The Becker Fertility Model:
Theory and Critique
Joseph Anthony Burke
Ave Maria University
Department of Economics and Business
Working Paper No. 1201
Contact Information:
Joseph.Burke@avemaria.edu
phone: 239-280-1613
fax: 239-280-1637
Department of Economics and Business
Ave Maria University
5050 Ave Maria Boulevard
Ave Maria, FL 34142
Draft dated May 24, 2012.
The Becker Fertility Model:
Theory and Critique
This paper is an exploration of the theoretical properties of the Becker fertility model.
I demonstrate that the comparative statics of the Becker fertility model with a general
budget constraint and its corresponding expenditure model can be expressed in terms of
the ordinary consumer expenditure function with a change of variable. When there are no
fixed-costs of quality per child, the Becker fertility model is equivalent to the ordinary
consumer model with restrictions on the form of the utility function. Solutions to the
Becker fertility model are provided with the Cobb-Douglas, CES, and AIDS
specifications under this assumption, Becker’s hypothesis that demand for quality per
child increases with income is not valid under any Cobb-Douglas or CES specification,
but can be tested with a valid estimation of the AIDS model. I also evaluate the role of
fixed-costs of quality per child in Becker’s model and show that they introduce a third
term into the Slutsky matrix, i.e. in addition to ordinary substitution and income effects,
but that these effects are small when the average fixed costs of quality per child is small
relative to the marginal cost of quality per child. When the Becker model is expressed in
terms of children, quality, and other goods, children must be a substitute for either their
quality or for other goods and either quality or other goods must be a luxury good.
Keywords: microeconomic theory, Becker, fertility
JEL Classification: D01, J11, J13, Z13
Introduction
This paper is an analysis and critique of the theoretical properties of the fertility
model introduced by Gary Becker in his Treatise on the Family. In this model, Becker
argues that, as income increases, households substitute fewer children of higher quality
for more children of fewer quality. This model is interpreted as both a model of
household demand as income increases and as a model of society experiencing economic
growth, where fertility rates fall overall as households move up the socioeconomic
ladder. Becker’s model is an argument against Westoff and Ryder (1977), who attribute
the decreases in fertility over the last fifty years to a “contraceptive revolution” that
began with the invention of the Pill in 1960.1
This paper is an exploration of the theoretical properties of the Becker fertility
model. First, I derive the properties of the Becker fertility model with his original
formulation of the budget constraint, and then the properties of his more general budget
constraint with two-part pricing of children and quality per child. I show that, when there
are no fixed-costs of quality per child, the Becker fertility model is equivalent to the
ordinary consumer problem with restrictions on the form of the utility function. Under
this assumption, solutions to the Becker fertility model are provided with the CobbDouglas, CES, and AIDS specifications. These examples show that the Becker model can
be easily solved with a simple substitution of variables and illustrate that Becker’s
hypothesis that demand for quality per child increases with income is not valid under any
Cobb-Douglas or CES specification, though it can be tested with a valid estimation of the
AIDS model. I then derive the comparative statics for the Becker fertility model with the
general budget constraint in terms of the ordinary consumer expenditure function. I show
that they the inclusion of fixed costs of quality per child introduces a term of indirect
price effects into the Slutsky matrix in addition to ordinary substitution and income
effects, but that these indirect price effects are small when the average fixed costs of
quality per child is small relative to the marginal cost of quality per child. Finally, I show
that, when the Becker model is expressed in terms of children, quality, and other goods,
children must be a substitute for either their quality or for other goods and either quality
or other goods must be a luxury good.
1
See Becker, p. 143-44.
Becker’s Model of Fertility
Let k be the number of children, g be quality per child (“goodness”), and x be
goods other than children. The price of quality per child is pc and the price of other goods
is px. Households have income m, and let λ be the Lagrange multiplier on the budget
constraint Becker’s model of fertility is
v*(p,m) ≡ max k,g,x u(k,g,x)
subject to
pcgk + pxx ≤ m
This is analogous to the consumer problem with children k replaced by the product gk,
which is quality-children or effective children. Like the consumer problem, all known
comparative statics results are derived from the expenditure problem. Let v be the target
utility level for the expenditure problem and θ be the Lagrange multiplier for the utility
constraint. The corresponding expenditure problem is
eH(p,v) ≡ min k,q,x pcqk + pxx
subject to
u(k,q,x) ≥ v
Let kH, gH, and xH be functions of prices and target utility that solve this expenditure
problem. The Envelope Theorem gives the following results
∂eH/∂px ≡ xH(p,v)
∂eH/∂pc ≡ gH(p,v) · kH(p,v)
∂eH/∂v ≡ θH(p,v)
Like the ordinary expenditure problem, the Hessian of eH(p,v) is negative semi-definite in
p = (pc, px), which implies the following comparative statics,
∂2eH / ∂px2 ≡ ∂xH / ∂px ≤ 0
∂2eH / ∂pc2 ≡ ∂(gHkH)/∂pc ≤ 0
The first equation shows that Hicksian demand for other goods decreases with its price,
px. The second equation shows Hicksian demand for effective children decreases with the
price of quality per child, pc. Mutltiplying out the second derivative shows
∂2eH / ∂pc2 ≡ (gH) ∂kH/∂pc + (kH) ∂gH/∂pc ≤ 0
This shows that Hicksian demand for children and quality per child are ambiguous in
sign. Hicksian demand for quality per child is not necessarily downward sloping in the
price of quality per child, pc. The symmetry of the Hessian matrix also implies the
symmetry result
∂2eH / ∂px∂pc ≡ ∂2eH / ∂pc∂px
∂xH/∂pc ≡ ∂(gHkH)/∂px ≡ (gH) ∂kH/∂px + (kH) ∂gH/∂px
All three of the partial derivatives in this identity are ambiguous in sign.
