The Review of Economic Studies Ltd.
Consumer Demand and the Life-Cycle Allocation of Household Expenditures
Author(s): Richard Blundell, Martin Browning, Costas Meghir
Source: The Review of Economic Studies, Vol. 61, No. 1 (Jan., 1994), pp. 57-80
Published by: Oxford University Press
Stable URL: http://www.jstor.org/stable/2297877 .
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Review of Economic Studies (1994) 61, 57-80
? 1994 The Review of Economic Studies Limited
Consumer
0034-6527/94/00050057$02.00
Demand
and
the
of
Life-Cycle
Allocation
Household
Expenditures
RICHARD BLUNDELL
University College London and Institute for Fiscal Studies
MARTIN BROWNING
McMaster University
and
COSTAS MEGHIR
University College London and Institute for Fiscal Studies
First version received June 1991; final version accepted March 1993 (Eds.)
The purpose of this paper is to estimate the parameters of household preferences that
determine the allocation of goods within the period and over the life cycle, using micro data. In
doing so we are able to identify important effects of demographics, labour market status and
other household characteristics on the intertemporal allocation of expenditure. We test the validity
of the life-cycle model using excess sensitivity tests and find that controlling for demographics
and labour market status variables can largely explain the excess sensitivity of consumption to
anticipated changes in income.
1. INTRODUCTION
The purpose of this paper is to estimatethe parametersof household preferencesthat
determinethe allocation of goods within the period and over the life-cycle, using micro
data. In doing so we areable to identifyimportanteffectsof demographics,labourmarket
statusand otherhouseholdcharacteristicson the intertemporalallocationof expenditure.
The distinctivefeatureof our approachis that it integratestraditionaldemandanalysis
with intertemporalsubstitutionmodels in a coherentway.
The organizingidea behind our analysis is the life-cycle hypothesis. The principal
implicationof this hypothesisis that householdswill allocate consumptionexpenditures
so as to try to keep the marginalutility of wealth (A) constantover time (see Heckman
(1974), Hall (1978), MaCurdy(1983) and Browning,Deaton and Irish (1985)). This is
true both in the short run (for example,saving to capturethe benefitsof high real rates
of interest)and in the long run (for example,savingfor retirementor for a "rainyday").
Of course, A is unobservableso that the empiricalconsequencesof this "smoothing"
have to be derivedfor expenditureson individualgoods. Althoughnonparametrictests
are availableundercertainconditions(see Browning(1989)) the usual procedure,which
we follow here, is to specify a parametricutility function and then to derive the Euler
equationthat governsindividualexpenditures.
The relationshipbetween A and expenditureson individualgoods dependson many
things. Amongstthese are the shape of Engel curves;the amountof substitutionbetween
57
58
REVIEW OF ECONOMIC STUDIES
goods; the demographiccompositionof the household and the labour marketstatus of
the household. Analyses that assume that agents buy only one good ("consumption")
and that preferencesover this good are additively separablefrom demographicsand
laboursupplyarepotentiallyquitemisleading. In this paperwe specifya set of household
preferencesthat allow us to modeljointly within-periodallocations(the demandsystem)
and between-periodallocations (the consumption function). Although our principal
interestis in the latterwe shall see that we also need to model the former.
We utilize a time series of U.K. cross-sectionscovering 70,292 households over a
17-yearperiodto investigatethe linkbetweenwithin-periodpreferencesand intertemporal
substitution. Micro-leveldata avoids the problemof aggregationbias and allows us to
analysethe importanceof demographicand labour supply variables.
Our empiricalresultsare based on a structuralmodel of demandand intertemporal
substitutionas well as on a simple iso-elasticspecificationwhich we extend to allow for
household characteristics.To meet the great many reservationsto the life-cycle model,
documented recently by Deaton (1992), we carry out a number of specificationtests
includingexcesssensitivitytests,testsof overidentifyingrestrictionsanda simplestructural
change test.
The mainconclusionis thathouseholdcharacteristicsare of fundamentalimportance
in explaining the growth of consumptionover a household's life cycle. We find that
controllingfor these characteristicsis sufficientto eliminateexcess sensitivityof consumption growthto predictableincome growth. The issue of whethercharacteristicscapture
taste effectsor are simply a proxy for the effectsof liquidityconstraintsremainsan open
question.
2. A THEORETICALFRAMEWORK
We assumethe consumermaximizesthe discountedsum of period-specificutilities. Our
estimationprocedureexploits the two-stage budgetingresults of Gorman (1959). The
within-periodallocation of total consumptionx to individual goods with prices p, is
completely characterizedby the indirect utility function V(p, x) and is invariantto
monotonic transformationsof utility V.' Intertemporalallocationsare then determined
by the period-specificutility function U = F[ V(p, x)] where F[*] is a strictlyincreasing
monotonictransformationsuch that U is strictlyconcave in x.
In what follows we partitiongoods into two groups. The firstis the groupof interest
whose expenditureis x; let this have quantityand pricevectors(q, p). The second group
containsthose goods that we do not model explicitly. For example,these include labour
marketstatus and demographicstructure. Denote the vector of such goods z and let
sub-vectorsbe given by zl, z2 and z3, where zi and zi may have common elements. We
representperiod specific preferencesby the conditionalindirectutility function
U(p, z, x) = F[V(p, zl, x),
z2]+ H(z3).
(2.1)
This function gives the maximum utility in the period for an agent who has total
expenditurex on the firstgroup of goods with prices p conditionalon other goods and
household characteristicsz.
This treatmentof conditioningfactorshas a naturalinterpretationfor our discussion
of intertemporalallocations. All those factorsin Z3 but not in zl nor z2 enterneitherthe
1. See Deaton and Muellbauer (1980b) for a detailed description of the properties of indirect utility
functions.
BLUNDELL ET AL.
HOUSEHOLD EXPENDITURES
59
demandsystemnor the consumptionfunction. They are explicitlyadditivefromall other
commodities. Factors in Z2 but not in z1 enter the consumptionfunction but not the
demandsystem. These are weakly separablefrom q. Those in z1 influencethe marginal
rate of substitutionbetween elementsin q and are thereforenot separable. Factorsin z'
but not in Z2 conditiondemanddirectlybut do not affectintertemporalallocationexcept
throughtheir effect on the parametersof demandused in the consumptionfunction.
2.1. A model for within-periodpreferences
The form of within-periodpreferencesis independentof the normalizationF[*] in (2.1).
Moreprecisely,the shape of Engelcurvesand the specificformof within-periodsubstitution are independentof the parametersdeterminingintertemporalsubstitution.Suppressing the conditioning variables z, the indirect utility representationof within-period
preferenceswe use is
x
(a(p))
1
b(p)
(.2
and
ln x -ln a(p)
=(
b(p)
(2.3)
where { } representsa Box-Cox transformation
J AI Y
y{A}
1
A
These two are members of the PIGL and PIGLOG classes respectively(Muellbauer
(1976)). To completethe specificationwe adopt the followingstandardspecificationsfor
the two price indices:
ln a(p) =ao+Ek
ln b(p)=
Yk Pk
ak lnPk+jEkEj
ykjln pk ln pj,
ln Pk*
(2.4a)
(2.4b)
To satisfy adding-up we require E ak = 1, E &k =0 and Ek Ykj=0, while homogeneity
implies >j ykj =0. Thus a( p) and b( p) are homogeneousprice indices of degreeone and
zero respectively.
Applying Roy's identity to (2.2) and using the above parameterizationswe obtain
the demand system
= ai +yj
Wit
yij ln pjt +f8i(x/a(
p)){o}+ vit
(2.5)
where i denotes the i'th good, t is the observationindex and vitis assumedto represent
unobservablecomponentsin demand.2The value of 0 determinesthe shape of the Engel
curve. Given our choice of a(p) and b(p), if 0=0 we have the Almost Ideal Demand
System of Deaton and Muellbauer(1980a). Alternatively,if 0= -1 we have quasihomotheticpreferenceswhilst 6 = + 1 gives a form of quadraticEngel curves in which
2. Note that the parameterization of the demand system implies that any stochastic variation of the
parameters would either enter in a non-additive way (i.e. through a(p)) or would interact with the endogenous
total expenditure term (if they enter through the ,3s). Moreover any random preferences would enter non-linearly
in the utility index V. Hence, in order to be consistent we have to assume that the stochastic specification
represents optimization errors in the allocation of budget shares. Adding-up implies that these errors sum to zero.
60
REVIEW OF ECONOMICSTUDIES
expendituresharesarelinearin x. If all of the ,8i'sarezerowe havehomotheticpreferences;
in this case all Engel curves are linear throughthe origin and 6 is not identifiedin the
demand system.
