the effects of male and female labor supply
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The Effects of Male and Female Labor Supply on Commodity DemandsAuthor(s): Martin Browning and Costas MeghirSource: Econometrica, Vol. 59, No. 4 (Jul., 1991), pp. 925-951Published by: The Econometric SocietyStable URL: http://www.jstor.org/stable/2938167
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Econometrica, Vol. 59, No. 4 (July,
1991), 925-951
THE
EFFECTS OF
MALE AND FEMALE
LABOR
SUPPLY
ON COMMODITY DEMANDS
BY
MARTIN ROWNINGND COSTASMEGHIR
We examine the effects of male and female labor
supply on household demands
and
present a simple and
robust test for the separability of commodity demands from
labor
supply. Using data on
individual households from six years of the UK FES we estimate
a
demand
system for
seven goods which includes hours
and participation dummies as
conditioning
variables. Allowance is made for the possible
endogeneity of these condition-
ing labor supply variables.
We find that separability is rejected. Furthermore, we present
evidence
that ignoring
the effects of labor supply leads to bias in the parameter estimates.
KEYWORDS:
Labor
supply, demand systems, separability,
conditional cost functions.
1.
INTRODUCTION
ALMOST
ALL EMPIRICAL
INVESTIGATIONS
of demand
and consumption
assume
that
preferences
over
goods
are
separable
from labor
supply.
Casual
observation
suggests
that -this
may
be
a
poor assumption;
obvious
examples
are the
(travel
and child care) costs of going to work and, say,
heating needs in
the
day.
Below
we
present
evidence
that
suggests strongly
that this casual
observation
is
correct.
Indeed,
it looks as
though participation and
hours have distinct
effects and
these effects differ as between men and women.
To address
these
questions we estimate
a demand system over seven
goods
using
UK
family expenditure
data from 1979 to 1984.
In this
system
we include
hours and
participation
variables
for
the
husband and
wife with
appropriate
allowance for
the fact that
these may
be
endogenous. This conditional
approach
contrasts
sharply
with the unconditional
approaches
adopted
in
Abbott
and
Ashenfelter
(1976),
Barnett
(1979),
and Blundell and Walker
(1982).
Each of
these
papers
estimates
a full
system
for several
goods
and
hours of
work;
that
is,
an unconditional
joint commodity demand
and labor
supply system.
The conditional approach we employ has a number of advantages. First, our
conclusions do
not
depend
on us
having
the correct model of
the
determination
of labor
force
participation
and hours
of
work.
Second,
we can use more
general
functional
forms for
preferences
and
provide
exact tests for
separability.
As
we
show below
the
previously
cited
investigators
had
to
trade
these
features off
against
each other.
1
We
are
grateful
to the
co-editor,
two
anonymous referees,
Richard
Blundell, Terence
Gorman,
Hashem
Pesaran,
Richard
Smith, Guglielmo Weber,
and
participants
in seminars
at
the 1988
Warwick Summer
Workshop, Cambridge, Nuffield,
Bristol, Stanford, UCLA,
and the
University
of
Washington for comments. Much of this research was done while Costas Meghir was at the Institute
for Fiscal Studies and Martin
Browningwas at
Stanford
University.
The
hospitality
and assistance of
both institutions
is
gratefully
acknowledged.
Financial assistance
was
also provided by
the
Canadian
SSHRC
Grant 410-87-0699 and
by
the
U.K.
ESRC. We
are grateful
to the Institute for Fiscal
Studies
and the
U.K.
Department
of
Employment
for
providing
the data.
They
are in
no way
responsible
for
the
use made of the data in this
paper.
925
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926
MARTIN
BROWNING AND COSTAS MEGHIR
If hours of work do affect
preferences between individual goods then demand
systems that take no account
of this dependence may give biased estimates. For
example, suppose we are interested in the effects of
children on demand.
If
we
ignore the interactions between demands and labor supply then we may mistak-
enly impute some of the effects of, say, female labor supply
on
demand
to the
presence of young children since the two are highly correlated.
Another area for which
the relationship between commodity demands and
labor supply is important is the optimal tax literature.
Many results here require
that
preferences
over
goods
be weakly separable
from
hours
of work
(see,
for
example, Atkinson and Stiglitz
(1981, Chapter 4). We should note, however,
that
the need for weak
separability
in
this context depends
somewhat on
the
way
similar problems
are
posed.
Alternative formulations of
the
optimal
tax
problem
lead one to other separability
structures (see, for
example, Deaton (1981b)
or
Besley and Jewitt (1987)). Often these alternative structures
are neither neces-
sary
nor sufficient for weak
separability
and
require
different
tests from those
described below.
Finally, we note that weak separability of goods
from labor is
a
necessary
condition for
additivity
between
goods and
labor
supply.
Since
most
dynamic
models of
labor
supply or
of
the
consumption function
assume that preferences
are
additive
between
goods
and leisure, this may lead
to
misspecification
if
goods
are
not separable.
For example, if the price of some complement to labor
supply (transport, say)
is correlated with wages, then
the
estimate
of the
wage
elasticity
from a
dynamic
labor supply function that leaves out this price will be
biased.2
This latter
may
also
have a
bearing
on the
question
of the
"excess
sensitivity"
of
consumption
to
anticipated changes
in income
(see,
for
example, Campbell
(1987)).
As
Blundell,
Browning,
and
Meghir (1989)
and
Attanasio
and
Browning
(1990) point out,
if
consumption
is not
additively separable
from labor
supply
and
the latter is correlated with
income
(as
it
surely
is),
then
in
a
life-cycle
model
we would
expect
to see
exactly
the "excess
sensitivity"
that
is
usually
regarded as evidence against
the
life-cycle hypothesis.
In
Section
2
we provide
the
theoretical underpinnings
for the empirical
investigation.
This uses the conditional demand
equation approach
of Pollak
(1969). This is a very natural way to model the dependence
of demands on labor
supply.
The
first
part
of
Section
2
lays
out the
general theory
as
developed
in
Pollak
(1969) and
Pollak
(1971).
In the second part of Section
2
we consider the
specifics
of
testing
for
separability
between
goods
and
leisure. This subsection
also details more
fully
the advantages of
the
conditional approach to which we
alluded above. The
last
part
of
this section deals with
the
incorporation of
fixed
costs of
working.
2
If we treat
consumption
as
a
composite commodity
and
include year
dummies in the labor
supply equation,
then this
problem
is
mitigated.
The
assumption
that
we
can
aggregate
across
goods
in this
way is very common presumably
because
it is
very convenient;
it can hardly be justified by an
appeal
to the
(obviously wrong)
sufficient conditions that validate it.
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MALE AND FEMALE LABOR SUPPLY
927
Section 3 deals
with two econometric issues that arise in the later
empirical
work.
The
first is the
need
to take account
of the possible endogeneity of some
of the right-hand
side variables in the conditional demand system that we
employ. The second
issue we address is how to decrease the costs
of working
with large data sets.
In our empirical work we use six UK Family
Expenditure
Surveys comprising
in
all
about
15,000
observations
(after
some sample selec-
tion). Estimating
a
seven-good demand system
on a
data
set
of such
a size can
be expensive, particularly
if we also wish to test and impose
a variety of
cross-equation restrictions and allow for
the endogeneity of some
of the right-
hand side variables.
Section
4
presents
our
results using
a
sample
of families headed by
a
married
couple
in
which
one, both,
or
neither
of
the couple are employed
in
the
labor
force.
Amongst
other things
we
show
that
the participation decision
and
the
hours decision have different and significant
effects; that is, we reject separabil-
ity in many directions. We also present some
evidence based on our estimates
that
ignoring
the effects of hours on demands
can
lead
to biases in the
estimates
of
the
parameters
of
demand.
A
final
section
concludes.