The ambiguity of these comparative statics results is propagated in the utility
problem. Let k*, g*, and x* be functions of prices and income that solve the Becker
Fertility model. Duality means
k*(p,eH(p,v)) ≡ kH(p,v)
g*(p,eH(p,v)) ≡ gH(p,v)
x*(p,eH(p,v)) ≡ xH(p,v)
Becker’s budget constraint alters the Slutsky equations so the income effect is now
multiplied by effective children,
∂k*/∂pc ≡ ∂kH/∂pc – gH kH (∂kH/∂m)
∂g*/∂pc ≡ ∂gH/∂pc – gH kH (∂gH/∂m)
The substitution effect, which is always negative in the ordinary expenditure problem, is
ambiguous in both of these equations. The income effect is also theoretically ambiguous
in both equations, and note that the factor producing the income effect is not children or
quality per child but effective children.
These equations can be combined to describe demand for effective children with
respect to the price of quality per child
g*(p,eH(p,v)) k*(p,eH(p,v)) ≡ gH(p,v) kH(p,v)
(k*) ∂g*/∂pc + (gH kH) (k*)(∂g*/∂m) + (g*) ∂k*/∂pc + (gH kH)(g*)(∂k*/∂m)
≡ gH (∂kH/∂pc) + kH (∂gH/∂pc)
(k*) ∂g*/∂pc + (g*) ∂k*/∂pc
≡ gH (∂kH/∂pc) + kH (∂gH/∂pc) – (gH kH)(g*)(∂k*/∂m) – (gH kH)(k*)(∂g*/∂m)
∂(g*k*)/∂pc ≡ ∂(gHkH)/∂pc – (gH kH) ∂(g*k*)/∂m
Becker’s thesis is that, as income increases, households substitute have fewer high quality
children instead of more children with lower quality. In terms of these derivatives, this
argument means that children are an inferior good, quality per child is a normal good, and
effective children are a normal good
∂(g*k*)/∂m ≡ (gH kH)(g*)(∂k*/∂m) + (gH kH)(k*)(∂g*/∂m) > 0
∂k*/∂m < 0
∂(g*k*)/∂m > 0
∂g*/∂m > 0
However, the signs of all three of these derivatives are theoretically ambiguous, though
the insight from economic tradition is that effective children should be normal, since it is
an aggregate good for which there is not close substitutes.
There are two main problems with this specification of the model. First, almost all
of the comparative statics results are theoretically ambiguous, leaving even fewer than
usual testable hypotheses. Second, Becker’s formulation of the budget constraint means
that children of quality zero (g = 0) are free. Given that even uneducated and poorly
behaved children need to eat, the budget constraint would seem to imply that children of
quality zero are dead.
Becker with Two-Part Pricing
In his more general model, Becker used a two-part pricing structure for both
children and quality per child that addresses the problems with the model.2 Here pk is the
fixed cost per child (of zero quality), and the pg is the fixed cost of quality per child. The
budget constraint is
pkk + pgg + pcgk + pxx ≤ m
With this slight modification, the expenditure problem is now
eH(p,v) ≡ min k,g,x pkk + pgg + pcgk + pxx
subject to
u(k,q,x) ≥ v
As before, the Hessian of the expenditure function for this problem is negative semidefinite, which, in conjunction with the Envelope Theorem, implies
∂2eH / ∂px2 ≡ ∂xH / ∂px ≤ 0
∂2eH / ∂pg2 ≡ ∂gH / ∂pg ≤ 0
∂2eH / ∂pc2 ≡ ∂(gHkH)/∂pc ≤ 0
∂2eH / ∂pk2 ≡ ∂kH / ∂pk ≤ 0
2
Becker, p. 149.
The addition of the fixed cost of children to the budget constraint now gives testable
comparative statics for Hicksian demand for both children and quality per child. Hicksian
demand for children decreases with the fixed price of children, and Hicksian demand for
quality per child decreases with the fixed price of quality.
The comparative statics results in a new Slutsky equation for Marshallian demand
for children
∂k*/∂pk ≡ ∂kH/∂pk – kH (∂kH/∂m)
∂g*/∂pg ≡ ∂gH/∂pg – gH (∂gH/∂m)
This is the ordinary Slutsky result decomposing an increase in pk and pg into substitution
and income effects. Becker notes that an increase in the fixed price per child may
represent “a reduction in child allowances or reduced costs of contraception.”3 The
marginal cost of a child of quality gH is (pk + pcgH). The substitution effect is no longer
theoretically ambiguous, and the income effect is now produced by children (kH) instead
of effective children as it was before. The Becker model with two-part pricing is
v*(p,m) ≡ max k,g,x u(k,g,x)
subject to
pkk + pgg + pcgk + pxx ≤ m
λ
To clarify Becker’s argument vis-à-vis Westoff and Ryder: Westoff and Ryder argue that
the reduction in fertility since 1960 can be attributed to the Pill, which is modeled as a
reduction in contraceptive costs and in increase in pk; Becker attributes the decrease in
fertility to increased demand for quality per child, which he claims is strongly driven by
increases in income, m, but may also be affected by reductions in the fixed and marginal
costs of quality per child, pg and pc.
3
Becker, p. 150.
Becker with a Quality Variable and pg = 0
I now consider a second modification to Becker’s formulation: specifying the
quality variable not on a per-child basis, but in total. Ordinarily the variables used in the
consumer problem are specified as totals, not per unit variables, so Becker’s specification
of g as quality per child is unusual. Assume for the moment that the fixed cost of quality
per child is zero, so pg = 0 and the budget constraint is
pkk + pcgk + pxx ≤ m
Let q be total quality, so q = gk and g = q/k, and let pq = pc. The Becker model with two
part pricing is then equal to
v*(p,m) ≡ max k,q,x u(k,q/k,x)
subject to
pkk + pcq + pxx ≤ m
In this model. the household chooses total quality instead of quality per child. For
u′(k,q,x) = u(k,q/k,x), the Becker model with two-part pricing is
v*(p,m) ≡ max k,q,x u′(k,q,x)
subject to
pkk + pqq + pxx ≤ m
This formulation shows that Becker’s model can be thought of as the ordinary consumer
problem with a restricted utility function. This has all of the ordinary properties of the
consumer problem. All Hicksian demands to the corresponding expenditure problem are
downward sloping and symmetric in their cross-price effects. Optimal quality per child in
the solution to the Becker model can be recovered from the solution to this problem with
the identity g*(p,m) = q*(p,m)/k*(p,m), and the corresponding expenditure problem can
be solved similarly with the identity g H(p,v) = q H(p,v) /kH(p,v). Becker’s argument about
the relationship between children and quality means that children and quality, should be
complements, so ∂k*/∂pq ≡ ∂q*/∂pk < 0.