One of our objectives is to assess the importanceof demographiccharacteristics,
labour marketvariablesand other taste shifters on intertemporalsubstitutionand consumptiongrowth. Since F[ *] and V( - ) in (2.1) are separatelyidentifiablewhen we use
data on within-periodand intertemporalallocations,omittingthese characteristicsfrom
the within-periodutility index V(*) may bias our conclusionsrelatingto intertemporal
allocations.Moreover,other studies on micro data sets have shown demographicsto be
very importantin demand systems.
2.2. The consumptionfunction
Our approachin derivingthe consumptionfunction is based on the Hall (1978) Euler
equationformulationimplementedon microdata by Heckmanand MaCurdy(1980) and
Altonji (1986); it follows closely the methodology of MaCurdy(1983). Defining A,=
OF,/Ox, as the marginalutility of an extra unit of expenditurein period t, we can write
the condition for optimal intertemporalbehaviourunderuncertaintyas
Et[(l + rt)At+l-At] = 0,
(2.6)
where rt is the nominalinterestrate earnedon assets held between t and t + 1 and Et{*}
representsthe expectationconditionalon informationavailableat time t.
We take the following functionalform for F(.)
(2.7)
F(Vt) = (1 + 8)-tqtVl+Pt}
where 8 is the rate of time preferenceand where ct representssome scaling of utility
which may depend on household characteristicsand other conditioningfactors and, as
before, the superscriptin { } denotes the Box-Cox transform.The parameterp will be
allowed to vary across households and across time accordingto movementsin demographicand other characteristics.Given this normalizationwe may writethe log of the
marginalutility of expenditure(in currentterms) for any household in period t as:
In At= In Ot- t In (1 + 8) + pt In Vt+In V'
(2.8)
where V' is the derivativeof Vtwith respectto xt.
To estimatePt we shall exploit the Eulerequation(2.6) which governsthe evolution
of At over time. To proceed, we re-write(2.6) as:
(1 + rt)At+l= Aktut+,
Et(ut+1)= 1.
(2.9)
Takinglogs through(2.9) we can then write (2.6) as:
A InAt+1+ In (I + rt)+ dt = -t+i,
where A refers to the first difference operator. Defining Et (ln ut+i) = -dt,
(2.10)
6t+1
is such
that Et(Et+1)= 0. If ut+1is lognormal then dt= 1, o2 being the variance of ln ut+1
conditionalon informationin time period t. In generaldt will also be a functionof higher
momentsof ln ut+1.
Substituting(2.8) in (2.10) and re-arrangingwe have:
-A In V+1 - ln (1 + rt)= Apt+,In Vt+1+ (dt- At)+ Et+,
(2.11)
in whichall parametersareidentifiablefromwithin-periodallocationsand aresummarised
BLUNDELL ET AL.
HOUSEHOLD EXPENDITURES
61
in the V and V' expressions. We haverewrittenA In (,t+1) -In (1 + 8) as -3, whichcould
be a function of levels and changes in household characteristics.To interpret(dt-t)
assume ut+1in (2.9) is lognormallydistributed;then the difference (0,- _,6t) can be
interpretedas capturingthe trade-offbetweenimpatienceand caution.To see the relationship betweenthis and intertemporalsubstitution,note thatincreasesin futureuncertainty,
decreasesin impatienceand increasesin the nominalrate rt all act in the same direction.
Thus d, can be thought of as capturingthe precautionarymotive for saving.
Using our definitionof indirectutility V from (2.2), the Euler equation (2.11) can
be rewritten
[(1 + 0)A ln Ct+1- it + A ln b(pt+1)] = A[pt+jl[n C{I1I}- ln b(pt+1)]] + (dt -At) + et+i,
(2.12)
where we have used Ct to representtotal real consumers'expenditurext/a(pt) and
it = ln (1 + rt) - A ln a(pt+i) is the real interestrate measureimplied by our specification
of individualpreferences.
In (2.12) the left-handside resemblesthe iso-elasticspecification;indeed if b(p,) is
constantover time and if Pt= 0 then the differencebetween consumptiongrowthscaled
by (1 + 6) and the real rate is just equal to dt- St and an innovation;this is preciselythe
log-linear version of the iso-elastic model. In general, given values for Pt and St,
intertemporalallocationswill depend on the set of conditioninggoods via the two price
indices a(pt) and b(pt) which characterisewithin-periodsubstitution. Nevertheless
demographicvariables may also affect intertemporalallocations directly since Pt (i.e.
F[ *] in (2.1)) may be a function of variables,such as the numberand age of children
and labour marketstatus. We thus specify:
Pt = Po+ 2k PkZkt.
(2.13)
We discuss alternativeinterpretationsof the results obtained from such a specification
in the empiricalsection.
The relationshipsgiven by (2.5) and (2.12) constitute our descriptionof the consumer'sallocationscheme. This parameterizationhas a recursivenature:there are some
parametersthat enter both the intratemporaland intertemporalallocation decisions i.e.
and b(pt). There is also a set of parameters that characterize
only intertemporalconsumptionallocations,namelyPt, dt and St. In particular,0, a(pt)
O and the parameters of a (pt)
and b(pt) determinethe shape of the underlyingEngel curveswhile, for any given value
of these, Pt determinesintertemporalsubstitution.If we changethe formerthen we shall
usually change the latter.
The intertemporalsubstitutionelasticity (ISE) a ln Ct/l ln (1+ it) implied by our
model takes the form
(Dt =
xtulf
Pt- (I
+ O)C{t}
(2.14)
where U' is the partial derivativeof U( ) in (2.1) with respect to x. If Pt =0 then
(D= -1/(1 +6) and (2.12) reducesto the standardiso-elasticEulerequation. Given our
concavityassumptionthe intertemporalelasticityshould be negativefor all values of Ct.
This requiresp < (1 + 0)CI01for all observationsin our data.
The specificationof the model implies various relationshipsbetween the ISE and
consumptiondependingon the estimatedparametersp(Zt) and 6. As C tends to 00, 4
tends to -1/(1 +9). (When 6 = -1,ID tends to 1/p). If p < 0 then the absolutevalue of
D increaseswith consumption,while if p > 0 the reverseis true.
62
REVIEW OF ECONOMIC STUDIES
3. ESTIMATION AND EMPIRICAL EVIDENCE
3.1. Data
In estimatingour demand system describingwithin-periodpreferenceswe have chosen
a group of seven broad commoditiesfor each household. These are: food, alcohol, fuel,
clothing,transport,servicesand othergoods. The resultsreportedhere referto a sample
of 70,292 households from 1970-1986inclusive, whose eldest adult is more than 18 and
less than 60 yearsof age and is not self-employed.These data are drawnfromthe annual
U.K. Family ExpenditureSurveyand are more fully describedin the data appendix at
the end of the paper. Our choice of goods clearly excludes some non-durableslike
tobacco as well as most durables,leisure and public goods. As describedin Section2.1
we allow for the effect of some of these on our allocation scheme by enteringthem as
conditioningvariablesin the demandsystemand consumptionmodel (see Browningand
Meghir(1991)).
We note firstthat our data, ratherthan being a panel followingthe same individuals
across time, is a time series of repeated cross-sections. As a result we construct a
pseudo-panel using cohort averages in order to estimate the model as suggested by
Browning,Deaton and Irish (1985). The cohortsare chosen in five-yearage bands. The
methodology which uses exact aggregation, and the assumptions underlying it are
describedbelow.
In this paperourfocus is on life-cyclepatternsof consumption.Westartby presenting
some of the principalfeaturesof our data that are salientfor consumption. In particular
we look at co-movements in consumption, income, demographicsand labour force
participation. It is our strong belief that since these are all determinedjointly by the
same household,looking at any pair in isolation may be quite misleading. In Figure1(a)
we present the life-cycle path of consumptionfor marriedcouples.3 In each of these
figures,each separateline representsthe evolution over time of the relevantvariablefor
one date of birth cohort defined over a five-yearband. This has the familiarpattern:
consumptionrises initially and then falls after the mid-forties. Figure1(b) shows the
life-cycle path of household income. Once again, the shape and the correlationwith
consumptionarefamiliar. If we identifythe causeof the correlationseen hereby assuming
that income is exogenousthen it looks like consumptiontracksincome very closely over
the life-cycle. The next two figures(1(c) and 1(d)) look at the paths of two potentially
importantdeterminantsof consumptionand income respectively:children and female
employment. As we expect, female employmentdrops during the child-bearingyears
and the numberof childrenin the householdpeaks soon thereafter.Whatis particularly
interestinghere is that althoughfemale participationfalls in the early years, household
income does not. One obvious explanationfor this is that householdschoose the timing
of births relativeto the husband'scareerprofile but this once again implicitlyassumes
that the latteris exogenous. It will be clear,however,thatthese figuresare also consistent
with the hypothesisthat husbands'income paths are chosen to facilitateparticularpaths
of births.