2. CONDITIONAL
DEMAND FUNCTIONS
2.1. General
Considerations
We divide all goods
into two exclusive
classes. Firstly,
we
have the "goods
of
interest;" we denote
the
quantity and
price vectors
of
these by q
and
p
respectively. The second
set of goods are "conditioning goods;"
these
may
affect
preferences over the
goods of interest but are
not
themselves
of
primary
interest. Denote the quantity and price
vectors of
these
goods
by
h
and
r
respectively. Finally,
we have some "demographic variables"
that
may
also affect
preferences
over the goods
of
interest;
denote the vectors
of these
by
a.
To
fix
ideas,
think
of the
goods
of interest as
a
group
of
nondurable
commodities and the set of
conditioning
goods
as other
nondurables, durables,
and
public goods,
as well
as
labor supply.
The demographic
variables
a
include
the
age
and
composition
of the
household
as
well as
things
like
education,
social
class,
and race.
If
preferences
over
all
goods
are
represented
by
the
utility
function
U(q, h, a),
then we define the conditional cost function
c(
p,
h,
a, u)
=
min(
pqIU(q, h,
a)
=
u).
q
Under conventional
assumptions
the
conditional cost
function
has
the
following
properties: (i)
it is
concave,
linear
homogeneous,
and
nondecreasing
in
p
for
fixed (h, a, u); (ii)
it is convex
in
h for fixed (p, a, u); (iii)
it is monotone
in h
(that is,
it
is
increasing
for
commodities
that lower
utility (for
example,
labor
supply)
and
decreasing
for
goods
which increase
utility (for
example, durables)).
Browning (1983) gives
a full
account
of the conditional
cost
function
and
its
relation to the
(unconditional)
cost function
which
is
defined
on
(p, r, a, u).
It
is
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928 MARTIN BROWNING AND COSTAS
MEGHIR
generally
a
matter of convenience
which preference representation we use; for
our purposes the conditional
cost function is the
most appropriate for reasons
that will be made explicit in the
course of this
and subsequent sections.
Given
a
conditional cost
function we can easily derive a conditional
demand
system. Firstly we
note
that
the gradient of the conditional cost
function with
respect to p gives conditional
compensated demands,
i.e. q
=
Vpc(p,
h, a, u). If
we
invert the
identity
cQ
, u)
=
x (where x is
the total expenditure on the goods
of interest
q)
to
derive
u in
terms of (p, h, a, x),
then we can substitute
this into
the
compensated
demands to give the uncompensated
demand
system
(2.1)
qi=fi(p,h,a,x)
(i=
1,2...n).
The first systematic discussion
of conditional demand functions
is due to Pollak
(1969).
In
the next
subsection we shall
give a number of reasons why
we might prefer
econometric modeling of the
conditional demand
system rather than the uncon-
ditional
system
q,
=gi(p,
r,a,
x*)
(i n1..
where
x*
is
the
total expenditure on (q, h).
We now
present
a result that relates the structure of conditional
cost
functions to the structure of the direct utility function. Generally
structure on
the direct
utility functions does
not have any obvious implications
for structure
on
dual
functions. As
the
proposition
below shows, this is not
the case for the
conditional
cost function; it is this fact which
makes this representation
useful in
the context of
testing
for
separability.
The goods of interest are
weakly separa-
ble from the
conditioning goods
if the direct utility function can
be written in
the form F(U(q,
a), h, a).
PROPOSITION:
The set
of goods
q is weakly separable from
h
if
and only if the
conditional
cost
function
takes the form
c(p,
a, g(h, a, u)).
The
proof
of this is
given
in the Appendix. Thus, under weak
separability,
conditioning goods
have
only
income effects. This result has the
corollary that
under
weak
separability
the conditional demand system for the
goods of interest
has the form:
qi =fj(p,
a,
x)
(i
=1,2
...
m).
This result is due to Pollak (1971).
Hence a
simple test of weak separability
consists of
testing
whether the
demands
qi
depend on the quantities
of goods h,
given
that we
have conditioned on
the
prices of the goods of interest
p, the total
expenditure on these goods, x, and on a.
One
thing
to note about the
demand system
in (2.1) is that this system is
unchanged
if
we
start
with the cost
function
c(p,
h, a, 11'(h,a,
u)) rather than
c(p, h, a, u)
where
'fr(-)
is any arbitrary
function that is
increasing
in u. Thus we
can choose
an
arbitrary
normalization for
the
utility
function which depends on
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MALE AND
FEMALE LABOR SUPPLY
929
h and a.
This means that
we cannot in general infer
anything about
preferences
over h and a from observing
demands
alone. Indeed, just about
the only thing
for which we can check
is separability
of q from h. In particular,
we can neither
test the integrability conditions on the conditioning variables nor test whether
even
the
simplest properties
(like
monotonicity) hold
for
them.
This is similar
to
the point made
in the
context
of constructing adult equivalent
scales by Pollak
and Wales (1979).
In our empirical work
we use the
following form for the
conditional cost
function (dropping the
demographic
variables a for presentational
convenience):
(2.2)
lnc(p,h,u)
=
lnr1q(p,h)
+
ub(p,h)
where
ln
denotes
natural log. To derive our demand
system we
take an Almost
Ideal parameterization
for
-q
and b (see Deaton
and Muellbauer
(1980)):
1
(2.3a)
ln-q(p,h)
=
Eak(h)
ln
Pk
+
-
E
EYkl
ln
Pk
ln
pl,
k /k
(2.3b)
ln
b(p,h)
=
Ef3k(h)
ln
Pk,
k
where
ak(h)
and
13k(h)
are functions
of the vector
of conditioning variables
h
and
the
Ykl's
are
specified
to be constant parameters
(this is
for the sake of
parsimony). We take
ak(h) and k(h) to
be linear in h. This gives
the following
equation for the budget share of good i:
(2.4)
wi =oi+ Eakihk+
Eyiln
PJ(+
oi+
Ekihk)ln(x/l7(P,h)),
where
-q(p,
h)
is defined in
(2.3).
Given
this
parameterization
the test
for
whether the
goods
of interest are
separable
from a
particular hk
reduces
to a
simple
test
of
whether
the relevant
aki
and
Oki
are zero
for all i.
2.2. Labor Supplyas a Conditioning Good
An
important
purpose
of this
paper
is
to test whether
household
commodity
demands are
weakly
separable
from male and
female labor
supply.
In all that
follows we shall restrict attention
to households consisting
of one married
couple
with or without
children.
We
specify
h
=
(hf,
hm)
where
hf
and
hm
denote
female
and
male hours of work
in the labor-force
respectively.
Thus our
conditional demand
system
has
the
general
form:
(2.5)
qi
=f
(p,
h,
a,
x1i)
(i
=
1,2
...
m),
where
a
is
a
vector of
conditioning
goods
other than hours and
Oi
is a vector of
parameters; thus each
equation
has the same
functional form
but different
parameters.
As shown in
the last subsection,
if
preferences
over
goods
are
weakly separable
from
labor
supply,
then each demand
function
is
independent
of h.
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930 MARTIN BROWNING AND COSTAS MEGHIR
There
are several
reasons why
we
might prefer
to
model demands using
conditional systems.
The first was noted
by
Pollak
(1969);
if some
good
is
given
in predetermined quantities (that is,
it is
rationed),
then
it is
appropriate
to
put
the level of the good on the right-hand side. As
an
example,
Deaton (1981a)
models housing in this way.
A second
advantage
of
the conditional demand approach
is that
testing
for
weak separability
is
very easy.
All
we have to do is
to
test whether
a
particular
set
of variables should
be
excluded
from the
right-hand
side of a
regression.
This is in marked contrast to the case for unconditional demand systems
which
do not typically
have
simple parametric
restrictions
that
are equivalent
to weak
separability, except
in the case of
quasihomothetic preferences (see
Gorman
and
Myles (1986)).
