heterogeneity and aggregationweb.mit.edu/tstoker/www/blundell_stoker_jel_05.pdf · blundell and...

347

Journal of Economic LiteratureVol XLIII (June 2005) pp 347ndash391

Heterogeneity and Aggregation

RICHARD BLUNDELL and THOMAS M STOKERlowast

lowast Blundell Institute for Fiscal Studies and Department ofEconomics University College London Stoker SloanSchool of Management Massachusetts Institute ofTechnology We are grateful for comments from numerouscolleagues especially Orazio Attanasio Martin BrowningJim Heckman Arthur Lewbel and the reviewers Thanksare also due to Howard Reed and Zoe Oldfield who helpedorganize the data for us Financial support from the ESRCCentre for the Micro-Economic Analysis of Fiscal Policy atIFS is gratefully acknowledged The usual disclaimer applies

1 Introduction

Different people do in fact behave dif-ferently To describe the behavior of a

group one must come to grips with this het-erogeneity In terms of empirical research ineconomics this means facing and resolvingaggregation problems

Aggregation problems are more than acloying annoyance in empirical work Theyexist at virtually every level from the initialissues of data construction and model speci-fication to the subsequent issues of how tousefully summarize and apply resultsBecause of their broad reach aggregationproblems have traditionally been kept clos-eted within the practice of empirical workalong with other issues for which there areno simple answers But recently this ischanging as there has been substantialprogress in dealing with aggregation prob-lems in applied research This survey coversmuch of this development

At the outset we must address a basicquestion why consider aggregation prob-

lems at all One could take the view thateconomics is mainly about the behavior ofindividuals or of individual markets andassert a methodological position that onlyanalysis of such individuals or individualmarkets makes any real sense Howeversuch a view eliminates the applicability ofthe tenets of economic behavior to some ofthe most important questions in economicsnamely those that concern economic aggre-gates Economic policy is most often con-cerned with prices interest rates aggregateconsumption and savings market demandand supply total tax revenues aggregatewages unemployment and so forth Thevaluation and allocation of scarce resourcesrequires that attention be paid to largegroups of individuals

It is important to study relationshipsamong economic aggregates and to bringindividual economic behavior to bear onthose relationships Addressing aggregationproblems just means creating a bridgebetween those behaving individuals and theeconomic aggregates After stating this goalhowever we immediately meet several vex-ing issues as to how to even start to thinkabout linking ldquoindividualrdquo and ldquoaggregaterdquo

At one extreme are the almost philosophi-cal issues of where or at what level to applythe strictures of economic theory Are we toassume that regularities associated withrationality apply to entire economies to

jn05_Article 1 61005 137 PM Page 347

348 Journal of Economic Literature Vol XLIII (June 2005)

1 J R Hicks (1956) p 552 This effort is underway primarily in work on psycho-

logical tendencies in economics finance and marketingThis work examines what is ldquoconsistent economic behav-iorrdquo and whether departures from rational behavior aresystematic or idiosyncratic

3 For instance to study inequality a relevant aggregatewould be the Gini coefficient The choice of aggregate mayeven be informed by empirical regularities in individualdata For example if an individual model is best specifiedwith the logarithm of observed income the geometricmean of income might be a more natural aggregate thantotal income or average income

ldquoreasonably homogeneousrdquo groups of house-holds firms or other types of economicagents or to Hicksrsquos Mr Brown or Mr Jonesas well as any of our own relatives or neigh-bors 1 To assert that there is a ldquocorrectrdquo indi-vidual level at which to apply a mathematicalmodel that is in line with rational behavior isto take a stand on those issues a stand whichcould only be properly validated by experi-mentation or much more extensive empiricalresearch than has been performed to date2

At the other extreme are questions per-taining to what the appropriate ldquoaggregaterdquois One typically considers sums or averagesas reported in national income accounts asthe relevant aggregates They are usually themost interpretable numbers and the mostrelevant for pricing or policy analysis Butwith large populations one could considermany other kinds of aggregates or statisticsfrom the population3

Once these issues are settledmdashwhat is therelevant ldquoindividual levelrdquo and what is therelevant ldquoaggregaterdquomdashthen aggregationproblems become purely practical For anyapplication a model must be specifiedwhich captures all important economiceffects allows for relevant individual hetero-geneity and bridges the gap between indi-vidual and aggregate facilitating analysis atboth levels

This survey covers specific solutions andrelated work in primarily three applicationareas consumer demand analysis consump-tion and saving analysis and analysis ofwages and labor market participation A keyissue is to identify what kinds of individual

4 This roughly coincides with the categorization of het-erogeneity discussed in Martin Browning Lars P Hansenand James J Heckman (1999 chapter 8)

differences or heterogeneity are relevantfor each application area As an organizingprinciple we consider (1) heterogeneity inindividual tastes and incomes (2) hetero-geneity in wealth and income risks faced byindividuals and (3) heterogeneity in marketparticipation4 There is a generic tensionbetween the degree of individual hetero-geneity accounted for and the ease withwhich one can draw implications for econom-ic aggregates We point out how differenttypes of heterogeneity are accommodated inthe different application areas

Our approach is practical and we hope toaddress many of the concerns faced byempirical research regarding aggregationWe take a ldquomicro-econometricrdquo view of theindividual modelmdashnamely an econometricmodel (obeying restrictions of economic the-ory) is applicable to individuals or house-holds We consider model specifications thatare typically used in empirical analysis ofindividual data in each application area Wetake the relevant ldquoaggregatesrdquo to be eithertotaled or per-capita (averaged) values of theindividual variables of interest coincidingwith aggregates as typically reported forregions or whole economies Whether suchaggregates are easy to model or not they arethe most interpretable and the most usefultypes of aggregates for interpretation policyanalysis or other uses of empirical results

We are concerned with models that strike abalance between realism (flexibility) adher-ence to restrictions from economic theoryand connections between individual behaviorand aggregate statistics We consider severalsettings where individual models are intrinsi-cally nonlinear and for those we must makespecific assumptions on the distributions ofrelevant heterogeneous characteristics Wepresent results that can be used to explore theimpact of heterogeneity in empirical applica-tions that assume reasonable (and hopefully


Blundell and Stoker Heterogeneity and Aggregation 349

plausible) parametrizations of both individualequations and distributions of heterogeneityWe do not go into details about estimationbut for each application area we will presentmodels with empirically plausible equationsfor individuals and consistent equations forthe relevant economic aggregates The pointis to have the ability to address empiricalissues at the individual (micro) level theaggregate (macro) level or both

We begin with our coverage of consumerdemand models in section 2 the area whichhas seen the most extensive development ofsolutions to aggregation problems The diffi-cult issues in consumer demand includeclear evidence of nonlinearity in incomeeffects (eg Engelrsquos Law for food) and per-vasive evidence of variations in demand withobservable characteristics of households Wediscuss each of these problems in turn anduse the discussion to cover traditional resultsas well as ldquoaggregation factorsrdquo as a methodof empirically studying aggregation bias Wecover recent empirical demand models andpresent aggregation factors computed fromdata on British households That is we coverthe standard issues faced by aggregatingover heterogeneous households in a staticdecision-making format and illustrate withapplication to empirical demand models incurrent use We close with a discussion ofrecent work that studies aggregate demandstructure without making specific behavioralassumptions on individual demands

In section 3 we discuss models of overallconsumption growth and wealth Here wemust consider heterogeneity in tastes butwe focus on the issues that arise from het-erogeneity in income shocks showing howdifferent types of shocks transmit to aggre-gate consumption We start with a discussionof quadratic preferences in order to focus onincome and wealth and then generalize torecent empirical models that permit precau-tionary saving Because of the log-linearform of these models we must make explic-it distributional assumptions to solve foraggregate equations We cover the types of

heterogeneity found in consumption rela-tionships as well as various other aspects ofour modeling illustrating with empiricaldata We follow this with a brief discussion ofmodeling liquidity constraints and theimpacts on aggregate consumption We closethis section with a discussion of recentprogress in general equilibrium modeling ofconsumption saving and wealth

Section 4 covers recent work on labor par-ticipation and aggregate wage rates Themain issues here concern how to interpretobserved variations in aggregate wagesmdasharethey due to changes in wages of individualsor to changes in the population of participat-ing workers We focus on the issues of het-erogeneity in market participation anddevelop a paradigm that allows isolation ofthe participation structure from the wagestructure This involves tracking the impactsof selection on the composition of the work-ing population the impacts of weightingindividual wage rates by hours in the con-struction of aggregate wages and the impactof observed wage heterogeneity We showhow accounting for these features gives asubstantively different picture of the wagesituation in Britain than that suggested byobserved aggregate wage patterns Here wehave a situation where there is substantialheterogeneity and substantial nonlinearityand we show how to address these issuesand draw conclusions relevant to economicpolicy

Section 5 concludes with some generalobservations on the status of work onaggregation in economics

This paper touches on many of the mainideas that arise in addressing aggregationproblems but it is by no means a comprehen-sive survey of all relevant topics or recentapproaches to such problems For instancewe limit our remarks on the basic nature ofaggregation problems or how it is senseless toascribe behavioral interpretations to estimatedrelationships among aggregate data without adetailed treatment of the links between indi-vidual and aggregate levels It is well known



5 See Thomas M Stoker (1986d 1993) and ArthurLewbel (1994) and others for examples of clear problemsin inferring behavioral reactions from time series results inthe presence of individual heterogeneity

that convenient constructs such as a ldquorepre-sentative agentrdquo have in fact no general justi-fication we will not further belabor their lackof foundation See the surveys by Thomas MStoker (1993) and Browning Hansen andHeckman (1998) for background on thesebasic problems It is useful to mention tworelated lines of research that we do not coverThe first is the work on how economic theoryprovides few restrictions on market excessdemandsmdashsee Hugo Sonnenschein (1972)and Wayne Shafer and Sonnenschein (1982)among others and Donald J Brown and RosaL Matzkin (1996) for a recent contributionThe second is the work on collective decisionmaking within households as pionerred byPierre-Andre Chiappori (1988) and FranccediloisBourguignon and Chiappori (1994)

We will also limit our attention to aggrega-tion over individuals and not discuss thevoluminous literature on aggregation overcommodities This latter literature concernsthe construction of aggregate ldquogoodsrdquo fromprimary commodities as well as the consis-tency of multistage budgeting and other sim-plifications of choice processes While veryimportant for empirical work the issues ofcommodity aggregation apply within decisionprocesses of individuals and as such wouldtake us too far afield of our main themes Seethe survey by Richard Blundell (1988) as wellas the book by Charles Blackorby DanielPrimont and R Robert Russell (1978) forbackground on commodity aggregation andmultistage budgeting Moreover we do notcover the growing literature on hedonicchar-acteristics models which can serve to facili-tate commodity aggregration or othersimplifications in decision making

Finally we do not cover in great detailwork that is associated with time seriesaggregation That work studies how the timeseries properties of aggregate statistics relateto the time series processes of associateddata series for individuals such as stationar-ity cointegration etc To permit such focusthat work relies on strictly linear models forindividual agents which again turn the dis-

6 Provided that intertemporal preferences are additivethis accords with a fairly general intertemporal model ofexpected utility maximization (see Angus S Deaton andJohn Muellbauer 1980b among others)

cussion away from heterogeneity in individ-ual reactions and other behavior We domake reference to time series properties ofincome processes as relevant to our discus-sion of individual and aggregate consump-tion but do not focus on time seriesproperties in any general way Interestedreaders can pursue Clive W J Granger(1980 1987 1990) and the book by MarioForni and Marco Lippi (1997) for morecomprehensive treatment of this literature5

2 Consumer Demand Analysis

We begin with a discussion of aggregationand consumer demand analysis Here theempirical problem is to characterize budgetallocation to several categories of commodi-ties The individual level is that of a house-hold which is traditional in demand analysisThe economic aggregates to be modeled areaverage (economywide per household)expenditures on the categories of commodi-ties We are interested in aggregate demandor how average category expenditures relateto prices and the distribution of total budgetsacross the economy