In Figure1(e) we presentthe path of consumptiondeflatingby "equivalent"household size which equals the numberof adults plus 0 4 times the numberof children.4As
can be seen, this removesmost (if not all) of the "hump"shape in consumption.
3. In our analysis below we include single-adult households with appropriate controls; here we use just
married couples to abstract from awkward composition effects.
4. Obviously we could use more sophisticated equivalence scales that allowed for economies of scale and
for age differences in children; this does not change things very much.
4q
4.5
(a)
3 28
63
HOUSEHOLD EXPENDITURES
BLUNDELL ET AL.
(b)
/
4
403
-
3-7 -42
36 -
41* -
3-5 -4
3-4-~~~~~~~~~~~~~~~~~.
3
_
_
20
_
_
25
_
_
_
30
Log real household
__
_
_
35
_
_
_
40
age
_
_
45
consumption
__
_
50
_
_
_
_
55
.5
_3
_
I
20
60
25
35
30
3 -(c)
40
45
age
50
S5
60
income over the life cycle
Log real household
over the life cycle
.8 -(d)
2-5.7
2
65
1-5
6
II..45-
.5
.40253
5 306
3540455
20
age
in the householdoverthe life cycle
Numberof children
35
25
40
45
50
55
age
Femaleemploymentoverthe life cycle
60
2.7
2-6
2.52-4
2-3
2.2-
(e)~~~~~~~~~~28
Lo osmto
e
dl
quvln
21
20
25
yl
vrte)lf
30
3
40
45
50
5
6
age
Log consumptionper equivalentadultoverthe life cycle
FIGURE 1
(a) Log real household consumption over the life cycle. (b) Log real household income over the life cycle.
(c) Number of childrenin the household over the life cycle. (d) Female employmentover the life cycle.
(e)
Log
consumptionper equivalentadult over the life cycle
The principalconclusionwe drawfromthese figuresis that it is verydifficultto infer
anythingabout life-cycle consumptionfrom looking at simple descriptivegraphs. Apart
from not being able to distinguishbetween anticipatedand unanticipatedcomponents
in the series,we cannotproperlytake accountof the possibleendogeneityof otherfactors
that affect consumption. Thus we must have recourseto an econometricanalysis that
addressesthese issues by conditioningon demographicsand labour marketstatus and
by using an instrumentalvariablesapproach.
REVIEW OF ECONOMIC STUDIES
64
Comparing Figures 1(a) and 1(e) also gives us some insight into how controlling for
life-cycle events may change our perceptions of year to year consumption changes. As
can be seen from l(a), for younger (older) households, consumption growth is largely
positive (respectively, negative). This life-cycle variation may mask a lot of year to year
variations and the correlation of the latter with common variables such as the real rate.
Thus not controlling for changes due to changes in "life-cycle" variables may make it
difficult to pick up high frequency intertemporal substitution effects.
In Figure 2 we plot the time path of real consumption growth and the smoothed
time path of the real interest rate.S As is well known the ex post real interest rate was
negative throughout the 1970s. On the other hand consumption growth has been positive
most of the time. In terms of the standard life-cycle model this must either imply that
the ex ante real interest rate was positive or that precautionary savings is an important
influence on consumption growth. In terms of our model this would mean that d, - 3,
was a positive number in the 1970s. After 1981, the real interest rate jumps, becoming
positive and higher than consumption growth. We investigate the implications of this
real rate jump in the empirical section.
3.2. Estimating within-periodpreferences
To estimate the parameters of the expenditure share system we adopt an iterative moment
estimator in which we allow for the endogeneity of total expenditure and iterate on the
a(p) price index and on the Engel curvature parameter 0. The estimation procedure and
*2
15
Real consumption growth rate
*1 -
-05 0-1
Real interest rate
-*15 -
-.2
I
I
I
1971
1973
1975
-
I
IiII
1979
1977
year
1981
1983
1985
FIGURE 2
Real consumptiongrowthand the "ex post" real interestrate
5. The consumptionpath in Figure2 is for all cohortsaveraged. In estimationwe use the three-month
treasurybill rateadjustedby the averagetax rate. We use the Stonepriceindexoverour subsetof commodities
to constructthe inflationrate with which the interestrate is deflated.
BLUNDELL ET AL.
HOUSEHOLD EXPENDITURES
65
TABLE I
Priceand incomeelasticities
Fuel
Transport
Services
0'019
0'553
-0'196
-0'568
-0'329
-0'142
-0'066
0-495
-0'331
-0'672
-0'708
0'341
0-058
-0'215
-0-435
-0-026
0'405
-0*901
0-006
0'383
-0-312
-0'093
-0'048
-0'237
0'081
0'630
-0'076
-0'513
-0'274
0'040
0'062
0'653
-0'086
-0'559
-0'593
0'717
0'143
-0 110
-0'271
0'049
0'481
-0'651
Average budget shares
0'067
0'347
0'102
0'086
0'176
0'117
Income elasticities
0'896
0'725
1.391
0'643
0'651
2'131
Food
Uncompensated
-0'619
-0'229
-0'460
0'107
-0'110
-0'311
Alcohol
Clothing
price elasticities
-0'068
-0'033
0'291
-1-508
0-158
-0'454
0'449
-0'159
0'203
-0-115
-0'202
-0'455
Compensated price elasticities
0-015
-0'368
-1-448
0'082
0'252
0'023
0'492
0'330
0'247
0'116
-0'060
0'429
the computationof the asymptoticcovariancematrixfollows from Browningand Meghir
(1991).
The complete specification,the parameterestimatesand the list of instrumentsare
all presentedin AppendixA. Thereis a clear indicationof the importantrole played by
demographicand labourmarketvariablesin the allocationof within-periodexpenditures.
It is also worth noting the significanceof the f3iparameterswhich imply a rejectionof
homotheticityand hence the b(p) index will vary with relative prices. The value of 0
was estimatedto be 0 54 with a standarderrorof 0 45. This would not rejectthe Almost
Ideal model of Deaton and Muellbauer(1980a) but we have not imposedthis restriction.
Table I presentsthe elasticitiesfor our model with 6 = 0 54. The overallresultsare
sensibleand conformwith the usual findingsin the literature:the budgetsharesfor food,
alcohol, fuel and transportdecline with increasing total expenditurewhile clothing,
servicesand other goods are luxuries. All own uncompensatedand compensatedprice
elasticitiesare negativeand this is in fact true almost everywherein the sample.6
3.3. Cohort aggregation and the estimation of intertemporalsubstitution
The estimationof the demandsystemprovidesestimatesfor the parametersthatdetermine
the two priceindicesa (Pt) and b(pt) as well as the curvatureparameter6; in the estimation
of the Euler equation which we now describe, we take these as given.7 The main
econometricproblemhere relatesto the fact that we do not have repeatedobservations
for any one individual: as noted above, our data consists of repeated cross-sections
spanning the seventeen-yearperiod 1970-1986. We thus use a groupingmethodology
used in Browning,Deaton and Irish (1985).
6. For a more detailed analysis of the intratemporalallocationson this data (albeit on shortertime
periods)see Browningand Meghir(1991) and Blundell,Pashardesand Weber(1993).
7. In AppendixB we show how the standarderrorsfor the Eulerequationcan be adjustedfor the fact
that we are conditioningon pre-estimatedparameters.This adjustmentmade verylittle differencehere.
66
REVIEW OF ECONOMIC STUDIES
The method relies on obtaininga randomsample of the same date of birth cohort
of individualsovertime and modellingthe averagebehaviourof the group. Thusconsider
the expectedvalue of the Eulerequationfor consumption(2.10), conditionalon a cohort
and on time period t. This is
AE[ln AitIiE c, t]+ln (l+rtl)+dt=-E[eitIie
c, t],
(3.1)
wherei standsfor individual,t fortimeperiodand c for a cohortandwherethe conditional
expectationof the log marginalutility of wealth is
E[ln AitIiE c, t]=-t ln (1+8)+E[p(zit) ln
c, t]+E[ln
VitIiE
(V't)IiE
c, t].
(3.2)
The estimationproceduremodels the behaviourof the cohort mean of the log of the
marginalutilityof wealth. Thuswe estimatethe groupedEulerequationanalogof (2.12).
Using (2.13) we have
(d-8) + [et].
(3.3)
In the above, the terms enclosed in square bracketswith a subscript"c" denote
sample averagesof these quantitieswithin a date of birthand time period cell. The lefthand side is the averagegrowthrate of consumptionminusthe averagereal interestrate
adjustedfor the price index b(p).8 The right-handside involves the growthrate of the
averagewithin-cohortcross-productof the characteristicsZkt with [ln C"' -ln b(pt)].