For
example,
Blackorby, Primont,
and
Russell
(1978,
Section
8.3) show
that this is the case for six
specifications
that
have been
suggested
in
the literature. Consequently, previous
investigators
of
separability
have had to
use
rather
simple specifications
for
unconditional
demand
systems
or
employ
approximations
to the
correct separability conditions.
As
an
example
of the
former
approach,
we
note that
the
test of
separability presented by
Blundell and
Walker
(1982)
is
a test of
additive
separability
between
goods
and
leisure,
in the
context of
quasihomothetic preferences.
As an
example
of the latter
approach,
Barnett (1979)
estimates a
Rotterdam
system for goods and leisure and derives
an
exact test
for
weak separability
which
depends
on unobservable variables.
Since this condition is
inherently
untestable Barnett
then has recourse to
an
approximation
which does
yield
a
testable condition.
The
conditional demand approach
allows us to test for weak separability
without
specifying
the
structure
of
preferences
for the
goods
that are
separable
under
the null.
Moreover,
we can use flexible
preference representations
for
the
goods
of interest. Thus
Deaton
(1981a)
estimates
a
seven
good
Al
demand
system
with
the
intercepts
as linear functions
of
the
quantity
of
housing
(that is,
(2.4)
with
f3ki
=
O
for
k
>
0).
Since the thrust of that
paper
is whether it is better
to model
housing
as rationed or
freely chosen,
no
test
of
separability
is
given.
Deaton does, however, report that the quantity of housing enters three
equa-
tions
significantly
when
homogeneity
is
imposed;
this
suggests
that
separability
would
be
rejected
in
this
case.
The
third advantage
of the conditional approach is that (2.5) is valid
whether
or not either of the hours
variables is equal to
zero.
Thus corner solutions
in the
conditioning goods
do not
lead
to
switching
in the
demand system.
This
gets
around
many
of
the
problems
raised by Lee and Pitt (1987).
A
fourth
advantage
is
that
we do not
need to model the determination of the
conditioning goods explicitly.
Indeed,
the
conditional demand approach
does
not
require
an
explicit
structural model for the conditioning good at all.
Moreover,
the
demand system (2.5)
will be correctly specified whether or not a
and
h are
chosen optimally. Additionally,
we do not need to model explicitly the
budget
constraint for the
conditioning
goods. This is particularly significant for
labor
supply and
for
durables.
The former requires modeling of the tax system
whilst the latter involves an unobservable rental
or user cost.
The
above provide
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MALE AND FEMALE LABOR
SUPPLY
931
an important
methodological
advantage: we may
study consumer demand
while
being agnostic
about
issues such as unemployment,
the determination of
hours
of work,
or the decision
to consume durable
services, while
at the same time
accounting
for their
possible influence on demand.
Conditional
demand func-
tions
are an economical way of
relaxing separability
and still maintaining
the
focus on the goods
of interest.
The advantages
of conditional
modeling are often compelling,
particularly
in
the case where
the
goods
on which we are focusing
may be nonseparable
from
say
durables, leisure,
or
goods
that are not consumed
by many households
(tobacco,
for example). There is,
however, one
disadvantage; all behavioral
and
policy implications
are
conditional
on the quantities
of the conditioning
goods
consumed.
To
see
this, suppose
that
we
are
interested
in the own price elasticity
for
good
i, conditional on
total expenditure in the group
(that is,
the Mar-
shallian own price
elasticity). This is defined
as (taking the case of
a single
conditioning good
for
simplicity)
dln
q/dlnpi
=ln
f/alnpi+
(ln
f/lh)(dh/alnpi).
The conditional
demand system will yield
estimates of the parameters
of the
first two
derivatives on the right-hand
side
but not of the third.
These latter
must
be estimated separately
unless the conditioning
variable is believed to be
genuinely
predetermined in which
case the
final
derivative
is zero.
Similar care
must be exercised in interpreting income effects.
2.3.
The
Effects
of Participation
If there are
specific
indivisible
costs associated
with
going
out to
work,
then
these
may
lead
to
a
discontinuity
in
demand
behavior
at zero hours of work. To
accommodate
such
fixed3 costs we
generalize
the
conditional
cost function
to
take the following
form
(where
h
is
taken to be
a
scalar for convenience):
c(p,a,h,u)
=cN(p,a,u)
+I(h>O)
x
(c'(p,a,
h,
(u,
a,
h))
-
cN(p,
a,
u)),
where
I
is
an
indicator
function for
positive
hours. Thus
cp(Q)
and
CN(.)
represent
preferences
when hours
are
positive
and
zero
respectively.
The above function
can
be
interpreted
as
a
conditional
cost function
despite
the
discontinuity
at
h
=
0,
since
it still is
upper
convex in
h.
Moreover,
the
power
of
the
conditional
approach
is
once
again
demonstrated
since
it
would
be
particularly
difficult
to
formulate
the unconditional
cost function
in the
pres-
ence of fixed costs. On the other hand, we cannot compute the value of fixed
costs,
since the
way
that the function
W(j)
depends
on h is not identifiable
unless
we
analyze
labor
supply
behavior
explicitly.
3This use of
terminology
does
not
preclude
the
possibility
that these sorts
of cost will also alter
the level
of
total
expenditure
as
well as
the
structure of demand.
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932
MARTIN BROWNING AND COSTAS MEGHIR
If
cN(p, a, u*)
=
C'(p, a,
0,
u*)
(where
u* is a common and
arbitrary
evel of
utility),
then
all that fixed costs of work can do is to
shift the function
TI(
)
upwards discontinuously
at h
=
0;
that
is, they
act
just
like an
income
loss. In
this case the goods q
are
weakly separable
from
participation
and the effect of
the latter on commodity
demands
is
fully
accounted
for
by
total
expenditure
x
(and
hours if
they
are
nonseparable).
If
this is not
the case then
fixed
costs of
work (for example,
for
clothing
and
travel)
will
affect the structure of
demand
directly;
that
is, cN(p, a, u)
is not
equal
to c
'(p,
a, 0, u).
To allow
for
fixed
costs we add
participation
dummies
(dm
and
df)
to the list
of
conditioning
variables
entering
the
specification
of a
and
f8
in
(2.3). Incorpo-
rating demographics
we have
the
following budget
share
equations:
(2.6)
wi
=
(aoi(a)
+
alihf
+
a2ihm
+
a3idf
+
a4idm)
+
nij
pn
+
(f3oi(a)
+
3,ihf+
t2ihm
+
t33idf+
f34idm)
xln(x/a(p,h,d,a)),
where
a(*)
is defined
by
ln
a(p,
h, d, a)
E (aOk(a)
+ alkhf +
a2khm+a3kdf
+a4kdm)
lnPk
k
+
?
E
EeYkl
ln
Pk
ln
Pl.
2k I
A
finding
hat the
participation
ummies
are
significant
n
the
demand
system
is consistent
with the
presence
of fixed costs but other
intepretations
are
possible.
For
example,
if the effect of hours on
demand
is continuous
at zero
hours (so
that there
are no
fixed
costs)
but
nonlinear,
then the
participation
dummies
may pick up
some
of the
nonlinearity.
3.
ECONOMETRIC CONSIDERATIONS
3.1. InstrumentalVariable
stimation
An
immediatereaction
to a
conditionaldemand
system
of
the
form
given
in
(2.6)
is
that it includesvariables
on
the
right-hand
ide that
may
be
endogenous
for
the
budget
share
equations.
This
raises
the issue of identification. n the
first
two subsectionsof this section we
discuss
the IV estimatorused and in the final
subsectionwe discuss
the
estimationof demand
systems
on
large
data
sets.
In the demand
system
we estimate
there
are five
(potentially)endogenous
variables.These are: maleand female hoursof work,male and femaleparticipa-
tion,
and
total
expenditure.
Thus
we need
at
least five identifyingassumptions.
For
the
conditioninggood
in a
conditionaldemand
system
a natural
nstru-
ment is
the
price
of the
good,
which in
this case would
be
the wage.
