper.
The derivation of the elasticities of the demand functions for number of
children (n) and child quality (q) can be carried in a direct fashion by differentiating the budget restraint and the first-order conditions. Because of the
nonlinearity of the budget restraint, however, if this direct mode of derivation
is followed, it is all too easy to lose sight of the underlying income and substitution elasticities in the utility function. Hence, we follow an indirect approach
that makes use of quite familiar propositions in demand theory.
First, we replace the curvilinear budget surface given in equation (10) by a
plane surface by adding nqis to both sides:
for y. However, since the
more than p,,, n would
plasticity of quantity with
nd 't tends to be greater
r quality. De Tray finds
he number of children by
dren.
ntity is a better substitute
ink is a plausible, though
induces a fall in n relative
ice elasticity of quantity
r conclusion for observed
•quality ordering for price
inexpected implication of
t
to the interaction
also to the quantity and
lications, and large numne elasticities of quantity
Dm the "true" elasticities.
nteraction between quan-
87
QUANTITY AND QUALITY OF CHILDREN
R
=
where
(Al)
+ yPy.
+
(A2)
(A3)
The two income concepts I and R differ by the nonlinear term nqir in the budget
restraint.
Define
kn±kq±ky= 1.
It is well known that the true income elasticities (ii,,,
the relation
+
+
1=
and
(A4)
'ly) must satisfy
(AS)
The observed income elasticities
obtained by changing full income
I while holding the it's constant, however, must satisfy
I— k
=—
+
=
+
(A6)
Equation (A6) may be verified by differentiating the budget restraint (10)
logarithmically with respect to I, holding the it's constant, and then using equations (A3) and (A4'i. Thus, the observed elasticities, on the average, are smaller
than the true elasticities in the ratio 1 — k = I/R.
We now define two household price indexes and in differential form as
follows:
y3t1,
from the budget restraint
Is of the more general form
E3t1,
—
any given percentage change
quantity. Conceivably then,
+
flitn
Eu.,,
+
+
nqut
Eu
1
but we consider this
1—k
L
(A7)
88
GARY S. BECKER AND H. GREGG LEWIS
Ep =
+
+
QUANTITY AND QUALIT
(A8)
1—k
where the symbol E denotes the natural logarithmic differential operator din.
Since
=
=
+
=
and
+
it follows that
—
L
Kq
(A9)
Substitute these results (A9) into (A8) and then use (A7) to obtain
+
Ep = (1 —
(AlO)
kEnqrt.
Now differentiate (A3) logarithmically:
ER
(1 —
k)EI
Subtract (Alo) from (All):
+ kEnqn.
(All)
= (1 —
(1 —
=
+
+
+ kQGnQEPq,
— (1 —
+
+ kqanq;
rather than (10), so that
(8) are simply
that section that needs
proposition that
Let
=
— (1 — kq)OqEPq, (A13)
(1 — kq)oq
=
—
= 0nq
—
1—k
(A12)
increases in the
When I is increased, holding the it's constant, real income
smaller ratio 1 — k = I/R. This is the economic basis of the downward bias in
the observed income elasticities relative to the true income elasticities.
We now turn to the derivation of the observed income and substitution elasticities. We make use of the well-known propositions that
En =
Eq =
D and (1 —
In the section on
where (1 — 2kcr) must be
on price effects, we assur
if
is sufficiently smi
of
We have no:
—
above).
We turn now to the in
with respect to the it's,
income elasticities. We co
+ knanq,
—
=—
where the cs's are the Allen partial elasticities of substitution in the utility
function. Notice that
is the average elasticity of substitution of n against y
and q and that
is the similar elasticity for q against y and n.
by letting I change
We first derive the observed income elasticities
and
while the ii's are constant. Because the at's are constant, it follows from (A9)
= (k3
—
k)
and (A12) that
When
=(1—k)E1.
=
=
—.