Here I assumed that fixed cost of quality per child was zero, pg = 0. If this
assumption is relaxed, then the price of quality is a function of children, pq = pq(k) = (pg /
k + pc). Insofar as Becker’s argument is about pc and not pg, models derived with the
substitution of q = gk under this restriction are still useful for illustrating Becker’s effects.
Properties of the model with the relaxed assumption are discussed later.
Example. The Cobb Douglas Utility Function (pg = 0)
Assume the utility function takes the Cobb-Douglas form, so u′(k,q,x) =
kaqbx1−a−b. The problem becomes
v*(p,m) ≡ max k,q,x kaqbx1−a−b
subject to
pkk + pqq + pxx ≤ m
Substituting q = gk, the utility function is u(g,k,x) = u′(k,gk,x) = ka(gk)bx1−a−b =
ka+bgbx1−a−b. With this substitution, the problem becomes
v*(p,m) ≡ max k,g,x ka+bgbx1−a−b
subject to
pkk + pcgk + pxx ≤ m
The solution to this problem is
k* = am / pk
q* = bm / pq
This implies
g* = q* / k* = (bm / pq) / (am / pk) = (bpk / apq)
This example shows that there is the income effect on quality per child is zero under a
Cobb-Douglas specification of the utility function. Optimal quality per child would not
increase with income as Becker hypothesizes, but would be independent of income.
Additionally, the solution with Becker’s formulation of the model with pk = 0 can be seen
by taking the limits of q*, k*, and g* as pk→0,
lim p(k)→0 q* = 0
lim p(k)→0 k* = ∞
lim p(k)→0 g* = 0
Under Cobb-Douglas utility function without two-part pricing, optimal quality per child
is zero. Households would choose an infinite number of children of zero quality without
the two-part pricing structure.
Example. The CES Utility Function (pg = 0)
Assume the utility function takes the CES form, so u′(k,q,x) = (ka + qa + xa)1/a.
The problem becomes
v*(p,m) ≡ max k,q,x (ka + qa + xa)1/a
subject to
pkk + pqq + pxx ≤ m
Substituting q = gk, the utility function is u(g,k,x) = u′(k,gk,x) = (ka + (gk)a + xa)1/a. With
this substitution, the problem becomes
v*(p,m) ≡ max k,g,x (ka + (gk)a + xa)1/a
subject to
pkk + pcgk + pxx ≤ m
Let b = a / (a – 1). The solution to this problem is
k* = pkb−1 m / (pkb + pqb + pxb)
q* = pqb−1 m / (pkb + pqb + pxb)
This implies
g* = q* / k* = [pqb−1 m / (pkb + pqb + pxb)] / [pkb−1 m / (pkb + pqb + pxb)] = (pk / pq)b−1
The income effect on quality per child is zero under a CES utility function. The solution
to the original Becker model can be found by taking the limit of q*, k*, and g* as pk→0,
which are
lim p(k)→0 q* = 0
lim p(k)→0 k* = ∞
lim p(k)→0 g* = 0
Like the Cobb-Douglas specification, optimal quality per child is zero. Households
choose an infinite number of zero-quality children.
The absence of any income effect on quality per child with either the CobbDouglas or the CES utility functions suggests that it may be difficult to find a simple
utility function that can illustrate the effects that Becker posits.
Example. The Almost Ideal Demand System (pg = 0)
Assume the utility function is that for the Almost Ideal Demand System.4 For
i=k,q,x, let wi be budget share of i, e.g. wk = pkk / m. Let P be a price index defined by
log P = a0 + ∑i ai log pi + ½ ∑i ∑j bqi log pi log pj
The solution to the AIDS problem is
wk* = pkk* / m = ak + ∑i bki log pi + ck log (m / P)
wq* = pqq* / m = aq + ∑i bqi log pi + cq log (m / P)
This implies
g* = q* / k* = pkwq* / pqwk*
= pk [aq + ∑i bqi log pi + cq log (m / P)] / pq [ak + ∑i bki log pi + ck log (m / P)]
Becker’s hypothesizes that ∂g*/ ∂m > 0. For the AIDS system, this implies
∂g*/ ∂m ≡ pkwq* / pqwk*
≡ [pqpkwk* (∂wq* / ∂m) − pkpqwq* (∂wk* / ∂m)] / [pqwk*]2 > 0
This implies
[pqpkwk* (cq) − pkpqwq* (ck)] > 0
pqpk [wk* (cq) − wq* (ck)] > 0
(pqpk / m) [pkk* (cq) − pqq* (ck)] > 0
pkk* (cq) − pqq* (ck) > 0
pkk* (cq) > pqq* (ck)
pkk* / ck > pqq* / cq
wk* / ck > wq* / cq
Becker also hypothesizes that ∂k*/ ∂m < 0.
∂wk* / ∂m = [mpk(∂k*/ ∂m) – pkk*] / m2= ck / m
mpk(∂k*/ ∂m) – pkk* = mck
4
Deaton and Muellbauer, p. 75.
∂k*/ ∂m = (1/pk)[ck – (pkk* / m)] = (1/pk) [ck –wk*]
Since pk > 0, the hypothesis that ∂k*/ ∂m < 0 is equivalent to
ck – wk* < 0
or
ck < wk*
Becker’s hypotheses can be tested with a valid estimation of the parameters ck and cq in
the AIDS model.
Theoretical Implications (pg = 0)
With the addition of the two-part pricing for children, Becker’s fertility model
specified in terms quality per child is equivalent to a theoretical restriction on the form
that the ordinary utility function, in terms of quality, can have. Specifically, for every
utility function u′(k,q,x) there exists another utility function u(k,q/k,x) such that
u′(k,q,x) ≡ u(k,q/k,x)
Utility functions are ordinarily nondecreasing in all of their arguments, so
∂u′/∂k ≡ ∂u/∂k – (q/k2) ∂u/∂g ≥ 0
∂u′/∂q ≡ (1/k) ∂u/∂g ≥ 0
∂u′/∂x ≡ ∂u/∂x ≥ 0
The first inequality leads to the restriction that
∂u/∂k ≥ (q/k2) ∂u/∂g
Multiplying by k/u and observing that g = q/k shows that this restriction is equivalent to
εu,k ≥ εu,g
The elasticity of the Becker utility function with respect to k must be greater than the
elasticity of the utility function with respect to g. If this is not true, the Becker utility
function would not be increasing in quality per child, g.