There
are,
however,several difficulties
with the use of the
wage
as an instrument.
n
the
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MALE
AND
FEMALE LABOR SUPPLY
933
first place, the wage may
be
endogenous,
an
issue
which is
frequently
debated
in
the
labor
literature
where the
evidence
is mixed
(see,
for
example,
Mroz
(1987)).
In the second place, wages are only observed for workers.
Finally, the wage may
not be correlated strongly with hours of work.
This
could occur under
certain
types of labor contracts or if job offers consist
of
fixed wage-hours packages.
Given those problems we prefer
to use a
general
IV estimator employing
instruments that are observed for
all
individuals (whether working or not).4
To identify the parameters in the system we assume
that, conditional on total
expenditure and the labor variables, asset
income
and
education do
not enter
the
demand system. Given this assumption
the model
is
identified (using
these
instruments and other functions of them as described
in the next
section).
The
exclusion of asset income is in the spirit of the two-stage budgeting approach
to
inter-
and
intra-temporal allocation. Education,
which we
use in lieu of
wages
as
an instrument for the labor variables is perhaps
more tenuous since
it
may
be
that education has a direct effect on tastes even when we condition
on
all of the
other
right-hand
side variables we
employ.
In
addition
to the instruments described
in the last
paragraph,
we also
use
year
dummies
(to pick up any
common "macro"
effects on
consumption
and
labor
supply)
and
male
age.
We
report Sargan
statistics for
the
overidentifying
restrictions
to evaluate the
validity
of our exclusion
restrictions.
The demand
system including separate participation
effects can
be written as:
(3.1)
Wk f(pk,
ak, hk, dk,
XklO)
+
Uk,
where
Wk
s the vector of
budget
shares for the
kth
household,
hk
and
dk
are
the vectors
of actual
hours and
participation,
and
ak
is
a vector of
demograph-
ics. 0 is
the
vector
of
unknown
parameters
to be
estimated.
We assume
that a
set
of
instruments
z is available such that
E(ulz)
=
0
and such
that
plim(N-1Ekzk
0fk(*)/00)
=
M,
where N is the total sample
size and
M is a
matrix with rank
equal
to the
dimension
of 0
(see
Amemiya (1974)
and
Hansen
(1982)).
Given these, a consistent estimator for the parameters
0
can be obtained by
minimizing u'(I
X
P)u
where u is the
stacked
vector of
error terms
and
P
=
Z(Z'Z)-
'Z'
(Z being
the matrix of
instruments).
The asymptotic
variance-
covariance
matrix of
this estimator is
(3.2) V(0)
=
(G'(I ?P)G) 'G'(I ?P)2(I ?P)G(G'(I
?P)G)',
where
Q
is the error covariance matrix
and
G
is the stacked
matrix of
derivatives
of each
budget
share function
with
respect
to the
parameters
of the
model for all observations.
In
estimating V(0)
we evaluate G at the estimated
parameter vector and we replace Q2 by the outer product of the estimated
4
In an earlier version of this
paper
we
also
presented
results on the
subsample
for
which both
male and female
participation
dummies
are
one, using
the
wage
as an instrument
(with appropriate
allowance for the
sample selection).
The results from this
investigation
were often different
in detail
from those
reported
below but the broad
qualitative
results were the same.
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MARTIN BROWNING AND COSTAS MEGHIR
residual vector, setting all cross observation terms to zero. This allows for
general forms of heteroskedasticitysee White
(1980)).
3.2. Testing or Exogeneity
For an IV estimator a natural exogeneity test is the Hausman (1978) test.
When
the
IV
estimator,
8A,
is
compared
o an estimator hat is consistent and
efficient under the null of exogeneity (relative to IV),
130,
then,
(pA
-
130)
a
N(0,
VA
-
V?),
VA and
V? being
the covariance
matrices
of the
respective
estimators. The
degrees
of freedom
of
the
test
are
equal
to
the
rank of
(VA
-
V?). This poses
certain
problems.Firstly,the
rank
may
not be
the
same
in
small samples and asymptotically. hus the test will
be
based
on an
estimate
of the
degrees
of freedom
equal
to the rank of the
estimate
of
(VA
-
VO).
Secondly,
n
practice,
t
is difficult
o determinethe rank of a matrix.
Here we use a modificationof the standardHausmantest which gets round
the
above
difficulties:we randomlydivide the sample in
half and
estimate
the
model under
the null and
alternative
on the
different subsamples.
The two
estimators are
by constructionasymptotically ndependent5
and
hence,
under
the null
hypothesis,
(pA
_0)
a
N(0, V?
+
VA). The
asymptotic
covariance
ma-
trix
(VO
+
VA)
as well
as its
estimate are
guaranteed
to be
positive
definite
providedthe individualasymptotic ovariancematrices and their estimates are
not singular.Denoting the differencebetween the two estimates by d, the test
statisticwe use is
d'(V0 +
VA)-'d.
Clearly
he
advantages
of
this
test are the a
priori knowledge
of the
degrees
of freedom and its
computational implicity.6
The
main
drawback
s
the
loss of
power
in relation to the Hausman test. In fact
the
difference
d is estimated
much less
precisely,
the loss of
precision being
2V0.
On the
other hand the
properties of the Hausman test
are
only
clear conditional on
knowing
the
correct
degrees
of freedom.
Nevertheless,
we
also
present
a
standard
Hausman
test
focusing
on
the
parameters
f
the variableswhose
exogeneity
we are
testing
and where
the
efficiency
oss
from IV
is
likely
to
be the
greatest.
The
covariance
of the differenceturns out to be positivedefinitein this case and we assumeit
remainsso asymptotically.
3.3.
Estimating Large Systems
The
demand systemwe estimate has a set of within-equation
nd
cross-equa-
tion restrictions hat we
wish
to
impose. These
are
homogeneity,which gives
rise
only to within-equationrestrictions, and symmetry,which gives rise to
cross-equation
estrictions.
Finally,
he
price
index
which
deflates
total
expendi-
ture dependson estimated parametersand is commonacrossall equationsand
5Assuming
of course
that
the observationsn the
survey
are
independent.
6The test
is
analogous
to a Chow test
of
structural
tability
and
can
be viewed as a
general
specification
est.
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MALE AND FEMALE LABOR SUPPLY
935
hence gives rise to an additional set of
within
and
cross-equation restrictions.
Given the size of the data set, it is practical
to
consider estimation
in
several
steps.
The first step is to estimate the model
without imposing symmetry but
allowing for the cross-equation restrictions
imposed by having the same
q(-)
in
every equation deflating total expenditure. To do this we could use conventional
nonlinear methods, but we prefer a simpler
iterative method that exploits the
structure of the
problem.
If the
value
of
-q7()
for each household is known, then
we have
a
straightforward
linear estimation problem with no cross-equation
restrictions. Thus as
a
first approximation
to
q(
) we compute household
specific
Stone
price
indices.7 We then estimate the system equation by equation
and
compute the price
index
-q(p,
h, d,
a) using the estimated parameters
and
re-estimate
the
model.
We iterate until this process has converged. For
our data
and system, convergence takes place in about 5 iterations.
The fact that there
are
important
differences
in
the
parameter
estimates obtained
in
the first
and
last iterations indicates that using
the Stone price
index
approximation
with no
iteration is not acceptable; this is in contrast
to the experience on aggregate
time
series
data.
This
procedure provides
consistent
parameter
estimates for the model
with-
out
symmetry imposed.
The
covariance matrix
for
this
IV estimator is
given
in
(3.2)
above and takes into account the
cross-equation
restrictions.
Given these
first step estimates, the symmetry restrictions
can be imposed using minimum
distance. Denote the
unrestricted parameters
by 0 and the symmetry restricted
parameters by /8. Then under the null
0
=
K:8
where K is a matrix of rank
1
-
m(m
-
1)/2,
and
/
is the
number of
unrestricted
parameters
in
the demand
equation system.