(A14)
Substitute (A14) into (A13) and collect terms to obtain
=-
I
1—
(1
Solve these two equations for
and
= (1—
(AlS)
Both observed "own p
the sign of
—
ER AND H. GREGG LEWIS
QUANTITY AND QUALITY OF CHILDREN
(A8)
(1 —
—k
c differential operator din.
it follows that
89
=
(1 —
—
(A16)
1.
k2(1
kq
(A9)
(A7) to obtain
(AlO)
D and (I —
must be positive by the second-order conditions.
In the section on income effects, we used the simpler budget restraint (2)
k; equations (7) and
rather than (10), so that we assumed that
(8) are simply equations (Al6) when
= kq = k. The only proposition in
that section that needs qualification when the budget restraint is (10) is the
proposition that
(All)
Let
— Tj,,
has the sign of
—
= 0nq = a in (A16). Then
=
unless aqy is much larger than
ka
.1—ic
icnicq
(A12)
increases in the
of the downward bias in
:ome
tome elasticities.
ome and substitution elas-
where (1 — 2ka) must be positive by the second-order conditions. 'In the section
on price effects, we assume that
—
is positive. But then
= nit,, —
if Tlq/fln is sufficiently smaller than k,/kq,
will have a sign opposite that
—
of Tlq — n,,. We
above).
(A13)
income elasticities. We consider first the elasticities with respect to it,, and itq:
=
kyGqy + knOnq,
ubstitution
have noted this qualification and commented on it (see n. 4
We turn now to the income-compensated elasticities of quantity and quality
with respect to the it's, deriving them in essentially the same manner as the
KEPn + kqonqEPq,
(1
(A17)
in the utility
—
bstitution of n against y
y and n.
nd
by letting I change
nt, it follows from (A9)
k)
ka,,q)
+
i'/jzn,q.
(A18)
a, these simplify to
When
—
-
(A14)
k
(k1k,
(1 —
D=(l—2ka)—
(AlS)
—
k( 1 —
+
—
is
= n, q,
(A19)
L
are negative, and
ambiguous even when it is assumed that
Both observed "own price" elasticities, 1j,I,.,,(') and
the sign of
i
90
any =
to Pq,
GARY S. BECKER AND H. GREGG LEWIS
a and that
— k5
> 0. However, if
is small relative
will exceed
The observed cross-elasticities,
are positive
exceeds
anq > 0, and
if
assume. The cross-partial derivatives, however, are equal:
=
aq
= nq Lr
— kany)
+
and
kq, as we
Child Quality
for Children
RnIIq
(A20)
Dennis N. De
where the derivatives are income-compensated.
The observed elasticities with respect to
=
— kany) —
=
RAND Corporation
are
+
i
j
= n, q,
(A21)
from which it follows that
= (Gny —
+
+
sizable portion of such
— kq).
(A22)
Thus, if the a's are equal and if kn > kq, the quantity elasticity (ijflTY(8)) exceeds
the quality elasticity
this difference is increased if
> 0qy' a special,
though plausible, assumption.
The observed elasticities with respect to equal percentage changes in
tq,
are simply those with respect to it,, but with signs changed. Thus, the
demand functions for a and q are homogeneous of degree zero in I and the it's,
just as they are in the shadow income (R) and the shadow prices (ft's).
and
Introduction
The past decade has b
of phenomena outside
(1 — kanq)
— kq2aqy)
I.
desired fertility and f
degree to which pure
completed fertility. Th
the way in which
services either by mci
creasing the resource
quantity and quality
production function
of desired family size
children and child qu
parameters and then
U.S. counties.
This paper is drawn lii
by the RAND
and criticisms on earlier;
Yoram Ben-Porath,
Finis Welch for their
rated in this paper. I, of
work for this paper was
the National Institute o
grant to the RAND Co
in this paper are mine
1 See,
for example,
(1970a), Nerlove and Sc
Familiarity with th
(Becker 1965; Lancaste
description of this mod