The negative semi-definiteness of the Hessian of the utility function implies
∂2u′/∂k2 ≡ ∂2u/∂k2 – 2(q/k2) ∂2u/∂g∂k + 2(q/k3) ∂u/∂g + (q2/k4) ∂2u/∂g2 ≤ 0
∂2u′/∂q2 ≡ (1/k2) ∂2u/∂g2 ≤ 0
∂2u′/∂x2 ≡ ∂2u/∂x2 ≤ 0
and symmetry properties.
The Becker Fertility Model with a General Budget Constraint
I now consider Becker’s model with a relaxation of the assumption that pg = 0.
Let the price of quality be a function of children, pq = pq(k) = (pg / k + pc). With two-part
pricing, the Becker fertility model can be expressed as
vB(p,m) ≡ max k,q,x u′(k,q,x)
subject to
pkk + pq(k)q + pxx ≤ m
The corresponding expenditure problem, which I will refer to as the Becker expenditure
problem, is
eBH(p,v) ≡ min k,q,x pkk + pq(k)q + pxx
subject to
u′(k,q,x) ≥ v
Here the price vector is p = [pk, pg, pc, px]. I will refer to the solution to the expenditure
problem as the Becker-Hicksian demands. Let eH(p′,v) and v*(p′,m) be the solution to
this Becker problems under the assumption that pg = 0, which are the ordinary consumer
and expenditure functions for p′ = [pk, pq, px]. The solutions to two expenditure problems
are identical at the point pq = (pg / kBH + pc), i.e. when the price of quality is equal to the
price function of quality evaluated at the optimal choice of children kBH. These functions
are related by the identities
eBH(p,v) ≡ eH([pk, (pg / kBH + pc), px] ,v)
kBH(p,v) ≡ kH([pk, (pg / kBH + pc), px] ,v)
qBH(p,v) ≡ qH([pk, (pg / kBH + pc), px] ,v)
xBH(p,v) ≡ xH([pk, (pg / kBH + pc), px] ,v)
Differentiating the second identity with respect to all variables except pg and evaluating
them where kBH = kH shows
∂kBH / ∂pk ≡ ∂kH / ∂pk – (pg/(kH)2)(∂kH / ∂pq) (∂kBH / ∂pk)
∂kBH / ∂pc ≡ ∂kH / ∂pq (∂pq/∂pc) – (pg/(kH)2)(∂kH / ∂pq) (∂kBH / ∂pc)
∂kBH / ∂px ≡ ∂kH / ∂px – (pg/(kH)2)(∂kH / ∂pq) (∂kBH / ∂px)
∂kBH / ∂v ≡ ∂kH / ∂v – (pg/(kH)2)(∂kH / ∂pq) (∂kBH / ∂v)
Note ∂pq/∂pc = 1 is the second equation. Let φkH ≡ (pg/(kH)2)(∂kH/∂pq). Note that ∂kH/∂pq
> 0, so φH > 0. The identities are differentiated at the point kBH = kH, so φH is a function
of (p′,v). Solving these equations shows
∂kBH / ∂pk ≡ (1 + φkH)−1 (∂kH / ∂pk)
∂kBH / ∂pc ≡ (1 + φkH)−1 (∂kH / ∂pq)
∂kBH / ∂px ≡ (1 + φkH)−1 (∂kH / ∂px)
∂kBH / ∂v ≡ (1 + φkH)−1 (∂kH / ∂v)
These equations show that the comparative statics of the Becker-Hicksian demand for
children are those of the ordinary Hicksian demand for children scaled by the factor (1 +
φkH)−1. Since φH > 0, the Becker-Hicksian demands share the signs of the ordinary
Hickisan demands for pk, pc = pq, px, and v, i.e. Becker-Hicksian demand for children is
downward sloping in pc. Differentiating the demand for children with respect to pg shows
∂kBH/∂pg ≡ [(∂kH/∂pq) (1/kBH) – (pg/(kBH)2) (∂kH/∂pq) (∂kBH / ∂pg)]
Solving this for ∂kBH/∂pg and evaluating it where kBH = kH shows
∂kBH/∂pg ≡ (1/kH) (1 + φkH)−1 (∂kH / ∂pq) ≡ (1/kH) (∂kBH/∂pc)
Let φqH ≡ (pg/(kH)2)(∂qH/∂pq) and Let φxH ≡ (pg/(kH)2)(∂xH/∂pq). Using these results, the
derivative of Becker-Hicksian demand for quality with respect to pk is
∂qBH / ∂pk ≡ ∂qH / ∂pk – (pg/(kH)2) (∂qH / ∂pq) (∂kBH / ∂pk)
≡ ∂qH / ∂pk – (φqH / (1 + φkH )) (∂kH / ∂pk)
The derivatives of the Becker-Hicksian demands with respect to the variables pk, pc = pq,
px, and v all have the same form
∂qBH / ∂pk ≡ ∂qH / ∂pk – (φqH / (1 + φkH)) (∂kH / ∂pk)
∂qBH / ∂pc ≡ ∂qH / ∂pq – (φqH / (1 + φkH)) (∂kH / ∂pq)
∂qBH / ∂px ≡ ∂qH / ∂px – (φqH / (1 + φkH)) (∂kH / ∂px)
∂qBH / ∂v ≡ ∂qH / ∂v – (φqH / (1 + φkH)) (∂kH / ∂v)
∂xBH / ∂pk ≡ ∂xH / ∂pk – (φxH / (1 + φkH)) (∂kH / ∂pk)
∂xBH / ∂pc ≡ ∂xH / ∂pq – (φxH / (1 + φkH)) (∂kH / ∂pq)
∂xBH / ∂px ≡ ∂xH / ∂px – (φxH / (1 + φkH)) (∂kH / ∂px)
∂xBH / ∂v ≡ ∂xH / ∂v – (φxH / (1 + φkH)) (∂kH / ∂v)
The second term in each of these derivatives is the indirect price effect. The indirect price
effect is the effect that any price change has price (marginal cost) of quality, which is a
function of the number of children chosen by the household in Becker’s budget
constraint.