The
symmetry
restricted
parameter
estimates
can be obtained
by minimizing
X2
= (0
-
K/8)'V- 1(0
-
K:8),
where 0 is
an
estimate
of the
unrestricted
parameter vector
and where
VD
is the inverse
of an
estimate of
the variance-covariance matrix of
0
(see
Ferguson (1958)
and
Rothenberg
(1973)).
The minimized value of
x2 follows
a Chi-squared distribution
with
degrees
of
freedom
equal
to
the number of restrictions
(here m(m
-
1)/2).
An
estimate
of the
covariance matrix of the
restricted estimator is
(K'V- 1K)l
In
breaking up
the estimation
problem
in this
way
we
are
able
to
implement
a
rather
large
and
complicated
estimation problem with just
a
simple sequence
of
instrumental variables
regressions
followed by
a
linear feasible GLS step.
This is
a
very
convenient
way
of
handling large data problems.
The minimum distance ideas can be exploited
in
many ways
when
estimating
demand systems on time series of cross sections. For example they can
be
used
for
testing stability
across time:
separate Engel
curves can
be
estimated for each
period
over
which
prices
are
constant
and
for each
good.
The
cross
time
restrictions can then be
expressed
either as /3
=
/ or
a,
=f(pt)
where
8t
are
7That is, Pk
defined
by logPk
=
E,Wik
logp,
where
W,k
is the
budget
share of
good
i for
householdk.
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MARTIN BROWNING AND COSTAS MEGHIR
the Engel curve parameters that
are assumed constant across time
while
at
are
the parameters which under the
null should only vary as a result of price
variation. The restrictions can be
imposed by minimizing the minimum
distance
criterion.
If the correct covariance matrix
is used as a weight matrix, then the value of
the minimized criterion will be
a
chi-squared statistic for the hypothesis
of time
stability. Moreover
if
the
individual
Engel
curves have been estimated effi-
ciently,
minimum distance is
asymptotically
efficient. Hence,
this
technique
not
only provides
a
test
of the underlying hypothesis of time stability, but also
reduces
a
large estimation problem
to many small ones without, potentially, any
loss of efficiency.
4. RESULTS
4.1. Data
Our
sample
is taken
from the UK Family Expenditure Surveys
from 1979
through 1984.8 From
these surveys we select households
in
which
the only
adults
are a
married couple (so
that the only other people
in
the household
are
children). Moreover, the respective
partners have to
be
between
the ages of 20
and the
retirement age (60
for
women
and
65
for
men)
and not classified as self
employed. Finally, we exclude households residing in Northern Ireland. Our
selected
sample
consists
of
15,010
households. Each household falls into
exactly
one of
the
four strata
given by
the possible
values for the
participation
dummies
(df,
dm)*
The goods we model
are
food, alcoholic beverages, fuel, clothing,
transport,
services,
and other
goods.
This excludes durables and
housing.
One
way
to
include these goods would
be
to
include
the
"stock" of each of them as further
conditioning goods.
Unfortunately,
these stocks are not well measured
in
our
survey;
hence their exclusion. The prices
for the seven
goods
modelled
are
taken from Economic Trends (1986) and are matched to the survey month. Thus
we
have
72
prices
for each
good
(January,
1979
through December,
1984).
The means
apd
standard deviations
of
the
principal
variables used
in
the
demand
system
are
provided
in Table
I.
As can
be
seen,
there
is
quite
a lot
of
variation in
expenditure patterns
across
these
regimes. For example,
the trans-
port
share for
a
family
where both
go
out to work
is
about 20%
whereas that for
a
family
where
neither
go
out to work is
only
12%.
As
another
example,
the
total
expenditure
of the latter
group
is
significantly
lower than that
of
the
former.
Our test
for
separability is,
of
course, simply
a
test
as
to whether the variation
in
budget
shares
across
participation
regimes
can be
"explained" solely by
the
variations
in
nonlabor variables
like children
and
total
expenditure.
8Although we have data from
1970 the FES does
not
record
education
before 1979.
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MALE AND
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LABOR SUPPLY
937
TABLE
I
SAMPLE
STATISTICS
BY PARTICIPATION GROUP
(df,ddm)
(0,
1)
(1,1) (1,0)
(0,0)
Sample Size 5432 8484 372 722
Variable
Food
Share
0.349
0.316
0.373
0.415
(0.12)
(0.11)
(0.12)
(0.13)
Alcohol Share
0.055 0.068
0.061
0.046
(0.06)
(0.07)
(0.08)
(0.07)
Fuel Share
0.094 0.079
0.121
0.151
(0.06)
(0.05) (0.08)
(0.09)
Clothing
Share
0.089
0.098 0.072
0.071
(0.09)
(0.10) (0.09)
(0.09)
Transport
Share
0.182
0.203
0.158
0.121
(0.14)
(0.15)
(0.14)
(0.13)
Services Share
0.116 0.132
0.110 0.089
(0.10)
(0.12) (0.09)
(0.08)
Other Goods
0.115 0.104
0.105
0.107
(0.11)
(0.11)
(0.10)
(0.13)
Children
0-5
0.819
0.219
0.156
0.651
(0.82)
(0.51) (0.43)
(0.87)
Children 5-18
0.726
0.754
0.548
0.953
(1.00)
(0.98)
(0.98)
(1.30)
Male
Hours 40.53
40.21
(7.28)
(6.53)
Female
Hours 26.20
27.14
(12.0)
(12.5)
Ln Total
9.068
9.217
8.816
8.690
Expenditure
(0.50)
(0.47)
(0.48)
(0.52)
Cohort
pre
1931 0.149 0.195
0.505
0.248
(0.36)
(0.40)
(0.50)
(0.43)
Cohort 1931-40
0.137
0.194
0.121
0.150
(0.34)
(0.39)
(0.33)
(0.36)
Cohort 1941-50
0.403 0.350
0.220
0.260
(0.49)
(0.48)
(0.42)
(0.44)
Notes: Standard deviations
in
parentheses.
The total
expenditure
variable
is
the average
of the
log
of
the nominal value.
4.2.
Separability
Tests
Rewriting (2.6)
we have the following
form for the typical budget
share
equation
in
our system:
(4.1)
wi
=
a
i(hf
h
m,
df
,
dm
m
children,
demographics)
+
Eyij
ln
pJ
+
13i(hf,
hm
5
df
5
dm,
children)
ln
y.
The childrenvariablesare
the numbersof children aged
0-4
and
5-18. The
demographic
ariables
are the
age of
the wife and dummies
for
three
year-of-
birth
cohorts,
three
seasons,and ten regions.
The variabley is total
expenditure
on our seven
goods
deflated
by
the
price
index
q(p,
h, d, a)
defined after (2.6).
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938 MARTIN BROWNING AND COSTAS MEGHIR
In all the estimates
reported below we impose homogeneity and symmetry.9 We
drop the "other goods" (OG) equation to accommodate adding up.
The complete set of instruments used are male and female education and
male age and the squares of these three variables, female age squared, the
numbers of children
in
each of four age groups (K1-K4), asset income and asset
income squared, both of these latter crossed with KI to K4, log prices crossed
with the number of
children,
and
year dummies.
As
well
we
include the
nonlabor variables from the right-hand side of (4.1) excluding, of course, ln y.
To focus attention on the parameters of interest,
in
Table
II
we present the
estimates of
the coefficients
on
the
hours
and
participation variables.10 Note
that hours are 100's of hours per week so that the effect of moving from, say, 20
to 40 hours is 0.2 times the estimate
given.
We postpone for the moment any discussion of the implications of these
estimates and concentrate on the tests of
separability
and
exogeneity. Just
eyeballing
the estimates of the hours and
participation
variables we see that
each of them is
significant
in
some
place.