In a bit more detail we assume that house-holds have a two-stage planning processwhere they set the total budget for the cur-rent period using a forward looking plan andthen allocate that current budget to the cat-egories of nondurable commodities6 Assuch we are not concerned with heterogene-ity in the risks faced by households in incomeand wealth levelsmdashthey have already beenprocessed by the household in their choice oftotal budget (and possibly in their stocks ofdurable goods) We consider commodity cat-egories that are sufficiently broad thathousehold expenditures are non-zero (food



categories clothing categories etc) and sowe are not concerned with zero responses orheterogeneity in market participation

We are concerned with heterogeneity intotal budgets and in needs and tastes It is awell-known empirical fact that categoryexpenditure allocations vary nonlinearly withtotal budget size (for instance Engelrsquos Lawwith regard to food expenditures) Earlyapplications of exact aggregation demand sys-tems had budget shares in semi-log form(with or without attributes) namely the pop-ular Translog models of Dale W JorgensonLawrence J Lau and Stoker (1980 1982)and Almost Ideal models of Angus S Deatonand John Muellbauer (1980a 1980b) respec-tively More recent empirical studies haveshown the need for further nonlinear termsin certain expenditure share equations Inparticular evidence suggests that quadraticlogarithmic income terms are required (seefor example Anthony B Atkinson JoannaGomulka and Nicholas H Stern 1990Herman J Bierens and Hettie A Pott-Buter1990 Jerry A Hausman Whitney K Neweyand James L Powell 1994 Wolfgang HaumlrdleWerner Hildenbrand and Michael Jerison1991 Arthur Lewbel 1991 and BlundellPanos Pashardes and Guglielmo Weber1993) This nonlinearity means that aggregatedemands will be affected by total budget sizeas well as the degree of inequality in budgetsacross consumers It is also well known thatcategory expenditures vary substantially withdemographic composition of householdssuch as how many children are present orwhether the head of household is young orelderly see Anton P Barten (1964) Robert APollak and Terence J Wales (1981) RanjanRay (1983) and Browning (1992)

Our aim is to understand how behavioraleffects for households impinge on priceeffects and distributional effects on aggre-gate demands Understanding these effectsis a key ingredient in understanding how thecomposition of the population affectsdemand growth over time and relative pricesacross the different commodity categories

7 It is common parlance in the demand literature torefer to ldquototal budget expenditurerdquo as ldquoincomerdquo as we dohere In the later section on consumption we return tousing ldquoincomerdquo more correctly as current consumptionexpenditures plus savings

8 For most of our discussion zit can be taken as observ-able When we discuss explicit empirical models we willinclude unobserved attributes random disturbances etc

21 Aggregation of Consumer DemandRelationships

Our framework requires accounting forindividuals (households) goods and timeperiods In each period t individual i choos-es demands qijt (or equivalently expenditurespjtqijt) for j = 1hellip J goods by maximizingpreferences subject to an income constraintwhere i = 1hellip nt Prices pjt are assumed tobe constant across individuals at any point intime with pt = (p1thellip pJt) summarizing allprices Individuals have total expenditurebudget mit = Σj pjtqijt or income for shortand are described by a vector of householdattributes zit such as composition and demo-graphic characteristics 7 The general formfor individual demands is written

(1)

This model reflects heterogeneity in incomemit and individual attributes zit Specificempirical models involve the specificationof these elements including a parametricformula for gjt

8

Economywide average demands and aver-age income are

(2)

We assume that the population of the econ-omy is sufficiently large to ignore samplingerror and represent these averages as theassociated population means

(3)

Our general framework will utilize variousother aggregates such as statistics on thedistribution of consumer characteristics zit

E q j J E mt ijt t it( ) = ( ) 1 hellip and

sum=

sumi ijt

t

i it

t

q

nj J

mn

1 hellip and

q g p m zijt jt t it it= ( )



9 The use of aggregation factors was first proposed byBlundell Panos Pashardes and Guglielmo Weber (1993)

211 Various Approaches ExactAggregation and DistributionalRestrictions

We begin by discussing various approach-es to aggregation in general terms From (1)aggregate demand is given formally as

(4)

where Ft(mitzit) is the cross-section distribu-tion of income and attributes at time t Atthe simplest level approaches to aggrega-tion seek a straightforward relationshipbetween average demand average incomeand average attribute values

(5)

The exact aggregation approach is basedon linearity restrictions on individual prefer-encesdemands gijt that allow the relation-ship Gjt to be derived in a particularly simpleway such that knowledge of Gjt is sufficientto identify (the parameters of) the individualdemand model Take for example

(6)

where we suppose zit is a single variable thathas zit = 1 for an elderly household and zit = 0otherwise Individual demand has a linearterm in income a nonlinear term in incomeand the slope of the linear term is differentfor elderly households All of these slopescan vary with pt Now aggregate demand is

(7)

times

which depends on average income E(mit) andtwo other statistics E(mit lnmit) and E(mitzit)The coefficients are the same in the individ-ual and aggregate models which is thebridge through which individual preferenceparameters manifest in aggregate demands(and can be recovered using aggregate data)

In order to judge the impact of aggrega-tion on demand it is convenient to use

E m mit it( ) +ln bb p E m zj t it it2 ( ) ( )

E q b p E m b pt ijt j t t it j t( ) = ( ) ( ) + ( )0 1

b p m mj t it itln+ ( ) +1 bb p m zj t it it2 ( )

g p m z b p mjt t it it j t it ( ) = ( )0

E q G p E m E zt ijt jt t t it t it( ) = ( ) ( )( )

E q g p m z dF m zt ijt jt t it it t it it( ) = ( ) ( )int

10 For instance in (8) b1j(pt) is the coefficient of mit 1n mit whereas b1j(pi)π1t is the coefficient ofE(mit)1n E(mit) If π1t is stable π1t = π0 then b1j(pi)π1t

is proportional to b1j(pi) In this sense the structure ofaggregate demand matches that of individual demandbut the use of aggregate data alone would estimate theindividual coefficient with a proportional bias of π0

aggregation factors9 Write aggregatedemand as

(8)

times

times

where

(9)

The factors π1t and π2t show how the coeffi-cients in (7) are adjusted if individualdemand is evaluated at average income andaverage attributes as in (8) π1t reflectsinequality in the income distributionthrough the entropy term E(mit lnmit) andπ2t reflects the distribution of income of theelderly as the ratio of the elderlyrsquos share inaggregate income E(mitzit)E(mit) to thepercentage of elderly E(zit) in the popula-tion Aggregation factors are useful for tworeasons First if they are stable then aggre-gate demand has similar structure to indi-vidual demand Second their average valueindicates how much bias is introduced inestimation using aggregate data alone10

In contrast the distributional approachconsiders restrictions on the heterogeneitydistribution Ft(mitzit) Suppose the densitydFt(mitzit) is assumed to be an explicit func-tion of Et(mit)E(zit) and other parameterssuch as variances and higher order momentsThen with a general nonlinear specificationof individual demands gijt we could solve (4)

π 2 titE m z

= iit

it itE m E z( )

( ) ( )

π1tit it

it it

E m mE m E m

=( )

( ) ( )ln

lnand

E m E zt it it( ) ( )2π

E m Et it( )1π ln mm b pit j t( ) + ( )2




11 Technically what is necessary for recoverability iscompleteness of the class of income-attribute distributionssee Stoker (1984a)

directly expressing aggregate demand Et(qijt)as a function of those distributional parame-ters Here recovery of individual demandparameters from aggregate demand wouldbe possible with sufficient variation in thedistribution Ft(mitzit) over t11

While conceptually different from exactaggregation the distributional approachshould not be thought of as a distinct alter-native in empirical modeling With distri-bution restrictions formulating a model viadirect integration in (4) may be difficult inpractice As such distributional restric-tions are often used together with exactaggregation restrictions combining simpli-fying regularities of the income-attributedistribution with linearity restrictions inindividual demands

One example is with mean scaling as dis-cussed in Lewbel (1990) where the distribu-tion of income does not change relativeshape but just scales up or down Mean scal-ing can arise with a redistribution mecha-nism where individual budgets are all scaledthe same as in mit = mit1(Et(mit)Et1(mit1))This structure allows distributional statisticssuch as those in (7) to be computed frommean income only

Another example arises from (distribu-tional) exclusion restrictions Certain attrib-utes can be excluded from aggregatedemand if their distribution conditional onincome is stable over time if

(10)

where ƒz(zit|mit) does not vary with t thenfrom (4)

(11)

times

That is zit and its distributional statistics areexcluded from the equation for aggregate

dF mt it( )lowast int= ( ) ( )lowast lowastg p m dF mjt t it t it

E q g p m z f z mt ijt jt t it it z it it( ) = ( ) ( ) intint

dF m z f z m dF mt it it z it it t it( ) = ( ) ( )lowast

demand Aggregate demand reflects hetero-geneity only through variation in the incomedistributionmdashthere is not enough variationin the zit distribution over t to recover theindividual effects from aggregate demandWe discuss various other examples of partialdistribution restrictions below

212 Demand and Budget Share Models

There has been a substantial amount ofwork on the precise structure of individ-ual preferences and demands consistentwith exact aggregation The most well-known result of this kind is in the extremecase where the aggregate model simplyrelates average demands Et(qijt) to the vec-tor of relative prices pt and average expen-diture Et(mit) William M Gorman (1953)showed that this required preferences tobe quasi-homothetic with individualdemands linear in mit

Omitting reference to attributes zit fornow the general formulation for exactaggregation has demands of the form

(12)

with aggregate demands given as

(13)

times

As above provided there is sufficient varia-tion in the statistics Et[h1(mit)]hellipEt[hM(mit)]the coefficients a0j(pt)b0j(pt)hellipbMj(pt) andhence individual demands can be fully recovered from aggregate data

Lau (1977 1982) originally proposed theexact aggregation framework and demon-strated that demands of the form (12) werenot only sufficient but also necessary forexact aggregation or aggregation without dis-tributional restrictions (cf Stoker 1993 andJorgenson Lau and Stoker 1982)Muellbauer (1975) studied a related prob-lem and established results for the special

E h m b pt it Mj t( )[ ] + +0 hellip (( ) ( )[ ]E h mt M it

E q a p b pt ijt j t j t( ) = ( ) + ( )0 0

b p h mMj t M it+ + ( ) (hellip ))

q a p b p h mijt j t j t it= ( ) + ( ) ( )0 0 0



13 See also Lewbel (1989b 1991 1993) and Stoker(1984a 1984b)

case of (12) with only two income terms12

They both showed several implications ofapplying integrability restrictions to (12) Ifdemands are zero at zero total expenditurethen a0j(pt) = 0 The budget constraintimplies that one can set h0(mit) = mit withoutloss of generality With homogeneity ofdegree zero in prices and incomes one canassert the forms of the remaining incometerms which include the entropy formh1(mit) = mitlnmit and the power form h1(mit) = mθ

it This theory provides the back-ground requirements for specific exactaggregation demand models such as thosewe discuss below13

The tradition in empirical demand analysisis to focus on relative allocations and esti-mate equations for budget shares The exactaggregation form (12) is applied to budgetshares for this purpose In particular if weset a0j(pt) = 0 and h0(mit) = mit in (12) thenbudget shares wijt = pjtqijtmit take on a similarlinear form We have

(14)

times

where b0j(pt)hellipbMj(pt) and h1(mit)helliphM(mit)are redefined in the obvious way If wedenote individual expenditure weights asmicroit = mitEt(mit) then aggregate budgetshares are

(15)

times

b p E h mMj t t it M it+ ( ) ( )( )micro

E h mt it it( )( ) +micro 1 hellip

E p q

E mE w

b p b

t jt ijt

t itt it ijt

j t j

( )( ) ( )

= ( ) +

micro

0 1 ppt( )

h m bit M( ) + +1 hellip jj t M itp h m( ) ( )

wp q

mb p b pijt

jt ijt

itj t j t= = ( ) + ( )0 1

14 This is Muellbauerrsquos (1975) PIGL form

The same remarks on recoverability applyhere the individual budget share coeffi-cients b1j(pt)hellipbMj(pt) can be identified withaggregate data with sufficient variation inthe distributional terms Et(microith1(mit))hellipEt(ithM(mit)) over time As above aggrega-tion factors can be used to gauge the differ-ence between aggregate shares andindividual shares We have

(16)

times

where by construction

(17)

are the aggregation factors These factorsgive a compact representation of the distrib-utional influences that cause the aggregatemodel and the elasticities derived from it todiffer from the individual model