The next set of termsinvolve differencesin the averageof some householdcharacteristics
A[(1+ 6) ln Ct+ ln b(pt)]c - [it-J]c =
S
pkA[zkf(ln
+
}-ln
b(pt))]t
C{,
which may enter the discount rate or the conditional variance term dt. Finally [E,]c is
the within-cohortaverageof the innovations.
In estimationwe use 5-yearcohortsobservedannually.At this level of aggregation
the cell is probablysufficientlylargeso as to reducethe importanceof measurementerror
(see Deaton (1985)); details on the groupeddata are given in TableII.
TABLE II
The structure of the grouped data
Cohort
Year of birth
Period of observation
Average cell size
1
2
3
4
5
6
7
8
9
1916-1920
1921-1925
1926-1930
1931-1935
1936-1940
1941-1945
1946-1950
1951-1955
1956-1960
1970-1975
1970-1980
1970-1985
1970-1986
1970-1986
1970-1986
1970-1986
1974-1986
1979-1986
518
601
522
500
512
555
591
479
405
For consistent estimationof the parametersof interestwe need to impose certain
conditions. First, our estimationprocedure(both for the demandsystem and the Euler
equation) assumes asymptoticson both N (individuals)and T (time periods). Given
this, the averageinnovations[et]c tend to zero unless there are commonshocksaffecting
8. The real interest rate i,_1 = In (1 + r,-1) - A In a(p,) is individual specific for two reasons. First, because
of the individual specific nature of the inflation rate A In a (p,) which depends on household characteristics.
In addition each household is observed at a different point in time. The resulting nominal interest rate is a
weighted average of the interest rates for the grouping period with the proportion of individuals observed at
each point in time as weights.
BLUNDELL ET AL.
HOUSEHOLD EXPENDITURES
67
all membersof the cohort. Moreover,any idiosyncraticrandompreferenceerrorsentering
throughthe discountfactor at the individuallevel averageout at the cohort level. Thus
our only sourceof stochasticvariationis cohortlevel shocks;these areassumedto average
out over our 17 years of data.
The natureof the model is such that [Et]Jwill be correlatedwith consumption,prices
and most choice variables dated t (unless they are subject to relativelylong decision
lags). Moreovertime-aggregationcan induce serial correlationand hence [Et], may also
be correlatedwith past values of the decision variablesand prices. Hence we must use
an instrumentalvariablesprocedureusing appropriatelags as instruments.The instruments,listed below, involvethe secondlag of the growthrateof incomeand consumption,
the interactionsof the latterwith characteristicsas well as the characteristicsthemselves.
The latterare eitherdated t - 2 (labourmarketstatus) or dated t (all other demographic
variables).9
Consistencyrequiresthe groupedinstrumentsto be orthogonalto the groupederror
termover time. This will be true if each memberof the cohorthas in his informationset
the average cohort values of the variables we use as instruments. If the individuals'
informationsets containonly values relatingto themselvesthen this orthogonalitycondition will not be satisfied and cohort aggregationwill not lead to consistent parameter
estimatesdespite our exact aggregationprocedure(see Pischke(1991)).
Finally,sufficientconditionsfor consistencyof our instrumentalvariablesprocedure
are
(a) PlimT, Et [qt]c[Et]c/ T= 0, where
[qt]c
is the cell average of the instrumental
variables,
(b) plimT,OEt[qt]LX'1c/T=Mqx
where rankMqx =rows of Xt and where Xt=
[A[z'(ln Cl-o0-ln b(pt))], A[zt]']' is the set of right-handside variablesin (3.3)
above, and
(c) plimT
Et [qt]c[qt]/
T = Mzz where Mzz is full column rank.
The stationarityassumptionsimplicit in the above do not relate directlyto any of
the individualseries of interestrate or pricesbut to functionsof these variables. Hansen
(1982) and Hansenand Singleton(1982) discuss the sufficientconditionsfor consistency
in the estimationof Euler equationswith long time series.
Given these conditions, the estimatorfor the unknown parametersin the Euler
equation (Pk, k = 0, . . ., K and (d - 8)) is
=
(X'Q(Q'QQ)-lQQx)-lxXQ(Q'flQ)-1Q
,
where Y is the left-handside of (3.3), X is the collectionof the right-handside regressors
and Q is the matrix of instruments. We replace Q'fQQby its estimate c
is an estimateof [Et]cfroma firstroundunweightedIV regression(see White
where[
(1980)). The covariancematrixof the estimatoris presentedin Appendix B.
3.4. Empirical resultsfor the intertemporalconsumption model
In Table III we presentestimatesof threevariantsof our base model. Thesethreemodels
illustratehow demographicsand laboursupply variablesinteractin the Eulerequation.
To interpretthe results note that the coefficientspresentedare those in (2.13); that is,
Pt = Po+ k PkZkt where the variablesZktare those listed in TableIII. Lowervalues of p
withlog realconsumptionwithin
9. Precisely,we constructthe averagecross-productof the characteristics
a cohortandthenwe takefirstdifferencesof this measure.Thesecondlag of this variableis thenourinstrument.
REVIEW OF ECONOMIC STUDIES
68
TABLE III
The Euler equation for consumption
Po
HWORK
WWORK
KI
NCHILD
OWNER
SINGLE
MULTIPLE
d -s8
Sargan
p-value
4)
(1)
(2)
(3)
-2-369
(3.923)
2-863
(1.457)
2-212
(1.500)
-0-872
(0.989)
0-466
(0-219)
0.091
(1-027)
4-280
(4.339)
1-432
(1.140)
0.011
(0.018)
22'5 (14)
7%
-0-75
(0.12)
-2-233
(3.785)
3*096
(1.439)
2-884
(1.082)
0-570
(3-273)
0*478
(0*206)
-0-291
(0.938)
3-700
(3.694)
1-204
(1.078)
0.021
(0.013)
24.72 (15)
5.4%
-0-77
(0-12)
-2-377
(0 724)
0-326
(0-212)
2-242
(0-762)
5*931
(2.702)
1*947
(0.865)
-0*028
(0.010)
28.7 (16)
2-6%
-0 75
(0. 11)
Notes: Asymptotic standard errors in parentheses. c1: Estimate of the intertemporal elasticity of substitution for consumption at the sample mean.
HWORK: Head of household in paid employment, WWORK: Wife in paid
employment, Kl: Number of children 0-2 years of age, NCHILD: Total
number of children, OWNER: Owner occupier, SINGLE: Single adult household, MULTIPLE: Multiple (>2) adult household. Sargan: Test of overidentifying restrictions (degrees of freedom in parentheses). Sample: 95
grouped observations.
Test for the joint significance of cohort dummies (x2): 12X3p-value 12%.
Instruments.
Age, Cohort Dummies, Kl, NCHILD, real interest rate lagged two years,
two year lags of the first difference in the interaction of real expenditure
with: Kl, NCHILD, HWORK, WWORK, SINGLE, MULTIPLE, OWNER
and the second lags of OWNER, real income growth and real consumption
growth.
imply less intertemporalsubstitutionfor the referencehousehold which is a childless
couple living in rented accommodationwith neither spouse in employment. The value
for the intertemporalsubstitutionelasticity, 1, presentedat the bottomof the table is the
value given by equation (2.14) evaluated at the sample mean. In all cases we use
instrumentslaggedtwo periods;the Sarganstatisticis a test of the orthogonalityconditions
(or over-identifyingrestrictions)for these instruments.
The first column of TableIII gives our most general base model; it includes both
laboursupply and demographicvariables. As can be seen, the test of the over-identifying
restrictionsis borderlinebut the actualp-valueis higherthanfor similartests on aggregate
data. The next two columns focus on the interactionbetween labour supply variables
and the presence of an infant in the household. As is well known there is a strong
correlationbetween the latter and female employment;the effect of this can be seen in
columns 2 and 3. Excludingthe labour supply variablesincreasesthe (absolutevalue)
of the coefficienton infants a great deal but leaves the estimateof the ISE unchanged.
BLUNDELL ET AL.