Derivatives of Becker-Hicksian demand for quality and other goods with respect
to pg are equivalent to the derivatives of the Becker-Hicksian demand with respect to pc
scaled by (1/kH)
∂qBH / ∂pg ≡ (1/kH) (∂qBH / ∂pc) ≡ (1/kH) [∂qH / ∂pq – (φqH / (1 + φkH)) (∂kH / ∂pq)]
∂xBH / ∂pg ≡ (1/kH) (∂xBH / ∂pc) ≡ (1/kH) [∂xH / ∂pq – (φxH / (1 + φkH)) (∂kH / ∂pq)]
Applying the Envelope Theorem to the Becker-Hicksian expenditure function shows
∂eBH / ∂pk ≡ kBH(p,v)
∂eBH / ∂pg ≡ qBH(p,v) / kBH(p,v)
∂eBH / ∂pc ≡ qBH(p,v)
∂eBH / ∂px ≡ xBH(p,v)
The negative semi-definiteness of the Hessian of the Becker-Hicksian expenditure
function implies
∂2eBH / ∂pk2 ≡ ∂kBH / ∂pk ≤ 0
∂2eBH / ∂pg ≡ (1 / (kBH)2) [(kBH) ∂qBH/ ∂pg − (qBH) ∂kBH/ ∂pg] ≤ 0
∂2eBH / ∂pc2 ≡ ∂qBH / ∂pc ≤ 0
∂2eBH / ∂px2 ≡ ∂xBH / ∂px ≤ 0
plus symmetry results. The second inequality implies
(kBH) (∂qBH/ ∂pg) ≤ (qBH) (∂kBH/ ∂pg)
εBHq, p(g) ≤ εBHk, p(g)
Where εBHq, p(g) and εBHk, p(g) are the Becker-Hicksian elasticities of demand for quality
and children with respect to pg. The inequality shows that elasticity of the BeckerHicksian demand for quality with respect to pg must be less than the Becker-Hicksian
demand for quality elasticity of children with respect to pg.
With these results, we can now derive the equations for the Becker fertility model
kB(p,eBH(p,v)) ≡ kBH(p,v)
qB(p,eBH(p,v)) ≡ qBH(p,v)
xB(p,eBH(p,v)) ≡ xBH(p,v)
Slutsky equations for Becker demand for children are
∂kB / ∂pk ≡ ∂kBH / ∂pk − (kBH) ∂kB / ∂m
∂kB / ∂pg ≡ ∂kBH / ∂pg − (qBH / kBH) ∂kB / ∂m
∂kB / ∂pc ≡ ∂kBH / ∂pc − (qBH) ∂kB / ∂m
∂kB / ∂px ≡ ∂kBH / ∂px − (xBH) ∂kB / ∂m
Slutsky equations for Becker demand for quality are
∂qB / ∂pk ≡ ∂qBH / ∂pk − (kBH) ∂qB / ∂m
∂qB / ∂pg ≡ ∂qBH / ∂pg − (qBH / kBH) ∂qB / ∂m
∂qB / ∂pc ≡ ∂qBH / ∂pc − (qBH) ∂qB / ∂m
∂qB / ∂px ≡ ∂qBH / ∂px − (xBH) ∂qB / ∂m
Slutsky equations for Becker demand for other goods are
∂xB / ∂pk ≡ ∂xBH / ∂pk − (kBH) ∂xB / ∂m
∂xB / ∂pg ≡ ∂xBH / ∂pg − (qBH / kBH) ∂xB / ∂m
∂xB / ∂pc ≡ ∂xBH / ∂pc − (qBH) ∂xB / ∂m
∂xB / ∂px ≡ ∂xBH / ∂px − (xBH) ∂xB / ∂m
Using the results from the Becker-Hicksian demand, these equations can all be written in
terms of the ordinary expenditure function. Furthermore, note that the derivatives are all
evaluated at the point where pq = (pg / kBH + pc), m = eBH(p,v), and dB = dBH = dH for dB =
[kB, qB, xB] and dBH, dH defined similarly. With these substitutions, the Slutsky equations
for Becker demand for children are
∂kB / ∂pk ≡ (1 + φkH)−1 (∂kH / ∂pk) − (kH) ∂kB / ∂m
∂kB / ∂pc ≡ (1 + φkH)−1 (∂kH / ∂pq) − (qH) ∂kB / ∂m
∂kB / ∂px ≡ (1 + φkH)−1 (∂kH / ∂px) − (xH) ∂kB / ∂m
∂kB / ∂pg ≡ (1/kH) (1 + φkH)−1 (∂kH / ∂pq) − (qH / kH) ∂kB / ∂m
Slutsky equations for Becker demand for quality are
∂qB / ∂pk ≡ [∂qH / ∂pk – (φqH / (1 + φkH)) (∂kH / ∂pk)] − (kH) ∂qB / ∂m
∂qB / ∂pc ≡ [∂qH / ∂pq – (φqH / (1 + φkH)) (∂kH / ∂pq)] − (qH) ∂qB / ∂m
∂qB / ∂px ≡ [∂qH / ∂px – (φqH / (1 + φkH)) (∂kH / ∂px)] − (xH) ∂qB / ∂m
∂qB / ∂pg ≡ (1/kH) [∂qH / ∂pq – (φqH / (1 + φkH)) (∂kH / ∂pq)] − (qH / kH) ∂qB / ∂m
Slutsky equations for Becker demand for other goods are
∂xB / ∂pk ≡ [∂xH / ∂pk – (φxH / (1 + φkH)) (∂kH / ∂pk)] − (kH) ∂xB / ∂m
∂xB / ∂pc ≡ [∂xH / ∂pq – (φxH / (1 + φkH)) (∂kH / ∂pq)] − (qH) ∂xB / ∂m
∂xB / ∂px ≡ [∂xH / ∂px – (φxH / (1 + φkH)) (∂kH / ∂px)] − (xH) ∂xB / ∂m
∂xB / ∂pg ≡ (1/kH) [∂xH / ∂pq – (φxH / (1 + φkH)) (∂kH / ∂pq)] − (qH / kH) ∂xB / ∂m
These relationships can be represented in matrix form. Let J be the Jacobian operation
and H be the Hessian matrix. The ordinary slutsky equations for the consumer problem
with prices p, income m, Hicksian demand xH and Marshallian demand x* is
Jpx* ≡ JpxH – [xH]TJmx* ≡ HpeH – [xH]TJmx*
Let dB be the column vector of Becker demand functions, dB = [kB, qB, xB]T, dH be the
Hicksian demand functions dH = [kH, qH, xH]T and let pd = [pk, pc, px]T, p′d = [pk, pq, px]T,
and let φH be a 3×1 column vector φH = [φkH, φqH, φxH]T. For the Becker demands, the
Slutsky equations for pd are
Jp(d)dB ≡ Jp′ (d)dH – (1 + φkH)−1 [φH] [Jp′ (d)kH]T – (dH)TJmx*
≡ Hp′ (d)eH – (1 + φkH)−1 [φH] [Jp′ (d)kH]T – (dH)TJmx*
This modified Slutsky equation has three terms: the first is the matrix of substitution
effects, the third is the matrix of income effects, and the second term is 3 × 3 matrix of
indirect price effects capturing how changes in the variables affect the pq(kB) through
changes in kB.