The relevant
(Wald type) chi-squared
test statistics for
separability along
with the
p-value
are
presented
after the
individual estimates.1"As can be seen, male hours and participation seem to be
more
"significant"
than female hours
and
participation.
Even for the
latter,
however, there does seem to
be a
significant effect.
We
conclude
that
separabil-
ity is rejected.
Given that separability is rejected, we consider the hypothesis that the male
and female labor supply variables enter the demand functions
in
the same way.
A
sufficient
(but
not
necessary)
condition for this to
be
true would
be
that
male
and female nonmarket times are
perfect
substitutes
in
the household
utility
function.
As can be seen this
hypothesis
is
strongly rejected
for both
participa-
tion
and
hours;
it matters who
goes
out to work.
Finally
in
Table
II
we
present
tests for the
necessity
of
instrumenting (the
exogeneity test)
and tests for the
validity
of
excluding
the instruments from
a
direct role
in
the demand
system (the Sargan
test
statistics).
The exogeneity hypothesis is not rejected strongly on the basis of the test that
looks at
all
parameters.12 We also present the results
of
carrying out the test
focusing
on the labor
supply
variables
only (see
Section
3.2
above).
These
reject
much
more
strongly.
The tests are not
directly comparable
since the alternatives
9Generally
homogeneity is not rejected on micro
data (in contrast
to
the experience with
aggregate data);
see,
for
example, Blundell
et
al.
(1988)
or
Browning (1988).
The evidence
on
symmetry is more
mixed.
10
The full
set of
parameter estimates for this and later models is given in the Appendix.
11
In
judging
whether a test rejects or not, we use a 0.1% level of significance; this is
probably
more appropriate for a sample of this size than the conventional 5%. However we also give the
probability (under the null) that a chi-squared
variable with the relevant degrees of freedom is
larger than the
reported test statistic so that readers can
judge for themselves.
12
Total
expenditure was instrumented both under the
null
and under the alternative. The null
differs from the alternative
in
that,
in
the
former,
we
included the
hours
and participation variables
among
the
instruments, both alone and interacted with other instruments.
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TABLE
II
SOME
PARAMETER
ESTIMATES
Intercept
of the Budget Shares
Parameter
Food Alcohol Fuel
Clothing Trans Services
Constant
31.0 7.6 12.2
8.1 21.2
10.2
(2.1) (2.5) (1.3) (2.1)
(3.3) (2.9)
Hm 62.4 91.4 - 3.9 139.0 - 224.0 - 165.0
(28.4) (24.1)
(15.0) (34.6) (48.7)
(39.8)
Hf
-6.8
25.8 -15.5 25.9 -8.7 -35.8
(12.4) (10.6)
(6.5)
(14.9) (21.6) (17.2)
Df
-0.1
-7.70
6.1 -14.3 11.7
13.0
(4.2) (3.5)
(2.2) (5.1) (7.2)
(5.8)
Dm -21.3
-33.7 -3.0 -53.0
87.8
62.9
(11.8) (10.0)
(6.2) (14.4) (20.2)
(16.5)
Total
Expenditure
Coefficients
Parameter Food Alcohol
Fuel Clothing
Trans Services
Constant
-14.7 8.1
- 7.23 2.7 2.9
2.3
(3.7) (3.2) (2.0) (4.6) (6.4) (5.2)
Hm
50.8 -51.0
13.6 -88.6
116.0 37.2
(29.9) (25.8)
(15.7) (35.7)
(50.7) (41.4)
Hf
-58.0
27.4 15.8
4.0 23.7
37.5
(24.1) (20.2) (12.6)
(29.9)
(41.8) (33.9)
Df 25.5
11.6 -11.3
14.9 -16.3 -27.0
(8.9) (7.5) (4.7)
(10.9) (15.3)
(12.5)
Dm
-28.4
12.3
3.9 33.1 -42.8
-5.4
(14.0) (12.1)
(7.4) (16.8) (23.9)
(19.5)
Tests of
Separability
Degrees
of Test P-Value
freedom
statistic
(%)
Separability
Femalehours
12 31.49
0.17
Male hours
12 61.77 1E-6
Joint hours
24
93.96
3E-8
Female
participation
12 33.87
0.07
Male
participation
12 61.06 1E-6
Tests of
Equality
of Male
and
Female
Parameters
Hours:
X2(12)
=
66.5
(probability 1.5E-7%)
Participation:
X2(12)
=
61.0
(probability 1.5E-6%)
Tests
of
Exogeneity
Exogeneity f HoursandParticipationX2 statistics)
Degrees
of freedom:37 for individual
quations;
22 for
joint.
Food Alco
Fuel
Clth
Tran Serv
Joint
Test statistic
77.1 51.5 48.6 64.7
54.7 54.7
336.8
P-Value(%)
0.01 5.7
9.6 0.3
3.0 3.0
1OE-4
Hausman
ests
on the
parameters
f
the hours
and
participation
ariables
degrees
of
freedom
=
8 for
individual quations,
8 for
joint).
Food
Alco Fuel
Clth Tran
Serv Joint
Test statistic
32.7 38.3 53.8
37.6 43.0
38.4 171.2
P-Value
(%)
7E-3
7E-4
8E-7
9E-4 9E-5
6E-4 1OE-14
Tests of Overidentifying Restrictions
Sargan
ests
for
orthogonality
f
instruments
df
=
24).
Food
Alco
Fuel Clth
Tran Serv
Sargan
tatistic 83.4 98.9
105.6 63.6
83.3
60.5
P-value
%)
2E-6
SE-9 3E-10 2E-3
2E-6
SE-3
Notes:
All
parameter
stimates
and standard
rrors
given
n
brackets)multipliedby
100.
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940
MARTIN BROWNING AND COSTAS MEGHIR
are different,
but in an intuitive
sense these results indicate that most of
the
bias, when the labor variables
are
not instrumented, is concentrated among the
parameters attached to those variables.
As can be seen, the Sargan statistics are rather poor. In fact the variables
included in the intercepts in (4.1) represent the "second round" of looking at
the over-identifying assumptions. Specifically,
in
the original version of this
paper
we had
regional dummies
and the
wife's age amongst the instruments but
not in
the intercepts.
This
gave very large Sargan statistics; examination of
the
regressions of the residuals on the instruments suggested including regional
dummies and wife's age amongst the regressors. This greatly improved the
Sargan statistics but had little effect on the estimates. We have also experi-
mented with including
the education
variables as
well as
male age.
The
inclusion of these variables did not improve the Sargan statistics significantly;
neither
were the estimates
changed significantly.
In
none
of
these
experiments
was there
any significant
movement
in
our most
important qualitative results:
separability
and
exogeneity
are
always rejected.
4.3. The
Effects of Labor Supply on Demands
Having
established that hours and
participation
do have a
significant
effect on
the
pattern of demand,
we
present
in Table III
some economic properties
of the
estimated model
with
particular emphasis
on the variation with
respect
to
different
participation regimes.
We
give
estimates
(and
standard
errors)
of
predicted budget shares, expenditure elasticities,
and own
(uncompensated)
price
elasticities. These
are
calculated
for
the
four
possible
combinations
of
d
(j
=
f
and
m)
=
0 or
1. The
remaining
variables
(except
for
hours)
are
set to
the
means
for the whole
sample.
When not zero the hours variables are
set
to
the
means for
working
women
and men
(= 27 and
41
hours, respectively).
There are important
differences in the
budget
share
predictions
for
most
goods
and some of these
are
different from those
for the means as
reported
in
Table I. There is no
strong pattern
to the differences between the four
strata;
in
particular
we find no
support
for
the idea that
strata
(1, 0)
and
(0, 1)
are
convex
combinations
of
strata
(1,
1)
and
(0, 0).
If
anything,
it
looks
like stratum
(1, 0)
is
the
outlier;
it has the
highest
shares
for
fuel
and
transport
and the
lowest
for
clothing
and other
goods.