HOUSEHOLD EXPENDITURES
69
TABLE IV
Excess sensitivity tests
PO
HWORK
WWORK
Kl
NCHILD
OWNER
SINGLE
MULTIPLE
INCG
(1)
(2)
(3)
(4)
(5)
(6)
-4-559
(4.123)
2-066
(1.609)
2-252
(1.368)
-0 444
(0.943)
0 439
(0.198)
0-180
(0.955)
1-869
(3.925)
1-235
(1.124)
0 545
(0.368)
-2-185
(3.741)
2-182
(1.363)
2-253
(0.904)
-5 474
(3.562)
-2-360
(2.832)
3 430
(1.219)
3-660
(0.862)
0 740
(0.797)
0-467
(0.188)
-0 353
(0.777)
-0-723
(2.258)
0-714
(0.898)
8-479
(1.776)
0-462
(0 662)
1-501
(0.455)
0 007
(0.407)
0-219
(0.089)
0-160
(0.362)
0 577
(1.111)
0 707
(0.401)
7-563
(2.137)
0-331
(0.710)
1-510
(0.450)
0 044
(0.398)
0-218
(0.087)
0-229
(0.369)
0-122
(1.248)
0-624
(0.402)
0.110
(0.172)
0-082
(0.007)
-0 039
(0.009)
9.6 (13)
73%
-1.11
(0.17)
1-127
(0.724)
0-267
(3.194)
0.091
(0.760)
0-276
(0.397)
-1-616
(0.655)
0 359
(0.157)
1-598
(0.584)
1-917
(2.216)
1-487
(0.779)
1.115
(0.305)
D1980
d -s
Sargan
p-value
4)
0*007
(0.018)
25.6 (13)
2*6%
-0-64
(0.11)
0 009
(0.011)
29.0 (15)
1-6%
-0-71
(0.14)
-0-025
(0.010)
28.4 (15)
1.9%
-0.55
(0.07)
0-041
(0.012)
25.7 (15)
4%
-0-78
(0.09)
0-084
(0.007)
-0 039
(0.009)
9.2 (14)
82%
-1-21
(0.13)
(7)
10-070
(1.29)
0-320
(0.102)
0 093
(0-0056)
-0*046
(0.0039)
26-4 (20)
15%
-1.17
(0.12)
Notes: INCG: Growth rate of household income. D1980: Dummy equal to one pre 1981. See also notes for
Table 3.
Instruments: Columns (1), (2) and (3) as in Table III. Columns (4), (5) and (6) include D1980.
Excess sensitivity test for model in column (4) 0-27 (0.48).
We prefer the model in the first column but since some will be sceptical of including
laboursupplyvariablesin the Eulerequationwe presentbelow estimateswithand without
them. The test statisticfor the exclusion of the laboursupply variablesin column 1 has
a p-value of 4-7%.
There are at least two good reasons why the labour supply variablesin Table III
might be significant. Within the maintained model the obvious explanation is that
intertemporalallocationdoes indeed depend on labourforce participation;for example,
there are positive costs of going to work. An alternativeexplanationwhich lies outside
the theoreticalmodel is that consumptiongrowthis correlatedwith anticipatedincome
growthwhich is in turn correlatedwith labourforce participation.To look at this issue
of "excess sensitivity"we present some estimates in TableIV of models that include
(instrumented)real income growthas an additionalvariable.10
Comparingthe first column of Table IV with the first column of Table III we see
that including anticipatedincome growth(the INCG variable)does not do very much;
the extra variableis itself insignificantand the other parameterestimatesare not much
affected. The result that there is no evidence of excess sensitivityif we condition on
demographicsand laboursupply has also been found by Attanasioand Browning(1991)
and Attanasioand Weber(1993). Excludingthe demographicvariablesas in column (2)
10. See Deaton (1987) and Hayashi (1987).
70
REVIEW OF ECONOMIC STUDIES
does little to change this result. If, however,we exclude the labour-supplyvariablesas
in column (3) then the coefficienton INCG does become "significant";this reflectsthe
strongcorrelationbetween these two sets of variables.
As shown in Figure2, in 1981the real interestratejumped to a higheroveralllevel
at which it stayed for the remainderof the 1980s. Moreover,real consumptiongrowth
showed no correspondingincrease,even allowing for a reasonabletime lag. A possible
interpretationis thatthe macroeconomicpolicies thattriggeredthe increasein the interest
rate also led to a fall in the conditionalvarianceof income relativeto the rate of time
preference(d - 8). To investigatethe effects of allowing for such an interceptshift we
includea dummy(D1980) whichtakesthe value 1 before 1981.11 The resultsarepresented
in TableIV:12 column (4) reproducescolumn (1) of TableIII with the 1980 dummy
included in the instruments. This does result in a shift of the parameters(with large
effects on those most impreciselyestimated) but the main effect seems to be a large
reductionin the standarderror. However,the ISE at the meanpoint displayslittle change.
Includingthe 1980 dummyin the equationhas a dramaticeffect. First,the estimate
of the ISE increasesby about 50%;this is what we would expect given the differences
beforeand after1980in Figure2. Second,includingthe 1980dummychangesthe estimate
of (d - 8); before it was constrainedto be equal for the whole period and was estimated
to be0O041.Now it is 0 045 before 1981and -0 039 after1980. Third,the Sarganstatistic
is muchlower;thus includingthe 1980dummyclearsup some of the correlationbetween
consumptiongrowthand lagged variablessuggestedby the Sarganstatisticsin TableIII.
The inclusion of the 1980 dummychanges some of the other parameterestimatesa
good deal (comparecolumns4 and 5 of TableIV). Havingnoted the impactof the 1980
dummy it is worth stressingthat it has no effect on the significanceof the anticipated
income variable;comparethe coefficienton INCG in columns 6 and 1 of TableIV. In
view of the impactof the D1980 dummy,we presentbelow elasticitiesderivedfromboth
specifications.
3.5. Comparisons with a simple model
Thispaperpresentsa numberof innovations:we have allowedfor intratemporalpreferences when estimatingpreferencesabout allocation over time and we have allowed the
latterto depend on demographics,laboursupplyvariablesand the level of consumption
itself. We now considerthe following question:how differentwould our conclusionsbe
if we took an alternativesimplerformulationand estimatedthem on our data?
We addressthis questionin three steps. First,we look at how importantit is to use
the price indices derivedfrom the demandsystemratherthan some other ad hoc indices
commonlyused. In the second step we considerhow much differenceit makesto allow
the ISE to varywiththe level of consumption,given we allow it to varywith demographics
and laboursupply variables. Finally,we investigatehow importantit is to conditionon
the lattervariables.
For the firststep of our investigationswe replacethe a() price index in (2.12) by
a weighted geometricmean of prices, where the weights are household-specificbudget
shares(that is, a household-specificStone price index); this allows for some dependence
of the deflatoron family compositionand relativeprices (in so far as they affectbudget
11. The asymptotics here require that the number of observations pre- and post-1980 is large enough to
average out aggregate shocks within each sub period.
12. We postpone discussion of column 7 of Table IV until the next sub-section.
BLUNDELL ET AL.
HOUSEHOLD EXPENDITURES
71
shares). To deal with the b(*) index we note that it depends only on relativeprices so
that it is likely to be of only second-orderimportance;we simplyset it equal to unity for
all households. Replacing the price indices in (2.12) in the way describedabove and
setting 6 to 0-54 leads to some minor changesin the parameterestimates(not reported)
but the broad outlines of our resultsremainunchanged. The only changeof note is that
the coefficienton INCG (the expectedincomegrowthvariable)increasesand its standard
errorfalls; not enough, however, to make the coefficient"significant".'3Fromthis we
conclude that there is not much sensitivityto the precise form of the price indices. In
particularA ln b(p) is very small at the cohort level relativeto the interestrate (see the
consumptionfunction (2.12)). Since A ln b(p) is close to zero and, since the Stone price
index at the cohortlevel is a good approximationof ln a (p), we concludethat all aspects
of intertemporalsubstitutioncan be directlyidentifiedusing total real consumption,at
least for our data set.
Acceptingthat for our sampleperiodwe do not need to use moresophisticatedprice
indices than those introducedin the last paragraphwe now go on to investigatewhether
a simpleiso-elasticformgivesmuchthe sameresultsas ourmorecomplicatedspecification.
To do this, we defineconsumptionC, as total nominalexpendituredividedby our Stone
priceindex and we estimatethe followingregressionusingthe sameinstrumentsas before:
A[ln Ct] , =
ln Ct], + (d -8) + [ut,]
where the variables in square brackets are again cohort means. The implied ISE is
= - Yo/(1 - k YkZk)- The results of this regressionare presentedin TableV, columns
1 to 4. In all cases the estimatedISE is lower than the mean ISE for the corresponding
specification;sometimesmuch lower (for example,comparecolumns 1 in TablesIII and
V). Thusit seemsthat if we constrainthe ISE to be constant,the estimatedvalueseriously
underestimatesthe mean of the ISE when we use a flexible specification. Also of note
is the fact that the 1980 dummy is less well-determinedand there appearsto be some
significantexcess sensitivitywhen we use this dummy. We conclude that constraining
the ISE to be constantmay lead to erroneousinference.
Our final experimentconsists in droppingthe conditioningvariables. These results
are reportedin column 7 of Table IV for our specificationand in columns 5 and 6 of
Table V for the iso-elastic specification.