These results can be extended to any consumer problem with a product of two
variables in the budget constraint. For any consumer problem with a budget constraint of
the form
pkk + pgg + pcgk + pxx ≤ m
The Slustky matrix corresponding to the utility maximization problem with this budget
constraint can be expressed in terms of the ordinary Hicksian demands with the
substitution of q = gk with corresponding constant price pq, i.e. the consumer problem
with a budget of the form
pkk + pqq + pxx ≤ m
The demands that solve this utility maximization problem with this constraint will always
have a additional term in the Slutsky equations because of the indirect effects of k on pq
thought the equation pq(k) = pg / k* + pc.
The Indirect Price Effect
The Slutsky equation describing the effect of an increase in pb = [pk, pq, px] on
Becker demand for d = [k, q, x] is
∂dB / ∂pb ≡ ∂dH / ∂pb – (φdH / (1 + φkH)) (∂kH / ∂pb) − (bH) ∂dB / ∂m
≡ ∂dH / ∂pb – [(pg / (kH)2) / (1 + (pg / (kH)2) (∂kH/∂pq))] (∂dH/∂pq)(∂kH / ∂pb)
− (bH) ∂dB / ∂m
The second term in the equation is the indirect price effect, i.e. for d and b = k, q, h, the
indirect price effect of a change in pb on Becker demand for d is
(φdH / (1 + φkH)) (∂kH / ∂pb)
≡ [(pg / (kH)2) / (1 + (pg / (kH)2) (∂kH/∂pq))] (∂dH/∂pq)(∂kH / ∂pb)
Let εfp,k be the elasticity of the price function with respect to k, εfp,k ≡ − (∂pq(k) / ∂k)(k /
pq) ≡ − (pg / k2)(k / pq). With this definition, product of the indirect price effect and (pb /
dB) is
(pb / dB)[pg / (kH + (pg)(∂kH/∂pq))] (∂dH/∂pq)(∂kH / ∂pb)
≡ (pb / dB)(pq kH / pq kH) [(pg / (kH)2) / (1 + (pg / (kH)2) (∂kH/∂pq))] (∂dH/∂pq)(∂kH / ∂pb)
≡ (kH / pq) [(pg / (kH)2) / (1 + (pg / (kH)2) (∂kH/∂pq))] (∂dH/∂pq)(pq / dB) (∂kH / ∂pb)(pb / kH)
≡ [(pg kH / (kH)2) / (pq) (1 + (pg / (kH)2) (∂kH/∂pq))] εHd,p(q) εHk,p(b)
≡ [− εfp,k / (1 − εfp,k εHk,p(q)] εHd,p(q) εHk,p(b)
Multiplying both sides of the Slutsky equation by pb / dB and the third term by (m / m)
yields
εBd,p(b) ≡ εHd,p(b) – [−εfp,k / (1 − εfp,k εHk,p(q)] εHd,p(q) εHk,p(b) − sb εBd,m
The elasticity of the price function of quality with respect to children can also be
expressed in terms of average fixed costs and marginal costs of quality per child, AFCq =
pgg / q = pg (q/k) / q = pgq / kq and marginal costs of quality per child MCq = (pg / k) + pc
εfp,k ≡ (∂pq(k)/∂k) (k / pq) ≡ − (pg / k2) (k / ((pg / k) + pc))
≡ − (pg / k) / ((pg / k) + pc) ≡ − [(pg / k) / ((pg / k) + pc)] (q / q)
≡ − (pgq / kq) / ((pg / k) + pc)
≡ − AFCq / MCq
With this substitution, the Slutsky equation is
εBd,p(b) ≡ εHd,p(b) – [(AFCq / MCq) / (1 – (−AFCq / MCq) εHk,p(q)] εHd,p(q) εHk,p(b) − sb εBd,m
≡ εHd,p(b) – [AFCq / (MCq + AFCq εHk,p(q))] εHd,p(q) εHk,p(b) − sb εBd,m
This equation shows that the magnitude of the indirect price effect of a change in pb on
demand for d increases with magnitudes of (a) the elasticity of the price function of
quality with respect to children, (b) the elasticity of Hicksian demand for d with respect
to the price of quality, (c) the elasticity of Hicksian demand for children with respect to
pb, and (d) decreases in magnitude with the elasticity of Hicksian demand for children
with respect to the price of quality. Because of the relatively large own- and cross-price
effects of children and quality, indirect price effects would be relatively larger for
demand for children and quality and relatively smaller for demand for other goods.
However, they are always small when εfp,k is small, i.e. when an increase in the number
of children has a relatively small effect on the marginal cost of quality, MCq = pq(k)
evaluated at k. This occurs when average fixed costs of quality per child are small
relative to the marginal cost of quality per child.