The budget share form (14) accommodatesexact aggregation through the separation ofincome and price terms in its additive formAs before when integrability restrictions areapplied to (14) the range of possible modelspecifications is strongly reduced A particu-larly strong result is due to Gorman (1981)who showed that homogeneity and symmetryrestrictions imply that the rank of theJ times (M + 1) matrix of coefficients bmj(pt) can beno greater than 3 Lau (1977) Lewbel (1991)and others have characterized the full range ofpossible forms for the income functions

213 Aggregation in Rank 2 and Rank 3Models

Early exact aggregation models were ofrank 2 (for a given value of attributes zit)With budget share equations of the form14

π ktt it k it

k t it

E h mh E m

k M=( )( )

( )( ) =micro

1 hellip

+ ( ) ( )( )πM t Mt M t itb p h E m

( )( ) +1t t ith E m hellip

b p b pt t= ( ) +0 1 (( )π1t

E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro

12 Muellbauer (1975) studied the conditions underwhich aggregate budget shares would depend only on asingle representative income value which turned out to beanalogous to the exact aggregation problem with only twoexpenditure terms



15 It is worthwhile to note that with the power formestimation of with aggregate data would be complicat-ed because the aggregation statistics would depend in acomplicated way on

(18)

preferences can be specified that give rise toeither the log-form h1(mit) = 1nmit and thepower form h1(mit) = mθ

it Typically the for-mer is adopted and this produces Engelcurves that are the same as those that under-lie the Almost Ideal model and the Translogmodel (without attributes)15 In this caseaggregate shares have the form

(19)

where the relevant aggregation factor is the following entropy measure for the mit

distribution

(20)

where we have recalled that microit = mitEt(mit)The deviation of π1t from unity describes thedegree of bias in recovering (individual)price and income elasticities from aggregatedata alone

Distribution restrictions can be used tofacilitate computation of the aggregate sta-tistics as well as studying the aggregation fac-tors For instance suppose income islognormally distributed with 1nmit distrib-uted normally with mean micromt and varianceσ 2

mt The aggregation factor (20) can easily beseen to be

(21)

To the extent that the log mean and varianceare in stable proportion π1t will be stable Ifthe log mean is positive then π1t gt 1 indi-cating positive bias from using 1nEt(mit)

πmicro σ1 2

11

2 1tmt mt

= + ( ) +

πmicro

1tt it it

t it

t it it

t it

E mE m

E m mE m

=( )

( ) =( )ln

lnln

(( ) ( )ln E mt it

b p b pj t j= ( ) +0 1 tt t t itE m( ) ( )π1 ln

E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro

w b p b p h mijt j t j t it= ( ) + ( ) ( )0 1 1

16 It is tempting to consider the case of c1 = 0 c2 = 1which would imply that the aggregation factor π1t = 1 (andno aggregation bias) However that case appears impossi-ble although we do not provide a proof For instance if mit

were lognormally distributed c1 = 0 c2 = 1 would onlyoccur if 1n mit had zero variance

Distribution restrictions can also facilitatethe more modest goal of a stable relationshipbetween aggregate budget shares and aggre-gate total expenditure For instance supposethat the total expenditure distribution obeys

(22)

Then aggregate budget shares are

(23)

so that a relationship of the form

(24)

would describe aggregate data wellHere integrability properties from indi-

vidual demands can impart similar restric-tions to the aggregate relationship Lewbel(1991) shows that if individual shares

(25)

satisfy symmetry additivity and homogeneityproperties then so will

(26)

The analogy of (23) and (26) makes clear thatif c2 = 1 then the aggregate model will satis-fy symmetry additivity and homogeneity Assuch some partial integrability restrictionsmay be applicable at the aggregate level16

w b p b p mijt j t j t it= ( ) + ( ) +( )0 1 κ ln

w b p b p mijt j t j t it= ( ) + ( )0 1 ln

p E mj t t it+ ( )b1 ln (( )

E p q

E mpt jt ijt

t itj t

( )( ) = ( )b0

b p c c Ej t t+ ( ) +1 1 2 ln mmit( )( )

E p q

E mb pt jt ijt

t itj t

( )( ) = ( )0

c E m E mt it t itln+ ( ) ( )2

E m m c E mt it it t itln( ) = ( )1



17 A simple linear transformation will not in general beconsistent with consumer optimization BlundellBrowning and Ian A Crawford (2003) show that if budg-et shares have a form that is additive in functions of 1n mit

and demographics then if (1) Slutsky symmetry holds and(2) the effects of demographics on budget shares areunrestricted then they have to be linear in 1n mit

As we discuss in section 22 below rank 2models of the form (18) fail on empiricalgrounds Evidence points to the need formore extensive income effects (for givendemographic attributes zit) such as availablefrom rank 3 exact aggregation specificationsIn particular rank 3 budget share systemsthat include terms in (lnmit)2 (as well as indi-vidual attributes) seem to do a good job offitting the data such as the QUAIDS systemof James Banks Blundell and Lewbel(1997) described further in section 22below In these cases corresponding to thequadratic term (1nmit)2 there will be anadditional aggregation factor to examine

(27)

In analogy to (22) one can define partial dis-tributional restrictions so that aggregateshares are well approximated as a quadraticfunction of lnEt(mit)

214 Heterogeneous Attributes

As we noted in our earlier discussion theempirical analysis of individual level datahas uncovered substantial demographiceffects on demand Here we reintroduceattributes zit into the equations to captureindividual heterogeneity not related toincome Since zit varies across consumersfor exact aggregation zit must be incorporat-ed in a similar fashion to total expendituremit The budget share form (14) is extendedgenerally to

(28)

Restrictions from integrability theory mustapply for each value of the characteristics zitFor instance Gormanrsquos rank theory impliesthat the share model can be rewritten with

b p hMj t M+ + ( )hellip mm zit it( )

w b p b p h m zijt j t j t it it= ( ) + ( ) ( )0 1 1

t it itE m m=

(ln ))( )( ) ( )( )

2

2E m E mt it t itln

πmicro

2

2

2t

t it it

t it

E m

E m=

( )( )( )( )ln

ln

two terms that depend on mit but there is noimmediate limit on the number of h termsthat depend only on characteristics zit

17

Budget share models that incorporateconsumer characteristics in this fashion werefirst introduced by Jorgenson Lau andStoker (1980 1982) Aggregation factorsarise for attribute terms that necessarilyinvolve interactions between income andattributes The simplest factors arise forterms that depend only on characteristics asin hj(mitzit) = zit namely

(29)

This can be seen as the ratio of the incomeweighted mean of zit to the unweightedmean of zit If zit is an indicator say zit = 1 forhouseholds with children and zit = 0 forhouseholds without children then πz

jt is thepercentage of expenditure accounted for byhouseholds with children Et(mitzit)E(mit)divided by the percentage of householdswith children Et(zit)

More complicated factors terms arise withexpenditure-characteristic effects for exam-ple if hj(mitzit) = zit1nmit then the relevantaggregation factor is

(30)

As before in analogy to (22) one can derivepartial distributional restrictions so thataggregate shares are well approximated as afunction of Et(mit) and Et(zit)

22 Empirical Evidence and theSpecification of Aggregate Demand

221 What do Individual Demands LookLike

Demand behavior at the individual house-hold level is nonlinear As we have men-tioned it is not realistic to assume that

t itE m z= iit it

t it t it t it

mE m E z E m

lnln

( )( ) ( ) ( )

πmicro

1tz t it it it

t it t it

E z mE z E m

=( )

( ) ( )ln

ln

πmicro

tz t it it

t it

t it it

t it t i

E zE z

E m zE m E z

=( )

( ) =( )

( ) tt( )



Figure 1 Nonparametric Engel Curve Food Share

3 4 5 6 70

01

02

03

04

05

06

07

Foo

d Sh

are

Log Expenditure

kernel regressionpolynomial regressionpointwise confidence bands

demands are linear in total expenditures andrelative prices To illustrate typical shapes ofincome structure of budget shares figures 1and 2 present estimates of Engel curves oftwo commodity groups for the demographicgroup of married couples without childrenin the British Family Expenditure Survey(FES)18 Each figure plots the fitted valuesof a polynomial (quadratic) regression in logtotal expenditure together with a nonpara-metric kernel regression We see that forfood expenditures an equation thatexpressed the food share as a linear functionof log expenditure would be roughly correctFor alcohol expenditures the income struc-ture is more complex requiring quadraticterms in log expenditure Moreover as onevaries the demographic group the shapes ofthe analogous Engel curves are similar butthey vary in level and slope

The QUAIDS model of (Banks Blundelland Lewbel 1997) seems to be sufficiently

18 The FES is a random sample of around 7000 house-holds per year The commodity groups are nondurableexpenditures grouped into food-in food-out electricitygas adult clothing childrenrsquos clothing and footwearhousehold services personal goods and services leisuregoods entertainment leisure services fares motoring andgasoline More precise definitions and descriptive statisticsare available on request

flexible to capture these empirical patternsIn the QUAIDS model expenditure shareshave the form

(31)

where a(pt) and c(pt)are given as

with = (1hellipN) = (1hellipN)λ =(λ1hellipλN) and

This generalizes the (linear) Almost Idealdemand system by allowing nonzero λi val-ues with the denominator c(pt) required tomaintain the integrability restrictionsBanks Blundell and Lewbel (1997) doextensive empirical analysis and establish theimportance of the quadratic log expenditureterms for many commodities Interestingly

=prime

prime

γ

γ

1

N

ln lnc p pt t( ) = primeβ

ln ln lna p p p pt t t t( ) = prime + primeα 12

ln

mj

it+minus

λln lnna p

c put

tijt

( )( )( ) +

2

w p m a pijt j j t j it t= + prime + minus ( )( )α γ βln ln ln



Figure 2 Nonparametric Engel Curve Alcohol Share

35 4 45 5 55 6 65 7


Alc

ohol

Sha

re

Log Expenditure

they find no evidence of the rejection ofintegrability restrictions associated withhomogeneity or symmetry

To include demographic attributes anattractive specification is the ldquoshape invariantrdquospecification of Blundell Alan Duncan andKrishna Pendakur (1998) Suppose thatg0

j(lnmi) denotes a ldquobaserdquo share equationthen a shape invariant model specifies budgetshares as

The shape invariant version of the QUAIDSmodel allows demographic variation in the j

terms In Banks Blundell and Lewbel (1997)the j j and λj terms in (31) are allowed tovary with many attributes zit

19 Family sizefamily composition labor market status occu-pation and education are all found to beimportant attributes for many commodities20

w g m z zijt j i it it j= minus prime( )( ) + prime0 ln φ θ ϕ

19 For instance α j + jzit is used in place of α j and sim-ilar specifications for β j and λ j terms

20 Various methods can be used to estimate theQUAIDS model with the iterated moment estimator ofBlundell and Jean-Marc Robin (2000) particularlystraightforward It is worth noting that Banks Blundelland Lewbel (1997) also deal with endogeneity of totalexpenditures using various instruments

222 The Implications for AggregateBehavior

The stability and interpretation of aggre-gate relationships can be assessed from exam-ining the appropriate aggregation factors Wecan compute the empirical counterparts tothe factors by replacing expectations withsample averages For instance π1t of (20) isestimated as

(32)

and π zt of (29) is estimated as

(33)

where we recall that the weights have theform ˆ microit = mit (sumimit nt) Similarly quadrat-ic terms in lnmit will require the analysis ofthe empirical counterpart to the term (27)Interactions of the and τ terms with demo-graphic attributes necessitates examinationof the empirical counterparts of terms of theform (30) We can also study aggregationfactors computed over different subgroupsof the population to see how aggregatedemand would vary over those subgroups

ˆˆ

πmicro

tz i it it t

i it t

z nz n

=sum

sum

ˆˆ

πmicro

1ti it it t

i it t

m nm n

=sum ( )

sum( )ln

ln



Figure 3 Aggregation Factor for Households With Children

1

105

11

115

12

125

13

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000

Figure 3 presents the estimated π zt term

for the impact of children on householddemands This shows a systematic rise in theshare of nondurable expenditure and servic-es attributable to families with children overthe 1980s and 1990s The aggregate biasassociated with using observed percentageof households with children (as opposed tothe income distribution across householdswith and without children) varies from 15percent to 25 percent The path of π z

t alsofollows the UK business cycle and the pathof aggregate expenditure with downturns in1981 and 1992