The next
panel
in Table III indicates
that although there
are
some
similarities
(for example,
food and fuel are
necessities and other
goods
are a
luxury
for all
groups),
there are also
large differences
in the
responses to
income
changes
as
between different
groups.
For
example, alcohol
and
clothing
are
luxuries for
(df,
dm)
=
(1,
0)
but
necessities
for
stratum
(0, 1).
On
the other hand there do
not
appear
to be
any significant
differences
in
the price responses;
this
is
perhaps
to be
expected given
that we have not allowed the
y's
in
(4.1)
above
to
vary
with
the labor force
variables.
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MALE AND
FEMALE
LABOR
SUPPLY
941
TABLE III
PREDICTED BUDGET SHARES AND ELASTICITIES
Predicted Budget Share (x 100)
(df, dm) Food Alcohol Fuel Clothing Transport Services Other
(1, 1) 26.44 7.79
11.02 11.40 14.38 15.31 13.66
(1.84)
(2.24) (1.06) (2.37)
(2.54) (1.94)
(1,0) 24.85 6.09 14.58 7.55
17.28 17.52
12.14
(2.37) (2.62) (1.40) (3.22)
(3.70) (2.97)
(0, 1) 23.95
7.04 11.15 13.68 8.61 18.28 17.30
(2.06) (2.45) (1.11) (2.42)
(2.89) (2.20)
(0,0) 23.38 6.10
13.95 10.94
10.65 18.92 16.06
(2.17)
(2.47) (1.21) (2.72) (3.21) (2.48)
Expenditure
Elasticities
(df,
dm
)
Food Alcohol Fuel Clothing Transport
Services Other
(1,1)
0.47
1.10
0.27 2.16 0.95
1.11 1.52
(0.08) (0.23) (0.10) (0.22)
(0.25) (0.19)
(1,0)
0.74 2.54
0.04 3.17
0.68 0.53 1.69
(0.21)
(0.73) (0.19) (0.86) (0.53) (0.42)
(0, 1)
0.00 0.51 0.90 0.79
2.06
2.02 1.20
(0.15)
(0.45) (0.17) (0.32) (0.73) (0.28)
(0,0)
0.30 1.85 0.50
1.03
1.41 1.46
1.29
(0.13)
(0.41) (0.11) (0.34) (0.48) (0.22)
Own Price Elasticities
(df,
dm)
Food Alcohol
Fuel
Clothing Transport
Services Other
(1,1)
-
0.10
-
0.59
-
0.53
- 1.28 - 0.96
-
0.94 - 0.87
(0.15) (0.61) (0.14) (0.12) (0.60) (0.38)
(1,0)
-
0.15
-
0.53
-
0.55
-
1.36
-
0.91
-
0.84 -0.85
(0.17)
(0.76) (0.11) (0.19)
(0.50) (0.33)
(0,1)
0.14 -
0.50 - 0.62
-
1.13
-
1.00
-
1.05 0.88
(0.19) (0.67) (0.14) (0.12)
(1.02) (0.32)
(0,0)
0.03
-
0.51
- 0.63 - 1.20
-
0.99
-
1.01
-
0.88
(0.18)
(0.76) (0.11) (0.14)
(0.82) (0.31)
4.4. The Bias From Ignoring Labor Supply
Taken together
the
results
in Tables
II
and
III
show that both the participa-
tion decision and the hours decision for
each
partner
have
significant
effects on
the estimates of parameters
and statistics of
interest. Unfortunately,
in
many
surveys
there is
not enough
information to condition
on
hours and/or participa-
tion. To see what the effect of this would be,
in Table IV we
present
the
results
of two
experiments: ignoring
the hours information
and
ignoring
all
the
labor
supply
information
respectively.
The
first set
(model
A) repeats the estimates allowing for hours and participa-
tion and
the
endogeneity
of
these. Thus
these
correspond
to Table
111.13
For
the
second set
(model B)
estimation
and
prediction
uses only
the
participation
variables
df
and
d,n.
This
is
of interest since
for some available family
13Everything
here
is
evaluated at the overall
means, including hours;
hence the differences
between the
values for model
A
and those in Table III.
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942
MARTIN BROWNING
AND
COSTAS
MEGHIR
TABLE IV
THE EFFECTS OF IGNORING LABOR SUPPLY
INFORMATION
Total
Expenditure Elasticities
Model Food Alcohol Fuel Clothing Transport Services
A 0.31 1.01 0.49 1.65 1.16 1.43
(0.04) (0.11) (0.04) (0.10) (0.12) (0.04)
B 0.45 0.63 0.59
1.42 1.32 1.67
(0.02) (0.13) (0.03)
(0.06) (0.06) (0.06)
C 0.54 0.43 0.67 1.31 1.59
1.97
(0.01) (0.24) (0.02) (0.05) (0.07)
(0.07)
Own Price Elasticities
Model Food Alcohol Fuel Clothing Transport Services
A -0.02
-0.53 -0.57
-1.25 -0.98 -1.00
(0.16) (0.68) (0.14) (0.13) (0.66) (0.34)
B -0.12 -1.66
- 0.66 -0.95 -1.08 - 0.30
(0.05) (0.38) (0.16) (0.14) (0.03) (0.09)
C
- 0.43 -1.61 - 0.70
-
0.90
-
0.83 0.02
(0.10) (1.50) (0.01)
(0.08) (0.09) (0.12)
Notes: Model A:
Hours
and
participation;
Model B:
participation only;
Model C:
no labor
variables.
expenditure surveys only
the
participation
decision
is
reported
and not the
hours worked. Thus it
represents
the
most
general
system
that could
practically
be estimated on such data. The final set of estimates (model C) ignores the
labor
supply
decision
altogether.
This is
of
interest since
it is what is
generally
done. The hours variables
are set to the
sample
means for model
A
and the
participation dummies
are switched on for models
A
and
B.
Although
the
expenditure
elasticities derived from all three models are
qualitatively
similar
(in the sense
of
classifying goods
as luxuries
or
necessities)
the
precise elasticity
estimates
are
generally significantly
different. Similar
remarks apply
to the estimates of own
price
elasticities;
if
anything
these show
even more bias from
ignoring
the labor
supply
information. For
example,
services have
a unit
elasticity
in
model A whereas model C has
a
zero
elasticity.
Table IV also
indicates
that the bias from
ignoring
labor
supply
information
gets
worse the more
information we
ignore (that
is
the
B
estimates
always
lie
between the A and C
estimates).
However
the bias from
ignoring
the hours
data
is
often
significant;
this has obvious
implications
for
demand analysis on surveys
that
only
record the
participation regimes.
Often we use demand estimates to
give things
other than
budget
share
predictions
and estimates
of elasticities. One area that we
conjectured
would be
subject
to biased inference
if
we
ignore
labor
supply
information was the effects
of
children
on
demand.
The
presence of children (and particularly young
children)
is
highly
correlated
with the
participation
regime. Given this,
we
would
expect
that
ignoring
the
participation
variable
would
cause
significant
changes
in
the coefficients
on children.