Comparingcolumns 6 and 7 of Table IV we see that droppingthe labour supply
variablesand demographicsdoes not change our estimateof the mean of the ISE very
much. On the other hand, as expected,the coefficienton INCG is now significant.For
the iso-elasticmodelwe see thatwhenwe dropthe conditioningvariablesthe 1980dummy
becomesinsignificantand thereis strongevidenceof excess sensitivity. Moreoverfor the
first time in all our specificationsthe ISE is not well-determined(if we allow for the
excess sensitivity). Referringback to the discussionat the end of Section 3.1 we see that
this may explain why investigatorsusing aggregatedata or using micro data without
controllingfor demographicsand laboursupply have usually found only weak evidence
of intertemporalsubstitution.
Our interpretationof these resultsis that in terms of estimatingthe ISE it may be
misleadingto use an iso-elasticspecification.On the otherhand, ignoringdemographics
and labour supply variablesdoes not seem to lead to much bias in the estimateof the
Yo[ it]c + YkvkA[Ztk
13. A disadvantage of our Box-Cox transformation is that even when A In b(p) is zero the 0 parameter
remains identified from the Euler equation and hence its choice can affect even then the estimated ISE. In our
experiments relating to the sensitivity to the choice of price indices we have just considered 0 = 054 as implied
by estimated curvature of the Engel curves. This retains comparability with our complete specification.
72
REVIEW OF ECONOMIC STUDIES
TABLE V
An isoelastic specification
Comparison
Table (col)
Real rate
HWORK
WWORK
Kl
NCHILD
OWNER
SINGLE
MULTIPLE
(2)
(3)
(4)
A5)
III (1)
0*178
(0.057)
0 309
(0-070)
0*068
(0.094)
0.045
(0.061)
0*034
(0-010)
-0*037
(0.056)
0*231
(0-262)
04024
(0.070)
IV (1)
0-193
(0.051)
0*208
(0.096)
0.101
(0.084)
0*061
(0.056)
0-023
(0-010)
0-004
(0.048)
0 054
(0.283)
0-067
(0.063)
0-143
(0.142)
IV (5)
0-607
(0'209)
0-244
(0.070)
0-096
(0.070)
0-034
(0.056)
0-038
(0-010)
-0-026
(0.056)
0-178
(0-188)
0-140
(0.064)
IV (6)
0*546
(0.207)
0'143
(0.081)
0.109
(0.064)
0-046
(0.050)
0*024
(0.010)
0-013
(0.046)
0*003
(0-182)
0*068
(0.051)
0*253
(0.120)
0*042
(0.023)
-0*015
(0.017)
10.0 (13)
69%
-0*73
(0.21)
IV (7)
0-451
(0-377)
1-05
(0-41)
0'537
(0.196)
0-035
(0.041)
-0-019
(0.021)
0 (1)
99%
-0-45
(0.38)
0-102
(0.043)
0 046
(0.023)
4-2 (2)
12%
-1-05
(0-41)
INCG
D1980
d -a
Sargan
p-value
(6)
(1)
0*009
(0.007)
14.6 (14)
40%
-0-20
(0.065)
0.009
(0.001)
11.4 (13)
58%
-0-21
(0.056)
0-051
(0.023)
-0-017
(0-018)
13-8 (14)
46%
-0-96
(0.36)
Notes: Dependent variable is real consumption growth deflated by a Stone price index defined on the included
goods.
Instruments: Columns (1) and (2) as in Table IV including (INCG). Columns (3) and (4) include D1980,
Columns (5) and (6) only include consumption growth, the real interest rate, real income growth all lagged
two years and D1980.
mean of the ISE. Nevertheless,the ISE does varysystematicallywith these variablesand
the results of our excess sensitivitytests depend criticallyon whetherwe include such
variables.
3.6. The implications of our estimates for intertemporalsubstitution
In Table VI we presentthe estimatesof the ISE implied by three of our specifications
above;the preferredmodel withoutthe 1980dummy(TableIII, column1) which we call
model 1; the preferredmodel with the 1980dummy(TableIV, column5)-model 2-and
the preferrediso-elastic model (Table V, column 4)-model 3.
Consideringthe top panel in Table VI we see that systematicallythe estimatesfrom
model 2 are higher (in absolute value) than those from model 1. For both models the
importantvariableseems to be female participation:generallythe ISE is higher if the
wife is in employment.
The middlepanel in TableVI givesthe distributionof the ISE over ourwhole sample
whereasthe lower panel gives the distributionallowing consumptionto vary but fixing
all demographicsand labour supply variablesfor each household at the sample mean.
BLUNDELL ET AL.
HOUSEHOLD EXPENDITURES
73
TABLE VI*
Elasticitiesof intertemporal
substitution(4)
(a) Elasticitiesat groupmeans
WWORK
HWORK
OWNER
C(61
r(l)
(2)
Iso-elastic
-0 75
(0.12)
-0-61
(0.12)
-0-61
(0.11)
-0-69
(0.10)
-0-68
(0.09)
-0-82
(0.23)
-0 74
(0.16)
-0-81
(0.14)
-1-21
(0.13)
-1-45
(0.32)
-1-22
(0.20)
-1-18
(0.14)
-1-08
(0.10)
-2-21
(0.82)
-1-43
(0.27)
-1-32
(0.16)
-0-96
(0.36)
-0 70
(0.24)
-0-66
(0.22)
-0 94
(0.37)
-0*88
(0.33)
-0-84
(0*26)
-0 77
(0.24)
-1.10
(0.45)
-0 77
-1.19
-1.01
0-67
0-89
0 58
14-65
0
0
0
10-68
0
0
1
12-75
0
1
0
13-71
0
1
1
15-49
1
0
0
9-83
1
0
1
12-74
1
1
0
13-99
1
1
1
15-75
(0.12)
(0.11)
(0.39)
(b) The distributionof the elasticitiesover the entiresample
e1(1)
(D(2)
M1o
M25
M50
M75
M90
%positive
-1.10
-2*8
-0*81
-1.8
-0 76
-1.4
-0 70
-1-14
-0-66
-0-96
5*0%
0%
(c) The distributionof the elasticitiesat averagehouseholdcharacteristics
4 (1)
(D(2)
M1o
M25
M50
M75
M90
%positive
-0*85
-0*80
-1.9
-0 77
-1*4
-0 74
-1*13
-0 72
-0 96
0%
5-7%
-2*9
Mi is the i-th percentile.
Notes: Asymptoticstandarderrorsin parentheses.Definitionsof variablesas in TableIII. Column(1) and
4 (1) referto the model in column (1) of Table III. Column(2) and 4 (2) referto the model in column(5)
of Table IV. "Iso-elastic"refersto the model in column (1) in TableV.
*
Threethings standout fromthese distributions.First,the estimateswiththe 1980dummy
are not only higherin meanbut also muchmorespreadout. Indeed,some of the estimates
are positive which violatesthe concavityof the utilityfunction. Second,the distributions
are fairly similar across the two panels; this suggests that most of the variationin the
ISE across the population is due to differencesin consumption(which can loosely be
thought of as a proxy for lifetime wealth) and not to differencesin demographicsand
laboursupplyvariables.This is consistentwiththe resultsreportedin the last sub-section.
4. SUMMARY AND CONCLUSIONS
The life-cyclemodelplays an importantpartin ourunderstandingof consumerbehaviour.
It providesa comprehensiveframeworkfor analyzingthe relationshipbetweenintertemporal consumptionand intratemporalexpenditureallocations. This paper provides a
rigorousanalysisof the interactionbetweenthese two stagesof the consumer'sbudgeting
74
REVIEW OF ECONOMIC STUDIES
problemusing a time-seriesof repeatedcross-sectionscoveringsome 70,000households
in the U.K. over the 1970's and 1980's. We identify intertemporalbehaviourusing a
pseudo-panelof groupedannualcohortdataconstructedfromour repeatedcross-section.
Our principalconclusionscan be summarizedas follows: '
(a) Althoughhomotheticityis stronglyrejectedat the demandsystemlevel, a single
price index is sufficientto describeintertemporalallocationsin our sample. This
is because the rate of change of the homogeneous-of-degree-zero
price index
b(p), is in fact very small. Moreover using a Stone price index to deflate
consumptiongives very similarresultsto using the exact price index estimated
from the within-periodallocations.
(b) Allowingthe intertemporalelasticityof substitutionto vary with consumption,
as opposed to using an iso-elastic specification,is importantfor the estimates.
This extra degree of flexibilityleads to higher estimatesof the ISE. Moreover
we find that the ISE is very well determined.
(c) An importantinnovationof our approachis the systematicanalysisof the effects
of demographiccharacteristicsand labour marketvariables on intertemporal
behaviour.We findthatdemographiccharacteristicsand labourmarketvariables
have significantbut relativelysmall effectson intertemporalallocations. This is
truedespitethe fact thatthe priceindex deflatingconsumptionis itself a function
of these conditioningvariables.
(d) In testing the validity of the life-cycle model we included the growth rate of
income as an explanatoryvariableand performedstandardexcess sensitivity
tests. Once we control for labourmarketstatuswe find no excess sensitivityin
our full specificationalthoughthereis some evidenceof excess sensitivityin the
iso-elasticmodel.