From earlier, the scalar (1 + φkH)−1 ≡ [AFCq / (MCq + AFCq εHk,p(q))] . In elasticity
form, the Slutsky equations for the Becker demands are
[εB d,p(b)] ≡ [εH d,p(b)] – (1 + φkH)−1 [εHd,p(q)] [εHk,p(b)]T – [εH d,m] [sb]T
Here [εB d,p(b)] is a 3×3 matrix of Becker price elasticities, [εH d,p(b)] is the 3×3 matrix of
Hicksian price elasticities, (1 + φkH)−1 is a scalar, [εHd,p(q)] is a 3×1 vector of elasticities of
demand for d = [k, q, x] with respect to the price of quality, pq, [εHk,p(b)] is a vector of
elasticities of demand for children with respect to pb = [pk, pq, px], [εH d,m] is a 3×1 vector
of income elasticities for d, and [sb] is a 3×1 vector of budget shares sb = [sk, sq, sx].
Are the average fixed costs of quality per child small relative to the marginal cost
of quality per child? If this is the case, then the indirect price effects are relatively small
and the ordinary consumer problem with a quality of children variable is an adequate
approximation. Among the fixed costs of quality per child, Becker describes fixed costs
per child as “expenditures that are largely independent of children because of the joint
consumption by different children (items like hand-me-down clothes and learning from
parents).”5 These expenses are trivial compared to marginal costs of quality per child,
which includes all subsequent expenses and forgone income spent on quality, including
education, health care, food, sports and other extracurricular activities, and housing.6 This
suggests that indirect price effects are relatively small compared and the fixed costs of
quality per child can be safely ignored without materially affecting Becker’s argument.
Discussion
I have shown that the Becker fertility model is equivalent to the ordinary
consumer problem when pg = 0 with the substitution q = gk and differs by a third term of
indirect price effects in the Slutsky matrix when pg > 0. When pg > 0, effects of changes
to pg on Hicksian demand for k, q, and x are the same as those as the changes to pc on
5
Becker, p. 149.
Housing costs would be split between fixed and marginal costs: more children need more space, but there
would likely be a fixed cost based on location for living in a good neighborhood.
6
Hicksian demand scaled by (1/k). The indirect price effect is the effect that any price
change has price (marginal cost) of quality, which is a function of the number of children
chosen by the household in Becker’s budget constraint. All of the effects of pg can be
expressed as changes to pc = pq and pk in the ordinary consumer problem. Furthermore, I
have argued that the effects of pg through the indirect price effects are likely relatively
small and can be safely ignored without substantially changing Becker’s argument.
Because of this, I will dispense with this term and refer to the simpler version the Becker
model under the assumption pg = 0.
For u′(k,q,x) = u(k,q/k,x) and pg = 0, the Becker model is
v*(p,m) ≡ max k,q,x u′(k,q,x)
subject to
pkk + pqq + pxx ≤ m
This is an ordinary consumer problem in k, q, and x with the ordinary Slutsky matrix
Jpx* ≡ JpxH – (xH)TJmx* ≡ HpeH – (xH)TJmx*
Becker’s argument is about the relationship between children, k, quality, q, and income,
m, in this model.
One serious problem with Becker’s model of fertility is that it accepts the thesis
that children are an inferior good. In Becker’s model, demand for effective children
increases with income, but, as income increases households substitute fewer high quality
children for more low quality children, demand for children decreases with income and
children are themselves an inferior good. While Becker’s model allows for the possibility
that children are normal goods, his argument and interpretation of the evidence is that
they are inferior. Becker’s model is motivated by the evidence that fertility and wealth
have historically been positively correlated until the mid-nineteenth century, but they
have been negatively correlated for approximately the last 150 years.7
With the substitution of the quality variable, the Becker model is equivalent to the
consumer model, so changes in demand for children or fertility is driven only by changes
in prices, income, or preferences. The Becker fertility model therefore allows only two
possible explanations for the long-term decrease in fertility: either children were
previously normal but are now inferior goods, or the fixed cost of children, pk, relative to
quality has been continuously increasing for the last century and a half. The first
explanation implies on an unsatisfactory exogenous change in preferences, which is
outside the scope of economics, and the second depends on an unlikely multigenerational increase in the relative price of children.
Could a long-run trend of an increasing price of children relative to quality
leading to a substitution of quality for children explain the long-run trend towards
decreasing fertility? The real prices of these aggregate goods over such a long time period
would tend towards the long-run minimum average costs, which are driven by
technological changes and tend to decrease over time, not increase. Thus, the real price of
children and quality should have decreased, not increased, over the last 150 years. For the
relative price of children in terms of quality to increase, the price of quality would have
to decrease faster than the price of children.
The price of children relative to quality would trend upwards increase if
technological improvements lowering the price of quality have led it to fall faster than the
price of children, i.e. technological innovation has favored quality (of children) over
children over the last century and a half. This seems exceedingly unlikely, for it is hard to
7
Becker, p. 144.
imagine a technological improvement or set of improvements that would have lowered
the price of quality to the same degree that improvements in health care have lowered the
infant and child mortality rates and thereby lowered the price of children. Becker himself
notes the magnitude of this change, stating that “prior to the nineteenth century, even in
advanced countries no more than half of all live births survived to age ten.”8 It is unlikely
that there has been a long-run trend towards an increasing price of children relative to the
price of their quality, so the long-run trends in fertility likely cannot be explained by price
effects.
What about the relationship between children and quality? Are children and
quality complements or substitutes? At first glance, it would seem that quality and
children should be complementary goods, not substitutes. Becker himself observes that
children are a good for which there are or seem to be no close substitutes, suggesting that,
if there is a relationship, then it is a complementary one. Households would want to
consume these goods together. Indeed, Becker’s variable is defined to be quality per
child, so it is by definition attached to a child, though the amount per child can vary in
quantity. In Becker’s utility function, g = q / k and u(k, g, x) = u(k, q/k, x), so quality
does not enter utility separately but in conjunction with the number of children. If
children and quality are complements, then
∂kH / ∂pq ≡ ∂qH / ∂pk ≤ 0
An increase in either the price of quality or the price of children would lead to a decrease
in Hicksian demand for both.
Upon further reflection, Becker’s argument implies seems to imply that quality
and children are not complements, but substitutes. His thesis is that, as household income
8
Becker, p. 142.
increases, they have fewer children of higher quality instead of more children of lower
quality. This relationship implies children are an inferior good and quality is normal.