Figure 4 presents the estimated π1t and π2t

terms relating to the 1nmit and (1nmit)2

expressions in the QUAIDS demand modelIt is immediately clear that these also displaysystematic time series variation but in com-parison to π z

t above they increase over thefirst period of our sample and fall toward theend The bias in aggregation exhibited forthe (1nmit)2 term is more than double thatexhibited for the 1nmit term

Figure 5 presents the aggregations factorfor 1nmit term delineated by certain house-hold types The baseline lnm line (ldquoallrdquo) isthe same as that in figure 4 The other twolines correspond to interactions for couplesas a group and for couples with childrenWhile the time pattern of aggregation fac-tors is similar they are at different levelsindicating different levels of bias associatedwith aggregation over these subgroups

Finally it is worthwhile to mention somecalculations we carried out on whether dis-tributional restrictions such as (22) are capa-ble of representing the aggregatemovements in total expenditure data Usingthe time series of distributional statisticsfrom the FES data we followed Lewbel(1991) and implemented each of theseapproximations as a regression With demo-graphic interaction terms the aggregatemodel will only simplify if these conditionsalso apply to each demographic subgroup Invirtually every case we found the fit of theappropriate regressions to be quite close (say



102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

log expenditure log expenditure squared

R2 in the range of 99) This gives support tothe idea that aggregate demand relation-ships are reasonably stable empiricallyHowever the evidence on the cj termsimplies that aggregation factors are substan-tially different than one so again estimatesof the price and income elasticities usingaggregate data alone will not be accurate

23 Aggregation of Demand withoutIndividual Structure

We close with discussion of a nontradi-tional approach given in Hildenbrand(1994) which is to study specific aspects ofaggregate demand structure without relyingon assumptions on the behavior of individualconsumers This work makes heavy use ofempirical regularities in the observed distri-bution of consumer expenditures across thepopulation

We can understand the nature of thisapproach from a simple example Suppose

we are interested in whether aggregatedemand for good j decreases when price pj

increases (obeying the ldquoLaw of Demandrdquo)and we omit reference to other goods andtime t for simplicity Denote the conditionalexpectation of demand qij for good j givenincome mi and price pj as gj(pjmi)Aggregate demand for good j is

and our interest in whether dGdpj lt 0Form this derivative applying the Slutskydecomposition to gj(pjm) as

(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )

E q G p g p m dF mij j j( ) = ( ) = ( ) ( )int

Figure 4 Aggregation Factors for Income Structure π1t and π2t



Figure 5 Aggregation Factors for Income Structure by Household Type π1zt

101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999

log expenditure couples with children couples without children

The price effect on aggregate demanddecomposes into the mean compensatedprice effect S and the mean income effectA If we take S as negative which is fairlyuncontroversial then we know that dGdpj lt 0 if the income effect A gt 0Looking once more at A we can see variousways of ascertaining whether A gt 0

(35)

Without making any structural assumptionson g(pjm) one could estimate A with thefirst expression using nonparametric esti-mates of g() and its derivative Or wecould examine directly whether theldquospreadrdquo g(pjm)2 is increasing with m andif so conclude that A gt 0

This gives the flavor of this work withoutdoing justice to the details The main con-tribution is to link up properties of aggre-gate demand directly with aspects of the

d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

dF mj= ( ) ( )intint

distribution of demands across the popula-tion Hildenbrand (1998) shows that increas-ing spread is a common phenomena in dataon British households and it is likely to bevalid generally More broadly this work hasstimulated extensive study of the distribu-tion of household expenditures with a dif-ferent perspective than traditional demandmodeling Using nonparametric methodsWolfgang Haumlrdle Hildenbrand and Jerison(1991) study aggregate income effects acrossa wide range of goods and conclude that theldquolaw of demandrdquo likely holds quite generallyHildenbrand and Alois Kneip (1993) obtainsimilar findings on income structure bydirectly examining the dimensionality of vectors of individual demands21 SeeHildenbrand (1994) for an overview of thiswork as well as Hildenbrand (1998) for an

21 This is related to transformation modeling structureof Jean-Michel Grandmont (1992) It is clear that thedimensionality of exact aggregation demand systems isgiven by the number of independent incomeattributeterms cf W E Diewert (1977) and Stoker (1984b)



examination of variations in the Britishexpenditure distribution within a similarframework

3 Consumption and Wealth

We now turn to a discussion of total con-sumption expenditures Here the empiricalproblem is to characterize consumptionexpenditures over time periods includinghow they relate to income and wealth Theindividual level is typically that of a house-hold (or an individual person depending ondata source) The economic aggregate to bemodeled is average consumption expendi-tures over time and we are interested inhow aggregate consumption and savingrelate to income and wealth across the econ-omy as well as interest rates This relation-ship is essential for understanding howinterest rates will evolve as the populationchanges demographically for instance

Consumption expenditures are deter-mined through a forward looking plan thattakes into account the needs of individualsover time as well as uncertainty in wealthlevels There is substantial evidence ofdemographic effects and nonlinearities inconsumption at the individual level so wewill need to consider heterogeneity in tastesas before 22 Accordingly aggregate con-sumption is affected by the structure ofhouseholds and especially the age distribu-tion and aggregate consumption will beaffected by inequality in the distribution ofwealth We are not concerned here with het-erogeneity in market participation per se aseveryone has non-zero consumption expen-ditures Later we discuss some issues raisedby liquidity constraints which have much incommon with market participation modelingas described in section 4

Our primary focus is on heterogeneitywith regard to risks in income and wealthlevels and how the forward planning process

22 See Attanasio and Weber (1993) and Attanasio andBrowning (1995) among many others

is affected by them We take into account thenature of the income and wealth shocks aswell as the nature of the credit markets thatprovide insurance against negative shocks

We consider four different types ofshocks delineated by whether the effects arepermanent or transitory and whether theyare aggregate affecting all consumers orindividual in nature Aggregate permanentshocks can refer to permanent changes inthe productive capability of the economymdashsuch as running out of a key natural resourceor skill-biased technical changemdashas well asto permanent changes in taxes or other poli-cies that affect savings Individual perma-nent shocks include permanent changes inan individualrsquos ability to earn income such aschronic bad health and long term changes intype and status of employment Aggregatetransitory shocks refer to temporary aggre-gate phenomena such as exchange rate vari-ation bad weather and so forth Individualtransitory shocks include temporary job lay-offs temporary illnesses etc Many differentsituations of uncertainty can be accountedfor by combinations of these four differenttypes of shocks

In terms of risk exposure and marketsthere are various scenarios to consider Withcomplete markets all risks are insured andan individualrsquos consumption path is unaffect-ed by the evolution of the individualrsquos incomeover time23 When markets are not com-plete the extent of available insurance mar-kets becomes important and determines thedegree to which different individual risks areimportant for aggregate consumption behav-ior For example in the absence of creditmarket constraints idiosyncratic risks maybe open to self-insurance But in that casethere may be little insurance available foraggregate shocks or even for permanent idio-syncratic shocks Our discussion takes into

23 See Andrew Atkeson and Masao Ogaki (1996) for amodel of aggregate expenditure allocation over time and toindividual goods based on addilog preferences assumingthat complete markets exist



account the type of income risks and howrisk exposure affects aggregate consumption

Most of our discussion focuses on individ-ual consumption plans and their implicationsfor aggregate consumption Beyond this wecan consider the feedback effects on con-sumption and wealth generated throughgeneral equilibrium For instance if a cer-tain group of consumers systematically savemore than others then in equilibrium thoseconsumers will be wealthier and their sav-ings behavior will be a dominant influenceon the evolution of aggregate wealth Thestudy of this important topic is in its infancyand has been analyzed primarily with cali-brated macroeconomic growth models Weinclude a discussion of some of this work

31 Consumption Growth and IncomeShocks

In our framework in each period t indi-vidual i chooses consumption expenditurescit by maximizing expected utility subject to an asset accumulation constraintIndividual i has heterogeneous attributes zit

that affect preferences There is a commonriskless interest rate rt We assume separa-bility between consumption and labor sup-ply in each time period and separability ofpreferences over time

We begin with a discussion of aggregationwith quadratic preferences This allows us tofocus on the issues of different types ofincome shocks and insurance without deal-ing with nonlinearity In section 32 we con-sider more realistic preferences that allowprecautionary saving

When individual within-period utilitiesare quadratic in current consumption wehave the familiar certainty-equivalent for-mulation in which there is no precautionarysaving Within-period utilities are given as

(36)

for cit lt ait We model individual heterogene-ity by connecting ait to individual attributes as

U c a cit it it it( ) = minus minus( )12

2

(37)

With the discount rate equal to the realinterest rate maximizing the expected sumof discounted utilities gives the followingoptimal plan for the consumer (Robert EHall 1978)

(38)

Defining Ωit1 as the information set forindividual i in period t minus 1 the consumptioninnovation it obeys

(39)

In what follows we will use a time super-script to denote this conditional expectationnamely Et1 equiv E[|Ωit1] to distinguish itfrom the population average in period t(which uses a time subscript as in Et)Notice the model (38) is linear in thechange in attributes ∆zit with constant coef-ficients plus the consumption innovationIn other words this model is in exact aggre-gation form with regard to the attributes zit

that affect preferences

311 Idiosyncratic Income Variation andAggregate Shocks

When the only uncertainty arises fromreal income the consumption innovation it

can be directly related to the stochasticprocess for income We begin by spelling outthe income process in a meaningful wayExpress income yit as the sum of transitoryand permanent components

(40)

and assume that the transitory component isserially independent We assume that thepermanent component follows a randomwalk

(41)

where the innovation Pit is serially inde-

pendentNext decompose these two components

into a common aggregate effect and an

y yitP

itP

itP= +minus1 η

y y yit itP

itT= +

E it i tξ minus[ ] =1 0

c zit it it it it= + = prime +α ξ β ξ

a zit it= + primeα β



24 If L is the time horizon then

Clearly τtrarr0 as rrarr0 Note that for a small interestrate we have τt asymp 0 so that the transitory shocks becomeirrelevant for consumption growth

τ tL tr r r= +( ) minus +( )( )

minus minus +( )1 1 1 1

idiosyncratic effect

(42)

(43)

Here t is the common aggregate permanentshock εit is the permanent shock at the indi-vidual level ut is the aggregate transitoryshock and vit is the individual transitoryshockmdashthe four types of income shocks dis-cussed above This mixture of permanentand transitory shocks has been found to pro-vide a good approximation to the panel dataprocess for log incomes see Thomas EMaCurdy (1982) and Costas Meghir andLuigi Pistaferri (2004) We assume that theindividual shocks are normalized to averageto zero across the population namely Et(εit) = 0 and Et(vit) = 0

The stochastic process for individualincome then takes the form

(44)

The stochastic process for aggregate incomehas the form

(45)

where again Et denotes expectation (associ-ated with averaging) across the population ofagents at time t

312 Income Shocks and Insurance

The first scenario is where individual (andaggregate) shocks are not insurable Here theoptimal consumption innovation it for theindividual will adjust fully to permanentincome shocks but only adjust to the annuityvalue of transitory shocks To see this againsuppose that real interest rates are constantand equal the discount rate Under quadraticpreferences (36) consumption growth can bewritten (Deaton and Christina Paxson 1994)

(46)

where τt is the annuitization rate for a tran-sitory shock with planning over a finite

c z u vit it t it t t it= prime + + + +( )β η τε

E y ut it t t( ) = +η

y u vit t it t it= + + +η ε

y u vitT

t it= +

η η εitP

t it= +

25 Aside from the drift term ∆Et(zit) aggregate consumption is a random walk In particular the orthogo-nality conditions Etndash1(ηt + τt∆ut) = E(ηt + τt∆ut|Ωitndash1) = 0hold at the individual level and therefore also hold at theaggregate level

horizon24 Clearly expected growth isdetermined by preference attributes as

(47)

Aggregate consumption has the form

(48)

Thus the aggregate data is described exactlyby a representative agent model with quad-ratic preferences and characteristics Et(zit)facing a permanenttransitory incomeprocess25