The last of these findings is open to a variety of interpretationswhich cannot be
distinguishedconvincinglywithin this framework. First, it is quite possible that the
importanceof labour marketvariablesin the intertemporalmodel does in fact reflect
shiftingtastesas a functionof labourmarketstatus;since labourmarketstatusand growth
rate of income are obviously correlatedignoringthe formermakes the latterspuriously
significant. For the same reasonsthe reverseis also possible: labour marketstatus is a
good predictorof income growthand thus labourmarketstatusmay just captureexcess
sensitivity. Nevertheless,we should note that the labour marketvariablesremain significanteven in the presenceof income growth.
BLUNDELL ET AL.
HOUSEHOLD EXPENDITURES
75
DATA APPENDIX
A full description of the Family Expenditure Sata codes used to construct the variables is available from the
authors on request.
TABLE D.1
Descriptive statistics for demographic characteristicsand expenditures
Year
Female works
Male works
Married
Age of head
Child 0-2
Child 2-5
Child 5-10
Child 11-16
Child 17-18
Number of adults
Expenditure (x)
Share of food
Share of alcohol
Share of clothing
Share of fuel
Share of transport
Share of services
1970-1974
1975-1979
1980-1984
1985-1986
0 597
(0.490)
0-931
(0.252)
0-827
(0.378)
40-843
(11 00
0-218
(0.470)
0-152
(0.380)
0 457
(0.780)
0-328
(0.650)
0 040
(0.206)
2-083
(0.648)
31-436
(20.40)
0-364
(0.122)
0-061
(0.074)
0 109
(0.105)
0-081
(0.059)
0-169
(0.142)
0-108
(0.096)
0-699
(0.458)
0 900
(0.299)
0-792
(0.405)
40-308
(11.00)
0-174
(0.428)
0-128
(0.353)
0-413
(0.726)
0 339
(0.664)
0 044
(0.215)
2-054
(0.680)
63 655
(42.40)
0 352
(0.126)
0-068
(0.078)
0.100
(0.102)
0-081
(0.058)
0-170
(0.140)
0.113
(0.102)
0-682
(0.465)
0-832
(0.373)
0 757
(0.428)
40273
(1080)
0*173
(0.433)
0-108
(0.320)
0-346
(0.658)
0 339
(0-651)
0-060
(0.253)
2-042
(0.710)
113-643
(76.40)
0-321
(0.120)
0-067
(0.077)
0.093
(0.097)
0 091
(0.065)
0-185
(0.148)
0.126
(0.109)
0-696
(0.459)
0 797
(0-402)
0-711
(0.452)
39-732
(10.60)
0-170
(0-424)
0-114
(0.337)
0-317
(0-640)
0-283
(0-585)
0-054
(0.239)
1.997
(0.740)
142-404
(103.0)
0 307
(0-117)
0-068
(0.078)
0-096
(0.100)
0-096
(0.069)
0-173
(0.140)
0.139
(0.119)
Sample standard deviations in parentheses
76
REVIEW OF ECONOMIC STUDIES
APPENDIX A. THE PARAMETERS OF THE ESTIMATED DEMAND SYSTEM
(x1OO)
The demand system has the form
Wih
= aih +Ej yij in Pjt + ih ( X/ a (p))
+ Vih
where
alh = aio+Yk
fi6h =
Zkhaik,
13i0+Yk
Vij = Vji
ZkhlPik,
TABLE A
Parameters and standard errors (x 100)
Food
Alcohol
Clothing
Fuel
Transport
Services
8*65
(1.31)
-1-87
(0.81)
-2-79
(1.20)
-0 55
(0.71)
-4-88
(1.43)
2-68
(1.02)
-3.45
(1.00)
1*92
(0.89)
3-62
(0.68)
3-13
(1.22)
-1-39
(0.95)
5-77
(1.49)
-1 50
(0*80)
-2-31
(1.51)
-4-72
(1.06)
3-32
(0.81)
-6 58
(1.04)
-0*01
(0.76)
3 49
(3.04)
7 57
(1.63)
0.25
(1.58)
46-39
(3.95)
9-89
(3.40)
4-93
(3.46)
12'13
(3.46)
7-12
(3.53)
-1.34
(1.96)
10*74
(1.99)
4-84
(1.93)
0-66
(0.27)
-0-27
(0.06)
2*19
(0.10)
2*22
(0.11)
2*72
(0.07)
2-76
(0. 10)
2*74
(0.23)
1-92
(0.28)
4-60
(3.29)
-2-32
(2.77)
-2-02
(2.81)
1-22
(2.82)
-3-55
(2.88)
1-30
(1.57)
4*20
(1.57)
-5.14
(1.53)
-0-73
(0.21)
-0*02
(0.05)
-1*09
(0.08)
-1*03
(0.09)
-0*95
(0.05)
-0*78
(0.08)
-1.10
(0*18)
2-50
(0*22)
8-64
(4.30)
-2.81
(3.64)
0-14
(3.69)
-4 93
(3.70)
-0 70
(3.78)
-1.47
(2.05)
2*70
(2.09)
-1.18
(2.01)
0-02
(0.29)
0 07
(0.06)
0*31
(0.10)
0.15
(0.12)
0*20
(0.07)
0.51
(0.11)
0*44
(0.24)
0-26
(0.29)
18-33
(2.35)
1-57
(1.98)
-0.17
(2.01)
-0*52
(2.01)
1-85
(2.05)
0*35
(1.15)
-9*25
(1.14)
-1.34
(1*12)
0-02
(0.15)
0-03
(0.03)
0 94
(0.06)
0.56
(0.07)
0 25
(0.04)
0*06
(0.06)
0*16
(0.14)
-0*56
(0.17)
6-08
(2.26)
-14 07
(4.14)
-6.32
(4.18)
-6.29
(3.98)
-11-26
(4.10)
5*82
(2.50)
15-83
(2.93)
4-86
(2.64)
-0-65
(0.40)
0.19
(0.09)
-1*88
(0.15)
-1*48
(0.17)
-1*21
(0.10)
-0*86
(0.15)
-0*37
(0.33)
1-30
(0.40)
3.93
(4.27)
7-14
(3.83)
1-46
(3.87)
-1-00
(3.85)
6-81
(3.93)
-2*03
(2.15)
-0*47
(2.20)
-2*20
(2.13)
0-68
(0.30)
0.10
(0.07)
-0*59
(0.11)
-0*29
(0.13)
-0*91
(0.07)
-1.79
(0.11)
-1*77
(0*26)
-4.55
(0.31)
y parameters
P Food
P Alcohol
P Clothing
P Fuel
P Transport
P Services
a parameters
Constant
Quarter 1
Quarter 2
Quarter 3
Oct and Nov
HWORK
WWORK
Married
Age
Age2
KI
K2
K3
K4
K5
Number of Adults
BLUNDELL ET AL.
