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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2/28

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


3/28

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.


4/28

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


5/28

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


6/28

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.


7/28

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


8/28

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.


9/28

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


10/28

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.


11/28

934


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.


12/28


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.


13/28

936


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.


14/28

MALE AND

FEMALE

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).


15/28


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.


16/28

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.


17/28

940


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.


18/28

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.


19/28

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


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


20/28

MALE AND

FEMALE LABOR SUPPLY

943

TABLE V

CHILDREN

AND

LABOR SUPPLY VARIABLES

Young

Children: Intercept (x 100)


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


21/28


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.


22/28


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)

The Effects of Male and Female Labor Supply

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