For the second scenario suppose individ-ual shocks can be fully insured eitherthrough informal processes or through cred-it markets Now individual consumptiongrowth depends only on aggregate shocks

(49)

Consequently with (48) we will have

(50)

Thus consumption growth at the individ-ual level equals aggregate consumptiongrowth plus an adjustment for individualpreferences

Finally the third scenario is where allshocks (aggregate and individual) are fullyinsurable Now individual consumptiongrowth will be the planned changes β∆zit

only and aggregate consumption growth willbe the mean of those changes β∆Et(zit) Thisis the most complete ldquorepresentative agentrdquocase as complete insurance has removed therelevance of all income risks

c z E z E cit it t it t it= prime minus ( )( ) + ( )β

c z uit it t t t= prime + +β η τ

E c E z ut it t it t t t( ) = prime ( ) + +β η τ

E c E c ztit it i t it

minusminus( ) = ( )1

1 13 β



313 Incomplete Information

It is interesting to note that in our sim-plest framework incomplete informationcan cause aggregate consumption to fail tohave random walk structure In particularsuppose individual shocks are not complete-ly insurable and consumers cannot distin-guish between individual and aggregateshocks To keep it simple also assume thatthere are no varying preference attributes zitFollowing Jorn-Steffen Pischke (1995) indi-vidual i will view the income process (44) asan MA(1) process

(51)

where the θ parameter is a function of therelative variances of the shocks

Changes in individual consumption aresimply

Note that it is still the case that Etndash1(∆cit) =E(∆cit|Ωit1) = 0 However from (51) wehave that

(52)

Replacing ∆yit by (44) and averaging overconsumers we find

(53)

so that aggregate consumption is clearly nota random walk

32 Aggregate Consumption Growth withPrecautionary Saving

With quadratic preferences consumptiongrowth can be written as linear in individualattributesmdashin exact aggregation formmdashandwe were able to isolate the impacts of differ-ent kinds of income shocks and insurancescenarios To allow for precautionary savingwe must also account for nonlinearity in thebasic consumption process For this we nowconsider the most standard consumptionmodel used in empirical work that based onConstant Relative Risk Aversion (CRRA)preferences

ut t+ minus( ) +( )θ η1

E c E ct it t it( ) = ( )minus minusθ 1 1

c c yit it itminus = minus( )minusθ θ1 1

cit it= minus( )1 θ ς

yit it it= minus minusς θς 1

321 Consumption Growth with CRRAPreferences

We assume that within-period utility is

(54)

where ait permits scaling in marginal utilitylevels (or individual subjective discountrates) and sit is the intertemporal elasticityof substitution reflecting the willingness ofindividual i to trade off todayrsquos consumptionfor future consumption As before we willmodel the heterogeneity in ait and sit viaindividual attributes zit

We now adopt a multiplicative stochasticincome process with the decompositionexpressed in log-form as

(55)

The permanent and transitory error compo-nents in the income process are decomposedinto aggregate and individual terms as in(44) As noted before this income growthspecification is closely in accord with thetypical panel data models of income or earn-ings and it will neatly complement ourequations for consumption growth withCRRA preferences In addition we assumethat the interest rate rt is small for simplici-ty and is not subject to unanticipatedshocks

With precautionary savings consumptiongrowth depends on the conditional variancesof the uninsurable components of shocks toincome Specifically with CRRA prefer-ences (54) and log income process (55) wehave the following log-linear approximationfor consumption growth

(56)

where σ tndash1it is the conditional variance of

idiosyncratic risk (conditional on t minus 1

k kitt

Att+ + +minus minusσ σ κ1

12

11εε κ ηit t+ 2

lnc r r zit t t it= + +( )primeρ β ϕ

lny u vit t it t it= + + +η ε

U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11



26 See Blundell and Stoker (1999) for a precise deriva-tion and discussion of this approximation

27 See Banks Blundell and Agar Brugiavini (2001) fora detailed empirical specification of consumption growthin this form

28 If we evaluate the individual model at aggregate val-ues we get

∆1nEt(cit) = rt + ( + ϕrt)Et(zit) + k2σ tAndasht1 + ω t

Here ω t is a ldquocatch-allrdquo term containing the featuresthat induce aggregation bias that will not satisfy theorthogonality condition Etndash1(ω t) = 0 It is also worthwhile tonote that empirical models of aggregate consumption alsotypically omit the terms Et(zit) and k2σ t

Andasht1

information Ωitndash1) and σ tndash1At is the condi-

tional variance of aggregate risk26 Theattributes zit represent the impact of het-erogeneity in ait or individual subjectivediscount rates and the intertemporal elas-ticity of substitution sit = ρ + ϕzitTypically in empirical applications zit willinclude levels and changes in observableattributes and unobserved factors may alsobe appropriate27 As before

(57)

(58)

To sum up in contrast to the quadratic pref-erence case the growth equation (56) isnonlinear in consumption and it includesconditional variance terms that capture theimportance of precautionary saving

A consistent aggregate of the individualmodel (56) is given by

(59)

where Et(∆1ncit) refers to the populationmean of the cross-section distribution of∆1ncit in period t and so on The t subscriptagain refers to averaging across the popula-tion of consumers and we have normalizedEt(εit) = 0 as before Provided Et(∆1ncitndash1) =Etndash1(∆1ncitndash1) equation (59) gives a model ofchanges over time in Et(1ncit) which is a nat-ural aggregate given the log form of themodel (56)

However Et(1ncit) is not the aggregate typi-cally observed nor is it of much policy interestOf central interest is per-capita consumptionEt(cit) or total consumption ntEt(cit) Derivingan equation for the appropriate aggregates

k Et itt+ ( )minusσ1

1 ++ +minusk Att

t21

2σ κ η

E c r r E zt it t t t itln( ) = + +( )prime ( )ρ β ϕ

E Eit i tt

itη η minusminus( ) = ( ) =1

1 0

E Eit i tt

itε ε minusminus( ) = ( ) =1

1 0

involves dealing with the ldquologrdquo nonlinearity towhich we now turn28

322 How is Consumption Distributed

Since the individual consumption growthequations are nonlinear we must make dis-tributional assumptions to be able to formu-late an equation for aggregate consumptionIn the following we will assume lognormali-ty of various elements of the consumptionprocess Here we point out that this is moti-vated by an important empirical regularitymdashnamely individual consumption does appearto be lognormally distributed at least indeveloped countries such as the UnitedStates and the United Kingdom

Figure 6 shows the distribution of log con-sumption using US consumer expendituredata across the last two decades Consumptionis taken as real expenditure on nondurablesand services and is plotted by five year bandsto achieve a reasonable sample size Each log-consumption distribution has a strikingresemblance to a normal density In the expe-rience of the authors this result is often repli-cated in more disaggregated data by year andvarious demographic categorizations such asbirth cohort and also in other countriesincluding in the Family Expenditure Surveydata for the United Kingdom Given this reg-ularity one would certainly start with log-nor-mality assumptions such as those we makebelow and any subsequent refinements wouldneed to preserve normality of the marginaldistribution of log consumption

323 Insurance and Aggregation withPrecautionary Saving

As with our previous discussion wemust consider aggregation under different



1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001

scenarios of insurance for income risksWe again assume that agents have thesame information set namely Ωit1 = Ωt1

for all itWe begin with the scenario in which there

is full insurance for individual risks or pool-ing of idiosyncratic risk across individualsHere insurance and credit markets are suffi-ciently complete to remove individual riskterms in individual income and consumptionstreams so εit = 0 and σ t1

it = 0 for all it Theindividual model (56) becomes

(60)

with Etndash1(ηt) = 0 The mean-log model (59) isnow written as

(61)

The relevant aggregate is per-capita con-sumption Et(cit) Per-capita consumption isgiven by

r E z kt t it+ +( )prime ( ) +β ϕ 221

2σ κ ηAtt

tminus +

E c E c rt it t it tln ln( ) minus ( ) =minus1 ρ

k Att

t+ +minusσ κ η21

2


(62)

with the impact of log-linearity arising inthe final term a weighted average ofattribute terms interacted with laggedconsumption citndash1

Of primary interest is aggregate consump-tion growth or the log-first difference inaggregate consumption

This is expressed as

(63) ∆ = + +minusln ( )E c r kt it t Att

tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1

tE ciit t itr zminus +( )prime( )

1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot

r z kt it Att+( )prime + 2β ϕ σ minusminus + )1

2κ ηt

E c E c rt it t it t( ) = + +minusexp ln 1 ρ(

Figure 6 The Distribution of Log Nondurable Consumption Expenditure US 1981ndash2001



29 This is because of the potential dependence of citndash1 onthe same factors as citndash2 and so forth

Aggregate consumption growth reflects theinterest and risk terms that are common toall consumers a weighted average of attrib-ute terms and the log-difference in theaverage of citndash1 at time t versus time t minus 1

Notice first that even if zit is normally dis-tributed we cannot conclude that 1ncit isnormal We also need (as a sufficient con-dition) that 1ncitndash1 is normal at time t tomake such a claim This would furtherseem to require normality of 1ncitndash2 at t minus 1and so forth into the distant past In anycase we cover this situation with the broadassumption

The distribution of citndash1 is the samein period t minus 1 and t

That is the population could grow or shrinkbut the distribution of citndash1 is unchangedUnder that assumption we can drop the lastterm in (63)

(65)

Lagging the individual model (60) givesan equation for citndash1 but there is no naturalway to incorporate that structure directlyinto the equation for aggregate current con-sumption Et(cit)

29 Therefore we furtherassume

(66)

where we have set t = ( + rt) Thisassumption says that

σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)

We can now solve for an explicit solution to(63) apply (65) (67) and (68) and rearrangeto get

(69)

times

This is the aggregate model of interestexpressing growth in per-capita consumptionas a function of the mean of z the condi-tional variance terms from income risk andthe covariances between attributes z andlagged consumption citndash1 Thus shows howindividual heterogeneity manifests itself inaggregate consumption through distribu-tional variance terms These variance termsvary with rt if the intertemporal elasticity ofsubstitution varies over the population

Now consider the scenario where someindividual risks are uninsurable This rein-troduces terms εit and σtndash1

it in consumptiongrowth at the individual model and we mustbe concerned with how those permanentrisks are distributed across the populationIn particular we assume in each period thateach individual draws idiosyncratic risk froma common conditional distribution so thatσtndash1

it = σtndash1it for all i The individual consump-

tion growth equation (56) now appears as

(70)

The mechanics for aggregation within thisformulation are similar to the previous caseincluding the normalization Et(εit) = 0 but weneed deal explicitly with how the permanent

k kItt


12

11εε κ ηit t+ 2


t zc tr+ +( )summinus

21

β ϕ

t zz t tr r+ +( )prime sum +( )

12 β ϕ β ϕ

( )E z kt it Att+ +minusσ κ η2

12 tt

ln ( )E c r rt it t t= + +( )primeρ β ϕ

lnc Nit c t c tminus minus minus( )1 1 12~ micro σ

c t tminus+

1

2σ θ sum + sum )minuszz t t t z c t θ θ2

1

ln ( )c z N E zit t it c t t t itminus minus+ +1 1θ micro θ ~ (



30 Stephen P Zeldes (1989b) points out how differingmarginal tax rates can cause interest rt to vary acrossconsumers

individual shocks εit covary with 1ncit Asabove we adopt a stability assumption (64)We then extend (66) to assume that (1ncitndash1(β+ ϕrt)zitεit) is joint normally distributed Thegrowth in aggregate average consumption isnow given by

(71)

where

While complex this formulation underlinesthe importance of the distribution of riskacross the population In contrast to the fullinformation model (69) there is a term σ 2

εt

in Λt that reflects the changing variance inconsumption growth The term σ tndash1

1t cap-tures how idiosyncratic risk varies based ont minus 1 information

We have not explicitly considered unantic-ipated shocks to the interest rate rt or het-erogeneity in rates across individuals30

Unanticipated shocks in interest would manifest as a correlation between rt andaggregate income shocks and would needtreatment via instruments in estimationHeterogeneity in rates could in principle beaccommodated as with heterogeneous attrib-utes This would be especially complicated ifthe overall distributional structure were toshift as interest rates increased or decreased