HOUSEHOLD EXPENDITURES
77
TABLE A-continued
NORTH
YORKSHIRE
EAST MIDLANDS
EAST ANGLIA
LONDON
SOUTH EAST
SOUTH WEST
WALES
WEST MIDLANDS
NORTH WEST
COUNCIL HOUSE
RENT UNFURN
RENT FURN
OWNER MORT
OWNER NO MORT
CENT HEAT
TREND
COH 1911-15
COH 1916-20
COH 1921-25
COH 1926-30
COH 1931-35
COH 1936-40
COH 1941-45
COH 1946-50
COH 1951-55
COH 1956-60
WHITE COLLAR
HOUSE DURS
CARS
Food
Alcohol
Clothing
-0*30
(0.20)
0 30
(0.19)
0-02
(0.19)
0*37
(0.23)
0-84
(0.17)
0 11
(0.17)
0-63
(0.18)
0-76
(0.18)
0 04
(0.19)
-0 13
(0.18)
0-88
(0.25)
0 73
(0.27)
-0 54
(0.32)
0.10
(0.24)
0-56
(0.27)
-0-76
(0.10)
-3 70
(0.56)
1.90
(1.40)
1-46
(1.27)
1-46
(1.16)
1-33
(1.04)
1-12
(0.93)
0 94
(0.83)
0 39
(0.72)
0 34
(0.64)
0-21
(0.56)
-0-15
(0.53)
-1-13
(0.12)
-1-43
(0.22)
-3-25
(0.17)
1*18
(0.16)
0 34
(0.15)
0.11
(0.15)
-1*10
(0*18)
-1 03
(0-13)
-1 47
(0.14)
-1.10
(0.14)
-0-86
(0.14)
-0-41
(0.15)
0-28
(0.14)
1 38
(0.19)
1-40
(0.22)
2-78
(0.25)
0 55
(0.19)
0-17
(0.21)
-0 39
(0.08)
0-68
(0.45)
0-62
(1.10)
0.55
(1.00)
0-32
(0.91)
0-52
(0.82)
0 39
(0.74)
0.50
(0.65)
0-60
(0.57)
0 44
(0.50)
0 73
(0.45)
0-36
(0.41)
-0-80
(0.09)
-0-83
(0.18)
-0-88
(0.13)
-0*14
(0.21)
-0 53
(0.19)
-0-80
(0.20)
-0 97
(0.24)
-0-80
(0-18)
-1 18
(0-18)
-0 90
(0.19)
-0-56
(0.19)
-0-66
(0.20)
-0-72
(0.19)
-0-06
(0 26)
-0.00
(0.29)
0 94
(0.33)
0-38
(0.26)
0-56
(0 28)
0-21
(0.10)
1-26
(0.66)
-3.39
(1.46)
-3-02
(1.33)
-2-88
(1.21)
-2-70
(109)
-2-60
(0.97)
-2-49
(0.86)
-2-24
(0.76)
-2-25
(0-66)
-2-01
(0.59)
-1-29
(0.55)
0-26
(0.12)
0.72
(0.24)
-1-74
(0.17)
Fuel
-057
(0.11)
-051
(0.11)
-0-22
(0.11)
0*01
(0.13)
-059
(0.10)
0-02
(0.10)
-0.30
(0.10)
-0-41
(0.10)
-0 30
(0.11)
-0-02
(0.10)
0-86
(0.14)
0.01
(0.15)
-2-32
(0.18)
0 55
(0.14)
0-32
(0.15)
0-13
(0.06)
0.50
(0.36)
3-29
(0.79)
2-80
(0.72)
2-64
(0.66)
2-46
(0.59)
2-33
(0-53)
2-23
(0.47)
2-06
(0-41)
1-92
(0.36)
1-39
(0.32)
1-14
(0.30)
0-06
(0.07)
0-25
(0.13)
-0-61
(0.10)
Transport
-056
(0.30)
-0 95
(0.28)
-0-83
(0.29)
-0-61
(0.35)
0 11
(0.25)
-0 09
(0 26)
0-06
(0.28)
-0-51
(0.27)
-0-48
(0.29)
-0 37
(0.27)
-1 63
(0.36)
-1V64
(0.41)
0-46
(0.48)
-1-31
(0.36)
-1-63
(0.40)
0 54
(0.15)
-0-21
(0.75)
-2-20
(2.06)
-1-56
(1.88)
-1-07
(1.70)
-0 99
(1.54)
-0 90
(1.37)
-0-97
(1-22)
-0-87
(1.07)
-0-91
(0.95)
-0.51
(0.84)
-0-17
(0.79)
-0-63
(0.17)
0-82
(0.33)
10-04
(0.24)
Services
017
(0.22)
0-52
(0.21)
0 59
(0.21)
0 43
(0*26)
0-52
(0.19)
0 54
(0.19)
0 55
(0.21)
0-20
(0.20)
0-52
(0.21)
0 35
(0.20)
-0*11
(0.27)
0-46
(0.30)
0-31
(0.36)
0*51
(0.27)
0-82
(0.30)
0 34
(0.11)
-0*45
(0-55)
-0-17
(1.56)
0-14
(1.42)
0-07
(1.29)
-0-24
(1.16)
0-16
(1.04)
0-24
(0.92)
0-17
(0.81)
0-40
(0.71)
0-24
(0.63)
0.55
(0.58)
1-61
(0.13)
-0-28
(0.25)
-2.28
(0.18)
REVIEW OF ECONOMIC STUDIES
78
TABLE A-continued
Food
,8 parameters (C
Alcohol
Clothing
-0-18
(0.30)
-0-03
(0.25)
-0*02
0*61
(0.40)
-0 09
(0-33)
-0-27
(0.33)
0-20
(0.33)
-0-08
(0.34)
0-22
(0.19)
-0-21
(0.24)
0 03
(0.21)
Fuel
Transport
Services
0-28
(0.29)
1-42
(0.37)
0 70
(0 37)
0-74
(0.36)
1'13
(0.37)
-0 45
(0.23)
-1-52
(0.34)
-0.50
(0.28)
1-54
(0.40)
-0-28
(0.34)
0-25
(0.34)
0-52
(0.34)
-0 37
(0-35)
0-17
(0-20)
-0-01
x/a(p))
=
C
-1-42
(0.36)
-0-67
(0.30)
-0-26
(0.31)
-0 93
(0.31)
-0-49
(0-32)
-0.01
(0.18)
1.00
(0.23)
-0-23
(0.20)
CxQUAR1
CxQUAR2
CxQUAR3
Cx(OCT-NOV)
CxWWORK
CxHWORK
CxMARRIED
(0.25)
-0-31
(0.25)
0*13
(0.26)
-0.06
(0.14)
-0-23
(0.18)
0 49
(0.16)
-1.11
(0.22)
0 04
(0.18)
0*15
(0.18)
0-04
(0.18)
-0 18
(0-18)
-0-11
(0.10)
0 75
(0.13)
0-11
(0.12)
(0.25)
0-10
(0.22)
0 54 (0.45)
0
Notes: Asymptotic standard errors in parentheses. All parameters and standard errors are multiplied by 100.
The parameters of the seventh commodity (OTHER) are implied by adding up. AGE = (AGE-40)/10. Default
region: Scotland. Default season: Christmas. Default housing: Rent Free.
Instruments: Seasonal dummies, regional dummies, cohort dummies, asset income, asset income squared,
married, age and age squared of head of household, male and female employment status, child dummies,
housing tenure dummies, number of adults, central heating dummy, occupation, durables and car ownership
dummy, trend, trend squared and interactions of asset income with child dummies, employment status, married
and seasonal dummies.
APPENDIX B. THE VARIANCE COVARIANCE MATRIX
OF THE ESTIMATOR
We represent the Euler equation by the regression function
y(C) = p'X(C)i, + uit
where we denote the parameter vector of the demand system by g and where
W()t = A
I
?jC
c[(1+O)(lnx,j-(In
a(p| )),j)+(In b(pI|r))j]
-[In (1+r,,)-A
(In a(p|
)X ],
(B.1)
=(A
f{[ln (x,j/a(p IC,j)]01--In
z=1(
b(p
I?1}).
(B.2)
In the above c denotes a cohort, j an individual and t a time period. Nct is the size of the cohort in time period
t and Z2 is the i-th characteristic included in the Euler equation, for the j-th individual in the t-th time period.
To compute the asymptotic standard errors we need the derivatives of y(C) and X(;) with respect to C.
We denote the conditioning characteristics in the demand system by some vector z1. The non-zero elements
of these derivatives have the following form:
C zIj lnPk)t
ay(C)tlaak
= OA
8Y(0)t/YSk
= OA(In pts InPtk)
(Nct
1
1+ 8ks
(B.3)
,~
(B.4)
BLUNDELL ET AL.
= A (I
dy(,/da
=
dx(r)il/8k
Ej
HOUSEHOLD EXPENDITURES
(B.6)
cIn Ctk)
-,A
Pjz
i
79
Pn
in
in
(B.7)
where 5ks iS the Kronecker-8and k is the goods index. The vectorof derivativesis completedby placingzeros
at the correct positions.
Theestimatorof;~hasan asymptoticcovariancematrixV;. Theerrortermof the modelwhenwe condition
on consistentparameterestimates;~can be approximatedto the firstorderby
uX
* = u-A[2X/o(](g-)
+[
ty/di(
p
)
= U+
Q(-p
).
(B.10)
Hence Eu*(u*)I = l;x+QVQp.
Finallythe asymptoticcovariancematrixof the GMM estimatorof p is
X ilk
where
Pk
(X=P-X)
i Z(Z'KrZneZk,rwhere
+ 2(X PX
pXP I[QV nQ]PtX(Xk PX)1,
(B.ll)
Z is the matrix of instruments used for the estimation of the Euler equation.
This covariancematrixis computedby replacingl; by a diagonal matrixwhose diagonal elementsare the
squared residuals from the second step GMM regression. (See White (1980)).
Acknowledgements. This paper was originally circulated as an IFS Working Paper under the title "A
Microeconometric Model of Intertemporal Substitution and Consumer Demand", September 1989. We should
like to thanktwo anonymousreferees,JamesBanks,John Burbidge,Angus Deaton,TerenceGorman,Fumio
Hayashi, Ian Jewitt, Michael Keen, Arthur Lewbel, John Muellbauerand Guglielmo Weber for helpful
suggestions. We are grateful to the Department of Employment for providing the data used in this study.
Finance for this research, under the auspices of the ESRC Research Centre for the Micro-Economic Analysis
of Policy at IFS and from the Canadian SSHRC, is also gratefully acknowledged.
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