Furthermore, as demand for quality increases, demand for children decreases. This would
suggest that children and quality are substitutes: higher income households substitute
quality for children. This leads to a contradiction: quality and children cannot be both
substitutes and complements.
Is it possible for quality to be normal, children to be inferior, and children and
quality to be complementary? More generally, for any two complementary goods, is it
possible for one to be normal and the other to be inferior?
Assume children and quality are complements, so εHk,p(q) ≤ 0 and εHq,p(k) ≤ 0. The
homgeneity equations for Hicksian demands and the signs of their terms imply
εHk,p(k) + εHk,p(q) + εHk,p(x) ≡ 0
(−)
(−)
(+)
εHq,p(k) + εHq,p(q) + εHq,p(x) ≡ 0
(−)
(−)
(+)
εHx,p(k) + εHx,p(q) + εHx,p(x) ≡ 0
(+)
(+)
(−)
If children and quality are complements, then both children and quality are substitutes for
other goods, so εHk,p(x), εHq,p(x), εHx,p(k), and εHx,p(q) are all greater than or equal to zero.
Assuming the Marshallian demands take the same sign as the Hicksian demands, then the
homogeneity equations for the Marshallian demands and the signs of their terms are9
ε*k,p(k) + ε*k,p(q) + ε*k,p(x) + ε*k,m ≡ 0
(−)
9
Silberberg and Suen, p. 291.
(−)
(+)
(−)
ε*q,p(k) + ε*q,p(q) + ε*q,p(x) + ε*q,m ≡ 0
(−)
(−)
(?)
(+)
ε*x,p(k) + ε*x,p(q) + ε*x,p(x) + ε*x,m ≡ 0
(+)
(?)
(−)
(?)
The first equation shows that the two assumptions taken together imply that ε*k,p(x) ≥ 0,
so children and other goods are substitutes. In the Becker fertility model, it is possible for
children and quality to be complements, children to be inferior, and quality to be normal.
However, if children are not substitutes for quality, then they must be substitutes for other
goods. Children cannot be complements to both quality and other goods.
Becker’s thesis that children are inferior goods implies that either quality or other
goods is a luxury. Quality and other goods cannot both be necessities, though one of them
could be an inferior good (in addition to children) if the other was the luxury good.
Differentiating the budget constraint with respect to m shows
skε*k,m + sqε*q,m + sxε*x,m ≡ 1
The sum of the budget shares times the income elasticities must sum to one, children are
inferior (ε*k,m < 0) and all budget shares are less than one, so either quality or other goods
must be luxury goods, i.e. either ε*x,m > 1 or ε*q,m > 1.
Becker’s thesis is that children are an inferior good, and here we have seen that
children must be a substitute with either quality or other goods. Inferior goods are inferior
members of broader classes of goods, so implies that children are either an inferior
member of quality or an inferior type of other good. But how are these two classes of
goods comparable to children? Is either relationship even possible? Quality is defined to
be the aggregate quality of children. How can children be inferior to their own quality?
Or is it even possible for children to be an inferior member of other goods, which is
defined to be all goods other than children? The most likely relationship between these
three goods is that children and quality are complementary, both are substitutes for other
goods, and all three classes of goods are normal. This implausibility of children being an
inferior member of the other two classes of goods is a serious problem with Becker’s
thesis.
Conclusion
Becker’s fertility model can be expressed as the ordinary consumer problem with
the substitution of a quality of children variable for the product of children and quality
per child in his original specification. Without a term for fixed costs of quality per child
in the budget constraint, quality per child is independent under either a Cobb-Douglas or
CES specification, but Becker’s hypothesis can be tested with a valid estimation of the
AIDS model. Mapping Becker’s model into the ordinary consumer problem shows the
limit the possible explanations for the long-run trend in declining fertility since the
nineteenth century to three. Is the long-run trend in declining fertility driven by
increasing levels of income, changes in the price of children relative to quality, or
exogenous changes to preferences?
The inclusion of fixed costs of quality per child introduces nonlinearity in the
budget constraint and adds indirect price effects to the substitution and income effects in
the Slutsky matrix. The indirect price effect is the effect that any price change has price
(marginal cost) of quality, which is a function of the number of children chosen by the
household in Becker’s budget constraint. These indirect price effects should be relatively
larger for children and quality and smaller for other goods. All of the effects of fixed
costs of quality per child can be understood through changes to marginal costs per child
and changes to Hicksian demand for children in the ordinary consumer problem.
Becker’s argument has several contestable points: his argument implies that
quality of children is a normal luxury good, children are an inferior good, and either
quality of children or other goods is a substitute for children. The more plausible
relationship between these three goods is that children and quality are complementary,
both are substitutes for other goods, and all three classes of goods are normal.
Additionally, as an explanation for the long-run trends in fertility since the midnineteenth century, Becker’s argument relies on either an exogenous change in
preferences or a long-run trend towards an increasing price of children relative to the
price of quality of children. Given the vast improvements in medical care of infants and
children, it is most likely that the real price of children has decreased, not increased, over
the last 150 years. If the long-run trend in fertility is driven by the price of children
relative to quality, then the real price of quality would have had to decrease faster than
the real price of children over this time period.
Valid estimation of the Becker fertility model using historical data would be the
ideal empirical approach for determining the extent to which the change in fertility can be
attributed increase in the price children relative to quality or income effects, but such an
approach would first require the construction of reliable aggregates of historical data on
prices and quantities of children, quality of children, and other goods. Both projects—
constructing the data and estimating the AIDS model—would be valuable extensions of
this paper.
References
Becker, Gary S., A Treatise on the Family, Cambridge, MA: Harvard University Press,
1993.
Deaton, A., and J. Muellbauer, Economics and Consumer Behavior, Cambridge:
Cambridge University Press, 1980.
Silberberg, Eugene, and Wing Suen, The Structure of Economics: A Mathematical
Analysis, 3rd edition, Boston: Irwin McGraw-Hill, 2000.
Westoff, Charles F., and Norman B. Ryder, The Contraceptive Revolution, Princeton:
Princeton University Press, 1977.