33 Empirical Evidence on Aggregating theConsumption Growth Relationship

There are two related aspects of empiricalresearch that are relevant for our analysis ofaggregation in consumption growth models

z t tr+ sum +( )12 κ β ϕ

tr+ +( )prime sumβ ϕ2 zz c t c tminus minus+

1 1 12 κ σε

t t zz t t tr r= +( )prime sum +( )+β ϕ β ϕ κ σ 12 2

ε

k kItt+ +minusσ σ1

12 AAt

tt t

minus + + ( )11

12

κ η

lnE c r r E zt it t t t it( ) = + +( )prime ( )ρ β ϕ

The first concerns the evidence on full insur-ance of individual risks How good an approx-imation would such an assumption be Tosettle this we need to examine whether thereis evidence of risk pooling across differentindividuals and different groups in the econ-omy For example does an unexpectedchange in pension rights specific to onecohort or generation get smoothed by trans-fers across generations Are idiosyncratichealth risks to income fully insured Eventhough we may be able to cite individualcases where this perfect insurance paradigmclearly fails is it nonetheless a reasonableapproximation when studying the time seriesof aggregate consumption

The second aspect of empirical evidenceconcerns the factors in the aggregate model(71) that are typically omitted in studies ofaggregate consumption From the point ofview of estimating the intertemporal elastic-ity parameter how important are theseaggregation factors How well do they cor-relate with typically chosen instruments andhow likely are they to contaminate tests ofexcess sensitivity performed with aggregatedata

331 Evidence on Full Insurance and RiskPooling Across Consumers

If the full insurance paradigm is a goodapproximation to reality then aggregation isconsiderably simplified and aggregate rela-tionships satisfying the standard optimalityconditions can be derived with various con-ditions on individual preferences There is areasonably large and expanding empirical lit-erature on the validity of the full insurancescenario as well as complete markets sce-nario This work is well reviewed inAttanasio (1998) and Browning Hansen andHeckman (1998) Here we present evidencedirectly related to our discussion of con-sumption growth above Two rather effectiveways of analyzing failures of the full insur-ance paradigm fit neatly with our discussion

One approach to evaluating the full insur-ance hypothesis is to look directly for evidence



that unexpected shocks in income across dif-ferent groups in the economy leads to differ-ences in consumption patterns (as consistentwith (56) which assumes no insurance) Thisis not a trivial empirical exercise First suchincome shocks have to be identified andmeasured Second there has to be a convinc-ing argument that they would not be correlat-ed with unobservable variables enteringmarginal utility or observables such as laborsupply (in a nonseparable framework)

Building on the earlier work by John HCochrane (1991) Barbara Mace (1991)Fumio Hayashi Joseph Altonji and LaurenceKotlikoff (1992) and Robert M Townsend(1994) the study by Attanasio and Steven JDavis (1996) presents rigorous and convinc-ing evidence against the full insurancehypothesis using this approach Low frequen-cy changes in wages across different educa-tion and date-of-birth cohorts are shown to becorrelated positively with systematic differ-ences in consumption growth More recentlyBlundell Pistaferri and Ian Preston (2003)use a combination of the Panel Survey of Income Dynamics (PSID) and theConsumers Expenditure Survey (CES) toinvestigate insurance of permanent and tran-sitory income shocks at the individual levelThey find almost complete insurance to tran-sitory shocks except among lower incomehouseholds They find some insurance to per-manent shocks particularly among theyounger and higher educated But theystrongly reject the complete insurance model

The second approach to evaluating fullinsurance is to assume risk averse prefer-ences and to model the evolution of idiosyn-cratic risk terms In terms of the model (56)this approach examines the relevance ofindividual risk terms (eg σ tndash1

i t ) once aggre-gate risk (σtndash1

At ) has been allowed for This isaddressed by looking across groups wherethe conditional variance of wealth shocks islikely to differ over time and to see whetherthis is reflected in differences in consump-tion growth Following earlier work byKaren E Dynan (1993) Blundell and Stoker

31 Attanasio and Weber (1993) also note a strong impactof omitting the cross-section variance of consumptiongrowth

(1999) Ricardo Caballero (1990) andJonathan Skinner (1988) the study byBanks Blundell and Agar Brugiavini (2001)presents evidence that differential variancesof income shocks across date-of-birthcohorts do induce important differences inconsumption growth paths

332 Aggregation Factors andConsumption Growth

There are two issues First if one estimatesa model with aggregate data alone is therelikely to be bias in the estimated parameters ofinterest Second will the omission of aggre-gation bias terms result in spurious inferenceconcerning the presence of excess sensitivityof consumption to transitory income shocks

With regard to bias we consider the elas-ticity of intertemporal substitution whichis normally a focus of studies of aggregateconsumption In figure 7 we plot theaggregation factor

(72)

for the sample of married couples from theBritish FES used to construct the aggrega-tion factors for demand of section 2 The fig-ure shows a systematic procyclical variationWe found the correlation coefficient betweenthe real interest series and this factor to besignificant This indicates that there will existan important aggregation bias in the estimat-ed intertemporal substitution parameter fromaggregate consumption data (with a log-lineargrowth model) This is confirmed in the studyby Attanasio and Weber (1993) where aggre-gate data was constructed from micro surveyinformation31 They find an elasticity esti-mate for aggregate data of around 35 andthe corresponding micro-level estimates weretwice this size

The study of excess sensitivity involves theuse of lagged information as instrumentalvariables in the estimation of the consumption

ln lnE c E ct it t it( ) minus ( )



32 Detailed regression results available on request

Figure 7 Aggregation Factor for Consumption Growth

-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

growth relationship Omitting aggregationbias terms can invalidate the instruments typ-ically used For the consumption data usedabove we computed the correlation of theaggregation factor with two typically usedinstrumental variables in consumption growthequationsmdashlagged real interest rates andlagged aggregate consumption The estimatedcorrelation coefficient between these seriesand the omitted bias term was found to bestrongly significant32

Together these results suggest that aggre-gation problems are likely to lead to seriousbias in estimated intertemporal substitutionparameters and also to exaggerate the pres-ence of excess sensitivity in consumptiongrowth regressions on aggregate dataAttanasio and Browning (1995) investigatethis excess sensitivity issue in more detail andfind that excess sensitivity still exists at themicro-data level but disappears once controls

for age labor supply variables and demo-graphics are introduced in a flexible wayMoreover these variables explain why excesssensitivity appears to vary systematically overthe cycle

It is an important finding that evidence ofexcess sensitivity vanishes once we move toindividual data and include observable vari-ables that are likely to impact preferences forthe allocation of consumption over time Ithas important consequences for our under-standing of liquidity constraints and for partialinsurance It has implications for understand-ing the path of consumption growth over thecycle It also has implications for the retire-ment-savings puzzle or how consumptiondrops much more at retirement than is pre-dicted by standard consumption growthequations Banks Blundell and Sarah Tanner(1998) find that once demographics and laborsupply variables are allowed to affect the mar-ginal utility of consumption nearly two thirdsof the retirement-savings puzzle disappears



34 Consumption and Liquidity Constraints

Our previous discussion has focused onheterogeneity in wealth and income risk as itimpinges on consumption We now turn to adiscussion of liquidity constraints on con-sumption which generate a different kindof aggregation structure The evidence forliquidity constraints is relatively limitedMost studies of consumption smoothing atthe individual level find it difficult to rejectthe standard model once adequate care istaken in allowing for demographic and labormarket interactions see Attanasio andWeber (1995) and Blundell Browning andMeghir (1994) for example Much of theexcess sensitivity found in aggregate studiescan be attributed to aggregation bias as doc-umented in Attanasio and Weber (1993)Marvin Goodfriend (1992) and Pischke(1995) However there is some evidencethat does point to the possibility that a frac-tion of consumers could be liquidity con-strained at particular points in the life-cycleand business cycle At the micro level someevidence can be found in the studies byFumio Hayashi (1987) Stephen P Zeldes(1989a) Tullio Jappelli (1990) Jappelli andMarco Pagano (1994) Meghir and Weber(1996) and Rob Alessie Michael PDevereaux and Weber (1997) As men-tioned earlier the Blundell Pistaferri andPreston (2003) study shows that the con-sumption of low income households in thePSID does react to transitory shocks toincome which suggests that such house-holds do not have access to credit markets tosmooth such shocks

For aggregation liquidity constraintsintroduce regime structure into the pop-ulation Namely liquidity constrained consumers constitute one regime uncon-strained consumers constitute anotherregime and aggregate consumption willdepend upon the relative distributionacross regimes This structure is particular-ly relevant for the reaction of consumptiongrowth to increases in current income

since constrained consumers will show astronger reaction than unconstrained con-sumers In this section we discuss thesebasic issues and indicate how a model ofaggregates can be constructed Blundelland Stoker (2003) works out the details foraggregate consumption models of this type

There is some subtlety in consideringwhat population groups are likely to be liq-uidity constrained Poor households with areasonably stable but low expected stream ofincome may have little reason to borrowMore likely to be constrained are young con-sumers who have much human capital butlittle financial wealthmdashcollege students orperhaps poor parents of able children Suchindividuals may want to borrow against theirfuture earned incomes but cannot in partbecause their eventual income is higher thanothers and the growth of their income withexperience is higher Clearly such con-sumers will react more than others to shocksin current income and wealth

We start with the basic consumption modeldiscussed earlier with permanent and transi-tory shocks to income As in (55) the changein current income for consumer i at time t is

(73)

where ηt + εit is the permanent componentand ∆ut + ∆vit is the transitory componentTo keep things simple we assume that per-manent income shocks are not insurablewith log consumption given as

(74)

where we assume the precautionary riskterms (σtndash1

it σtndash1At ) are included with the zit

effects Note that (74) gives the consump-tion growth plan (rt + (β + ϕrt)zit) as well ashow consumption reacts to permanentshocks in income (here ηt + εit)

Liquidity constraints affect the ability ofconsumers to finance their desired con-sumption growth path We follow anapproach similar to Zeldes (1989a) wherethe incidence of liquidity constraints

lnc r r zit t t it t it= + +( )prime + +ρ β ϕ η ε




33 Various approaches have been applied to account forthe jump term mit in studies of micro level data See Zeldes(1989a) Tullio Jappelli Jorn-Steffen Pischke andNicholas S Souleles (1998) Rene Garcia AnnamariaLusardi and Serena Ng (1997) Rob Alessie BertrandMelenberg and Weber (1988) Alessie Michael PDevereux and Weber (1997) and Attanasio and Weber(1998)

depends on the degree of consumptiongrowth the consumer is trying to finance andthe existing stock of assets In particular liq-uidity constraints enter the growth plan onlyif they are binding in planning period t minus 1and then the best response will always be toincrease consumption growth so as to ldquojumprdquoback up to the optimal path If this responseis further frustrated by a binding constraintin period t consumption will simply grow bythe amount of resources available

This response structure is captured byadditional terms in the equation (74) Let Iit

denote the indicator

(75)

and suppose that a consumer who is con-strained in period t minus 1 needs to increaseconsumption growth by mit to return to theoptimal growth plan33 Then consumptiongrowth for unconstrained consumers is

(76)

We now model the constraints as well asconsumption growth for constrained con-sumers With growth in income of ∆1nyitconsumer i needs to finance a growth rate of

for consumption at time t to be on thegrowth plan To model liquidity constraintsat time t suppose that consumer i faces aborrowing constraint that is associated with amaximum rate of increase of consumption of

γ δ ζ+ +Ait it

ρrt= + ββ ϕ+( )prime + minus minusminusr z I m u vt it it t itit1

ρ β ϕ η εr r z I m yt t it it t it itit+ +( )prime + + + minusminus1 ln

I mit t itit+ + +minus η ε1


in period tt minus 1]I iit = 1 [consumer is constrained

where Ait is (say) accumulated financialwealth Consumer i is liquidity constrained inperiod t or cannot maintain the consumptiongrowth plan if

(77)

which we indicate by Iit = 1 as above In thiscase we assume that consumption growth isas large as possible namely

(78)

In terms of permanent and transitory termsof income growth (78) may be rewritten

(79)

This is consumption growth for constrainedconsumers The constraints have an impactas consumption growth clearly depends ontransitory income shocks and wealth levels