In
Table
V
we
present
the estimates of the
intercept
coefficients
for
children
for models
A-C. As
might
be
expected
it looks like the bias is
greater
for
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MALE AND
FEMALE LABOR SUPPLY
943
TABLE V
CHILDREN
AND
LABOR SUPPLY VARIABLES
Young
Children: Intercept (x 100)
Model Food Alcohol Fuel Clothing Transport Services
A
1.62 -0.32 1.19 -0.91 0.70
-1.36
(0.71) (0.61) (0.37) (0.86)
(1.23) (1.00)
B 2.32 -1.18 1.50 - 2.00 1.76
- 0.53
(0.55) (0.45)
(0.32) (0.71) (0.95) (0.75)
C
2.40 -1.99 2.18 -0.60 -1.87
0.14
(0.37) (0.36)
(0.28) (0.40) (0.59) (0.52)
Older Children: Intercept (x 100)
Model Food Alcohol Fuel Clothing Transport
Services
A
4.51 0.71
-0.36 2.94 -3.11 -5.40
(0.62) (0.52)
(0.32) (0.77) (1.08) (0.87)
B 4.86 0.17 -0.10 1.84 - 2.65 - 4.22
(0.49) (0.40)
(0.28) (0.63)
(0.85) (0.67)
C 4.06 -0.34 0.47
1.41 - 2.36
- 3.38
(0.49) (0.48) (0.36) (0.53)
(0.78) (0.68)
Notes: Model
A: Hours and
participation;
Model
B:
participation only;
Model C:
no labor
variables.
younger children
than for older children. As
an
example
of how sensitive the
estimated effects of
young
children
are
to
the exclusion of
labor
supply
vari-
ables,
consider the
transport equation.
In our preferred specification (model A)
young children have no significant effect on transport; in model B (participation
only) they
have a
positive
effect,
and
in
model C
they
have
a
negative effect.
This reflects the fact that female participation is negatively
correlated
with the
presence of young children and
positively correlated with the budget share
for
transport.14
This
is clear
evidence that
any
estimates of
the effects of
young
children
on demand
patterns
that do
not take into account labor supply
are
likely
to
be
seriously
biased.
15
5.
CONCLUSIONS
The interaction between household labor supply and
commodity
demands
is
of
significance
both for
policy purposes
and
for
analyzing
consumer
demand,
labor
supply,
and
consumption.
In this
paper
we have
presented
a
methodology
for
testing
for
separability
and for
estimating
demand
systems
under the
alternative of
nonseparability.
The
distinguishing
feature
of our
approach
is
precisely the fact that the null hypothesis is that of weak separability
of goods
from leisure;
no other restrictions need be
imposed
either
implicitly
or
explic-
itly.16 Moreover,
our
approach
allows for
testing
and estimation under
the
alternative with little
knowledge
as to the
process determining
the
conditioning
14
Note as well that the bias from excluding children is also large for alcohol;
this has obvious
implications for users of the "Rothbarth" method of imputing adult equivalence
scales from
household expenditures on "adults goods" (see, for example, Deaton et al. (1989)).
15As
a matter of fact, most studies of the effects of children on demand do ignore labor supply
(see Browning (1990)); this is so common that it would be invidious to quote examples.
16
Except,
of
course, those implicit
in
any parametric system
-
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944 MARTIN BROWNING AND COSTAS MEGHIR
goods. This is a considerable
advantage
in
the
cases where these conditioning
goods are hours
of
work,
durables,
or
public goods.
We
apply
our
methodology
to estimate
a
conditional demand system
for seven
goods and to test for separability of goods from male and female hours of work
in the UK Family Expenditure
Survey (FES)
1979-1984. From our empirical
results we draw the following conclusions:
1. Separability
for male and female hours and
for the respective participation
decisions are
rejected.
2. Male and female hours and participation
are
not substitutable
in
their
effect on the demands modeled
in this
study.
3. The hours and
participation decisions are
endogenous for the demand
system.
4. Estimates of demand that only take account of participation (and not
hours)
are
likely
to be biased. Estimates
of demand that
ignore
labor
supply
altogether
are
subject
to even
more bias.
5. Ignoring participation
biases
the
estimated
effects of
young
children on
demand.
All
of
these
are,
of
course,
conditional
on our
maintained
hypotheses.
Of
these we
conjecture
that the most
important
are:
the
goods
we model
are
separable
from the
goods
not modeled here
(principally public goods,
tobacco,
durables,
and
housing
services);
there are no
significant
habits or
durability
and
there are
no
significant
measurement
or
specification
errors. Our
feeling
is that
conclusion
1
will be
robust to almost
any
alternative
specification.
In
this sense
we offer it as our most important
and robust
finding.
Dept. of
Economics,
McMaster
University, Hamilton, Ontario,
L8S
4M4,
Canada.
and
Institute
for
Fiscal Studies,
and
Dept. of Economics, University College
London,
London, WCJE 6BT,
England.
ManuscriptreceivedMarch, 1989; final revisionreceivedJuly, 1990.
APPENDIX
PROOF OF PROPOSITION: Define
D weak
separability
(DWS): v(q, a, h)
=
F(G(q, a), a, h)
and
C
weak
separability (CWS): c*(p, a, h, u)
=
c(p,
a, g(a, h, u)).
We assume that
c*(*)
is
concave
in
p
and convex
in h
and that the
utility
function is
quasi-concave.
DWS =*CWS.
c*(p,a,h,u)
=
min
{pq:
F(G(q,a),a,h)
=
u}
q
=
min
{pq: G(q, a)
=
g(a, h, u)}
q
=
c(p,
a, g(a, h, u)).
CWS =DWS.
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8/11/2019 The Effects of Male and Female Labor Supply
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MALE AND FEMALE LABOR SUPPLY
945
Let 1(a,
h,
t)
be the
inverse
of
g(a,
h,
u)
on
u
(so
that
u =
l(a,
h, g(a, h, u))
by
definition). Note
that this inverse
always
exists
since
g(*)
is
strictly
increasing
on u and that
1(
)
is
strictly
increasing
in t. Then
v(q,
a, h)
=
min
{u:
c(p, a, g(a, h,
u))
=pq
for all
p}
u
=
min{l(a,h,t):
c(p,a,t)=pq
for
all
p}
=
l(a, h, min
{t:
c(p, a, t) =pq
for
all
p}
=
l(a,
h,G(q, a)).
TABLE Al
MODEL WITH
HOURS
AND
PARTICIPATION
EFFECTSa
Intercept Parameters
Food
Alco
Fuel
Cith
Tran
Serv
Int 31.04
7.64
12.17
8.07
21.21
10.15
(2.1)
(2.5)
(1.3)
(2.2)
(3.3)
(2.9)
K1
1.62 -
0.32 1.19
-
0.91
0.70 -
1.36
(0.78)
(0.6)
(0.4)
(0.9)
(1.2)
(1.0)
K2
4.51
0.71
-0.36
2.94
-3.11
-5.40
(0.6)
(0.5)
(0.3)
(0.8)
(1.1)
(0.9)
Hm
62.40
91.40 -3.85
138.86
-224.03
-164.92
(28.0)
(24.1)
(15.0)
(34.6)
(48.7)
(39.8)
Hf
-
6.84
25.79
-
15.45
25.87 -
8.69 -
35.77
(12.4) (10.6) (6.5) (14.9) (21.6) (17.2)
Df -0.13
-7.70 6.10
- 14.32
11.71
12.98
(4.1)
(3.5)
(2.2)
(5.1)
(7.2)
(5.8)
Dm
-21.3
- 33.7
-
3.0
-53.0
87.8 62.90
(11.8)
(10.0)
(6.2)
(14.4)
(20.2)
(16.5)
Pf
19.41
(4.1)
Pa -
7.15
3.2
(3.7)
(4.7)
Pf
1
2.37
1.31 4.06
(1.9)
(2.3)
(1.6)
Pc
-
1.43
-0.23
- 1.11
-2.16
(1.6) (1.5) (0.8) (1.4)
Pt
-3.08
11.30
-10.00
3.60
0.51
(4.3)
(4.3)
(2.4)
(2.4)
(8.6)
Ps
-
2.14
-
8.21
0.06
0.10
2.26
1.20
(3.5)
(3.7)
(2.2)
(1.8)
(5.1)
(5.8)
Fage 0.26
-0.07
0.14
-0.25
0.04
0.12
(0.3)
(0.2)
(0.14)
(0.3)
(0.5)
(0.5)
C1
1.0
-
0.91
-
0.15 0.84 -
2.18
0.19
(1.1)
(0.9)
(0.6)
(1.3)
(1.9)