Aggregate consumption growth will clear-ly depend on the proportion of consumerswho are constrained and the proportion thatare not Consumers who were constrainedlast period will have a boost in their con-sumption growth to return to the optimalpath This regime switching structure is non-linear in character Therefore to modelaggregate consumption growth we wouldneed to specify distributional structure forall the elements that are heterogeneousacross the population We then aggregateover the population of unconstrained indi-viduals with consumption growth (76) andthe population of constrained individualswith consumption growth (79) Using log-normality assumptions we carry out thisdevelopment in Blundell and Stoker (2003)It is clear how aggregate consumption isaffected by transitory income shocks as wellas the distribution of wealth

35 Equilibrium Effects

As we mentioned at the start one use ofaggregate consumption equations is to study

u v At it it it+ + + + +γ δ ζ lncit t it= +η ε

ln lnc y Ait it it it= + + +γ δ ζ

γ δ ζu v At it it iminus minus gt + + tt

ρ β ϕr r z I mt t it it it+ +( )prime + minus1



and understand the evolution of aggregateconsumption and saving by themselvesAnother important use is in studying equilib-rium price and interest rate paths over timeThis is an exercise in general equilibriumanalysis and every feature that we have dis-cussed above is relevantmdashconsumer hetero-geneity heterogeneity in income and wealthrisks liquidity constraints and the distribu-tion of wealth Further complicating thiseffort is the dynamic feedback that occurswherein the level and distribution of wealthevolves as a result of the level and distribu-tion of savings These difficulties make it veryhard to obtain analytical results on equilibri-um Nevertheless it is extremely importantto understand the nature of equilibrium hereincluding implications on prices and interestrates We now discuss some recent progressthat has been made using calibrated stochas-tic growth models A leading example of thiseffort is provided by Per Krusell and AnthonyA Smith (1998) although the approach datesfrom at least S Rao Aiyagari (1994) and JohnHeaton and Deborah J Lucas (1996)

The KrusellndashSmith setup has the followingfeatures Consumers are infinitely-livedwith identical (within-period) CRRA prefer-ences but they are heterogeneous withregard to discount rates Each consumer hasa probability of being unemployed each peri-od providing transitory idiosyncratic incomeshocks Production arises from a constantreturns-to-scale technology in labor and cap-ital and productivity shocks provide transito-ry aggregate shocks Consumers can insureby investing in capital only so that insurancemarkets are incomplete and consumersrsquocapital holdings cannot be negative (liquidityconstraint) This setup is rich but in manyways is very simple Nevertheless in princi-ple in order to predict future prices eachconsumer must keep track of the evolutionof the entire distribution of wealth holdings

Krusell and Smithrsquos simulations show arather remarkable simplification to this fore-casting problem For computing equilibriumand for consumer planning it is only necessary

34 Christopher D Carroll (2000) makes a similar argu-ment with emphasis on the role of precautionary savings

for consumers to keep track of two things themean of the wealth distribution and the aggre-gate productivity shock Thus there is an infor-mational economy afforded in a similarfashion to a formal aggregation result oncemean wealth is known the information con-tained in the distribution of wealth does notappear to improve forecasting very much Thisis true even with heterogeneity of many typesincluding individual and aggregate incomeshocks (albeit transitory)

The reason for this is clear once the natureof equilibrium is examined Most consumersespecially those with lowest discount ratessave enough to insure their risk to the pointwhere their propensity to save out of wealthis essentially constant and unaffected by cur-rent income or output Those consumersalso account for a large fraction of thewealth Therefore saving is essentially a lin-ear function in wealth and only the mean ofwealth matters to how much aggregate sav-ing is done each period The same is not trueof aggregate consumption There are manylow wealth consumers who become unem-ployed and encounter liquidity constraintsTheir consumption is much more sensitive tocurrent output than that of wealthier con-sumers In essence what is happening here isthat the dynamics of the savings process con-centrates wealth in the hands of a group thatbehaves in a homogeneous way with a con-stant marginal propensity to save This(endogenous) simplification allows planningto occur on the basis of mean wealth only

It is certainly not clear how applicable thisfinding is beyond the context of this studyThis is a computational finding that dependsheavily on the specifics of this particular set-up34 Nonetheless this form of feedback hassome appeal as a explanation of the smoothevolution of wealth distribution as well aswhy forecasting equations (that fit well) areso often much simpler than one would expectfrom the process that underlies the data The



rich are different (and in this model the dif-ference makes them rich) but what is impor-tant for forecasting is how similar the rich areto one another With equal saving propensi-ties it does not matter which group of richpeople hold the most wealth

In any case the study of equilibriumeffects is in its infancy and it will certainlygenerate many valuable insights

4 Wages and Labor Participation

Our final topic area is the analysis of wagesand labor participation Here the empiricalproblem is to understand the determinantsof wages separately from the determinantsof participation The individual level is thatof an individual worker The economicaggregates to be modeled are aggregatewages and the aggregate participation rateor one minus the unemployment rate Thesestatistics are central indicators for macroeco-nomic policy and for the measurement ofeconomic well-being

Our analysis is based on a familiar para-digm from labor supply Potential wages aredetermined through human capital andlabor participation is determined by com-paring potential wages to a reservation wagelevel Empirically there is substantial het-erogeneity in the determinants of wagesand substantial heterogeneity in the factorsdetermining labor participation and bothprocesses are nonlinear In particular it istypical to specify wage equations for individ-uals in log-form and there is much evidenceof age and cohort effects in wages andemployment As with demand and con-sumption we will need to be concernedwith heterogeneity in individual attributesTo keep things as simple as possible we donot consider forward looking aspects ofemployment choice and so are not con-cerned with heterogeneity in income andwealth risks

Our primary focus is on heterogeneity inmarket participation Aggregate wagesdepend on the rate of participation and theimportant issues involve separation of the

35 Heckman and Guilherme Sedlacek (1985) providean important generalization of this framework to multiplesector See also Heckman and Bo E Honore (1990)

wage process from the participation decisionTo put it very simply suppose aggregatewages are increasing through time Is thisbecause typical wages for workers areincreasing Or is it because low wage indi-viduals are becoming unemployed Do thesources of aggregate wage growth vary otherthe business cycle The aggregation problemmust be addressed to answer these questions

We now turn to our basic model of wagesthat permits us to highlight these effects Wethen show the size of these effects for aggre-gate wages in the United Kingdom a countrywhere there has been large and systematicchanges in the composition of the workforceand in hours of work A more extensive ver-sion of this model and the application isgiven in Blundell Howard Reed and Stoker(2003) They also summarize derivations ofall aggregate equations given below

41 Individual Wages and Participation

We begin with a model of individual wagesin the style of Andrew D Roy (1951) wherewages are based on human capital or skilllevels and any two workers with the samehuman capital level are paid the same wageOur framework is consistent with the pro-portionality hypothesis of Heckman andGuilherme Sedlacek (1990) where there isno comparative advantage no sectoral dif-ferences in wages for workers with the samehuman capital level and the return tohuman capital is not a function of humancapital endowments 35

We assume that each worker i possesses ahuman capital (skill) level of Hi Supposehuman capital is nondifferentiated in that itcommands a single price rt in each time peri-od t The wage paid to worker i at time t is

(80)

Human capital Hi is distributed across thepopulation with mean

w r Hti t i=



Et(1nHi) = js

where δjs is a level that varies with cohort jto which i belongs and education level s ofworker i In other words the log wageequation has the additive form

(81)

where ζit has mean 036 We will connect js toobservable attributes below

To model participation we assume thatreservation wages wlowast

it are lognormal

(82)

where js has mean 0 and where Bit is anexogenous income (welfare benefit) levelthat varies with individual characteristics andtime Participation occurs if wit ge wlowast

it or with

(83)

We represent the participation decision bythe indicator Iit = 1[wit ge wlowast

it]For aggregation over hours of work it is

useful to make one of two assumptions Oneis to assume that the distribution of hours isfixed over time The other is to assume thatdesired hours hit are chosen by utility maxi-mization where reservation wages aredefined as hit(wlowast) = h0 and h0 is the mini-mum number of hours available for full-timework We assume hit(w) is normal for each wand approximate desired hours by

(84)

This is our base level specification It issimple to extend this model to allow differ-entiated human capital or differentialcohort effects due to different labor marketexperience which permits a wide range of

= + minus + minus + minus( )h r Bt it js js it it0 γ α δ η ε ζln ln

h h w wit it it= + minus( )lowast0 γ ln ln

ln lnr Bt it js js it itminus + minus + minus geα δ η ε ζ 0

ln lnw Bit it js itlowast = + +α η ζ

ln lnw rti t js it= + +δ ε

educationcohorttime effects to be includ-ed (cf Blundell Reed and Stoker 2003)Because our examples involve log-linearequations and participation (or selection)we summarize the basic framework as

(85)

Here xit denotes education demographic(cohort etc) and time effects zit includesout-of-work benefit variables and Iit = 1denotes participation It is clear that the scaleof is not identified separately from the par-ticipation index 0 + zit + νit however weretain to distinguish between the fixed hourscase = 0 and the variable hours case ne 0

Our notation distinguishes two types ofindividual heterogeneity in (85) The vari-ables xit and zit are observable at the individ-ual level while εit and νit are unobservableAnalysis of data on wages and participationat the individual level requires assumptionson the distribution of those unobservableelements a process familiar from the litera-ture on labor supply and selection bias Wenow review some standard selection formu-lae here for later comparison with the aggre-gate formulations Start with the assumptionthat the unobserved elements are normallydistributed

(86)

This allows us to apply some well knownselection formulae (given in virtually everytextbook of econometrics) The micro par-ticipation regression or the proportion ofparticipants given xit and zit is a probitmodel

(87)

The micro log-wage regression for partici-pants is

E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2

h h zit it it= + + prime +( )0 0γ α α νmiddot

I zit it it= + prime + gt[ ]1 00α α ν

lnw xit it it= + prime +β β ε0

36 Clearly there is an indeterminacy in the scaling of rt

and Ht Therefore to study rt we will normalize rt for someyear t = 0 (say to r0 = 1) We could equivalently set one ofthe s to zero



37 Here Φ() is the normal cumulative distribution func-tion and λ() = φ()Φ()where φ() is the normal densityfunction

38 Assuming that the linear indices are normal is muchweaker than assuming that xit and zit are themselves jointmultivariate normal Such a strong structure would eliminatemany important regressors such as qualitative variables

(88)

reflecting the typical (Heckman style) selec-tion term which adjusts the log-wage equationto the group of participating workers37

42 Aggregate Wages and Employment

The aggregate of interest is average hourlyearnings where aggregation occurs over allworkers namely

(89)

where i isin (I = 1) denotes a participant(worker) hitwit is the earnings of individual iin period t and microit are the hours-weights

Modeling the aggregate wage (89)requires dealing with log-nonlinearity of thebasic wage equation dealing with participa-tion and dealing with the hours-weightingAll of these features require that distribu-tional assumptions be made for (observable)individual heterogeneity In particular wemake the following normality assumption forxit and zit

(90)

or that the indices determining log-wage andemployment are joint normally distributed38

xx xz

xz zz

prime sum prime sumprime sum prime sum

β β α ββ α α α

β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )

isin =( ) isin =( )sum1

1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2

E w I x zt it it it itln =[ ] =1 We now discuss some aggregate analogues

of the micro regression equations and thenour final equation for the aggregate wageThe aggregate participation (employment)rate is

(91)

using a formula originally due to DanielMcFadden and F Reid (1975) Aggregateparticipation has the same form as the microparticipation regression (87) with zit replacedby E(zit) and the spread parameter σν

replaced by the larger value

reflecting the influence of heterogeneity inthe individual attributes that affect the par-ticipation decision The mean of log-wagesfor participating (employed) workers is

(92)

using a formula originally derived byMcCurdy (1987) This matches the microlog-wage regression (88) with xit replaced byE(xit|I = 1) zit replaced by E(zit) and thespread parameter changed from σν to

This is an interesting result

but doesnrsquot deliver an equation for theaggregate wagewt

Blundell Reed and Stoker (2003) derivesuch an equation The aggregate wage isgiven as

(93)

where the aggregation bias is comprised ofa spread term

(94) t xx= prime +[ ]sum1

22β β σε

(E x= + prime0β β iit t t t) [ ]+ + +

ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )

E w I E x It it it itln ( )=[ ] = + prime =1 10β β

prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2



40 Comparing 1nwndashit to mean log wage E(1nwit) is in linewith the tradition of measuring ldquoreturnsrdquo from coefficientsin log wage equations estimated with individual data cfGary Solon Robert Barsky and Jonathan A Parker (1994)Other comparisons are possible and some may be prefer-able on economic grounds For instance if aggregate pro-duction in the economy has total human capital (ΣiHi) asan input then the appropriate price for that input is rt soone might want to compare wndasht to 1nrt for a more effectiveinterpretation In any case it is useful to point out that ifE(1nHi) is constant over time then comparing 1nrt to wndasht isthe same as comparing E(1nwit) to wndasht

plus two terms Ψt and Λt which representseparate sources of bias but have verycomplicated expressions39

What these terms represent can be seenmost easily by the following constructionBegin with the individual wage equationevaluated at mean attributes 0 + E(xit) =Et(lnwit) or overall mean log wage AddingΩt adjusts for log-nonlinearity as

Adding Ψt adjusts for participation as

(95)

Finally adding Λt adjusts for hours-weight-ing as

(96)

Thus the bias expressions are complicatedbut the roles of Ωt Ψt and Λt are clear Inwords the term Ωt captures the variance ofreturns observable and unobservable Theterm Ψt reflects composition changes withinthe selected sample of workers from whichmeasured wages are recorded The term Λt

reflects changes in the composition of hoursand depends on the size of the covariancebetween wages and hours

The formulation (93) of the log aggregatewage lnwt thus captures four important

ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt

ln lnE w I E wit it t it t=[ ] = ( ) +1

ln lnE w E wt it t it t( ) = ( ) +

sources of variation First aggregate wagesincrease if the distribution of log wages shiftsto the right which is a the typical ldquowell-beingrdquo interpretation of aggregate wagemovements40 This source is reflected by themean 0 + E(xit) of log wages Secondbecause individual wages are given in logform aggregate wages will increase withincreased spread of the log wage distributionas reflected by the heterogeneity term ΩtThird aggregate wages will increase if thebenefit threshold increases causing morelower wage individuals to decide not to par-ticipate This is reflected in the participationterm Ψt Fourth aggregate wages willincrease if the hours of higher wage individu-als increase relative to lower wage individualswhich is captured by the hours adjustmentterm Λt The aggregate model (93) permitsestimation of these separate effects

This framework could be relaxed in manyways We can allow all variance terms to betime varying as well as many of the basicbehavioral parameters If the normalityassumption on the overall log wage and par-ticipation index is not accurate for the wholepopulation the population can be segment-ed with separate aggregate equations devel-oped for each segment These variationsamong others are discussed in BlundellReed and Stoker (2003)

43 Empirical Analysis of British Wages

The different sources of aggregate wagevariation bear directly on the issue of whetheraggregate wages are procyclical or not In par-ticular the participation effect works counter

39 In particular we have

+γσε γγ α α σ λ

γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a

+ + prime + prime sum +t

it xzE z ln

h ( )0 γα γα γβ α0

γα γα+ + prime +ith E z0 0 ( )

+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=

+ prime + prime sum +

prime sum +

ln

( ) ( )α α β α σ

α α σε0

2



Figure 8 Male Hourly Wages

log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22

41 We exclude individuals classified as self-employedThis could introduce some composition bias given that asignificant number of workers moved into self employ-ment in the 1980s However given that we have no data onhours and relatively poor data on earnings for this groupthere is little alternative but to exclude them They are alsotypically excluded in aggregate figures

to a normal cyclical variation of aggregatewagesmdashdecreases in participation can lead toaggregate wage increases when there isessentially no change in individual wage levelsor distribution We now turn to an analysis ofBritish wages that shows these features

Our microeconomic data is again takenfrom the UK Family Expenditure Survey(FES) for the years 1978 to 1996 The FESis a repeated continuous cross-section sur-vey which contains consistently definedmicro data on wages hours of work employ-ment status and education for each yearsince 1978 Our sample consists of all menaged between 19 and 59 (inclusive)41 The

participating group consists of employeesthe nonparticipating group includes individ-uals categorized as searching for work aswell as the unoccupied The hours measurefor employees in FES is defined as usualweekly hours including usual overtimehours and weekly earnings includes over-time pay We divide nominal weekly earn-ings by weekly hours to construct an hourlywage measure which is deflated by thequarterly UK retail price index to obtainreal hourly wages

Individual attributes include educationlevel and cohort effects Individuals areclassified into three educational groupsthose who left full-time education at age 16or lower those who left aged 17 or 18 andthose who left aged 19 or over Dummyvariables capture effects of five date-of-birth cohorts (b1919ndash34 b1935ndash44b1945ndash54 b1955ndash64 and b1965ndash77) Weinclude various trend variables to account



Figure 9 Male Employment Rate

78 80 82 84 86 88 90 92 94 96075

080

085

090

year

42 In fact such a conclusion has been trumpeted byBritish newspapers

for a common business cycle effect Finallyour measure of benefit income (income atzero-hours) is constructed for each individ-ual as described in Blundell Reed andStoker (2003) After making the sampleselections described above our samplecontains 40988 observations of which33658 are employed or 821 percent of thetotal sample

431 Real Wages and Employment

Figure 8 shows log average wages inBritain from 1978 to 1996 These show astrong trend increase over the whole periodThe trend appears for more disaggregategroups Blundell Reed and Stoker (2003)present a more detailed breakdown bycohort region and education group andshow that the trend holds widely includingfor the least educated group

Figure 9 shows the overall male laboremployment rate for the same periodClearly there has been a large fall in the par-ticipation rate of men Figure 10 presentsthe employment rate for those with loweducation For this group there is a contin-ued and much steeper decline in employ-ment This period also included two deeprecessions in which there have been largefluctuations in male employment

Considering figures 8ndash10 together onecan understand the basic importance of sort-ing out wage growth at the individual levelfrom changes in participation The strongtrend of aggregate wages is suggestive ofgreat progress at increasing the well-being oflaborers in general42 However great



year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

increases in unemployment are likely associ-ated with unemployment of workers withlowest wages or workers from the poorestgroups It is very important to understandhow much of the upward trend in averagewages is due to the elimination of low wageearners from employment

There have also been well documentedchanges in real benefit income over time andacross different groups of individuals Whileit is unlikely that variation in real value ofbenefit income relative to real earnings canexplain all of the variation in participationrates the changes in real benefits act as animportant ldquoinstrumental variablerdquo for sepa-rating participation decisions from determi-nants of wages Again to the extent thatchanges in benefit income have discouraged(or encouraged) participation it is essentialto learn the size of this impact relative to theother factors driving changes in wages

432 Aggregation Results

The Blundell Reed and Stoker (2003)study considers a number of possible specifi-cations for our individual level wage equa-tions which relate to the variousspecifications In the simplest of our specifi-cations the full proportionality hypothesis isimposed on the (nondifferentiated) humancapital model together with trend terms toreflect the business cycle effects on skillprice This specification was strongly reject-ed by the data The preferred model had fullinteractions of cohort trend region andeducation These additional variables couldreflect many differences in minimum educa-tional standards across cohorts such as thesystematic raising of the minimum schoolleaving age over the postwar period in theUnited Kingdom The prices of different(education level) skills are allowed to evolve

Figure 10 Employment Rate for the Low Educated



Figure 11 Hourly Wages

-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e

Log aggregate wage with corrections

Aggregated micro log wage

in different ways by including an interactionbetween high education and the trendterms These coefficients are marginally sig-nificant and show an increasing trend amonggroups with higher levels of human capitalThe impact of adjusting for participation isvery important43 To see the impact of theseresults on aggregate wages we turn tographical analysis

Figure 11 displays the (raw) log aggregatewage the log aggregate wage minus the esti-mated aggregation bias terms and the meanof the log wage from the selectivity adjustedmicro model We have plotted the returnlines from a common point at the start of thetime series rebased to zero for 1984 to high-light the changes in trend growth in wagesindicated by our corrections There is a clear

downward shift in the trend and an increasedcyclical component in wage growth shown byboth the corrected aggregate series and theestimated micro model

This procedure is repeated for the lowereducation group in Blundell Reed andStoker (2003) Several features of this analy-sis are worth mentioning here For instanceeven the direction of movement of the uncor-rected log aggregate wage does not alwaysmirror that of the mean micro log wageThere is a reasonably close correspondencebetween the two in the 1984ndash88 period butthe 1990ndash93 period is different In 1990ndash93log aggregate wages are increasing but themean micro log wage (and the correctedaggregate wage) is decreasingmdashprecisely theperiod where there is a big decline in partic-ipation What is remarkable is that the aggre-gate data show reasonable growth in realwages but such growth is virtually absentfrom the corrected series We are left with amuch more cyclical profile of wages

43 Blundell Howard Reed and Stoker (2003) examinedthe impact of our normality assumptions by estimating withsemiparametric methods The estimated wage coefficientswere hardly affected by this generalization



44 To get an idea of the precision of these resultsBlundell Reed and Stoker (2003) present bootstrap 95percent confidence bands for the corrected log wage esti-mates for various groups These plots show that the micromodel prediction and the corrections to the log aggregatewage are both quite tightly estimated In all cases themicro model prediction and the corrections to the aggre-gate wage plot are significantly different from the rawaggregate wage measure and not significantly differentfrom each other This gives us confidence that we haveidentified compositional biases in the measured real wagewith a reasonable degree of precision

45 There are many stories told in the economics profes-sion about what giants of our field thought were the great-est contributions to social science In this spirit we relatethe following In the mid-1980s one of the authors askedPaul Samuelson what he felt was the greatest failure ineconomics Without hesitation his answer was ldquomacroeco-nomics and econometricsrdquo The reason for this is that therehad been an enormous anticipation in the 1940s 1950sand 1960s that simple empirical macroeconomic modelswould in fact be accurate enough to allow real economiesto be guided and controlled much like an automobile or aspacecraft That this turned out to not be possible was asource of great disappointment

If the model is exactly correct the resultsfrom aggregating the selectivity-adjustedmicro model estimates should match the cor-rected aggregate series They show a closecorrespondence in figure 11 and a similarclose correspondence is noted by BlundellReed and Stoker (2003) for more disaggre-gated groups44 In any case we view the cor-respondence between the corrected logaggregate micro wage and the mean microlog wage as striking validation of the frame-work This model specification which pro-vides a good and parsimonious specificationof the evolution of log real wages also seemsto work well in terms of the specification ofaggregation factors

5 Conclusion

Macroeconomics is one of the mostimportant fields of economics It has perhapsthe grandest goal of all economic studywhich is to advise policymakers who are try-ing to improve the economic well-being ofentire populations of people In the mid-twentieth century say from 1940 to 1970macroeconomics had an orientation towardits role much like an oracle giving advicewhile peering down from the top of a moun-tain That is while economists could see peo-ple making detailed decisions about buyingproducts investing their wealth choosingjobs or career paths etc macroeconomicmodels were extremely simple For instancedescribing the aggregate consumption of anentire economy could be done with takinginto account just a few variables aggregate

income lagged aggregate consumption etcSuch equations often fit aggregate dataextremely well Unfortunately such modelscould not predict future aggregate variableswith sufficient precision to dictate optimalpolicies Even with great statistical fit therewas too much uncertainty as to what theunderlying processes were that drove theaggregate data and for policy prescriptionsit is crucial to know something about thoseprocesses45

What economists could get a handle onwas how rational individuals and firms wouldbehave in various economic environmentsProblems like how to allocate onersquos budgethow much to save and invest or whether towork hard or not so hard are sufficientlyfamiliar that their essence could be capturedwith some mathematics and economistscould describe and prescribe optimal reac-tions Economists could settle how someonebeing really smart and clear-headed wouldbehave Notwithstanding the anomaliespointed out recently by behavioral econo-mists the predictive power of economicsrests on the notion that people facing a famil-iar situation will behave in their interestsFoolish self-destructive or purely randombehavior will not be repeated once it is con-sciously seen to be less good than anothercourse The transformation of economicanalysis by mathematics occurred throughthe systematic understanding of rational andlearning behavior by individuals and firmsand the overall implications of that for marketinteractions



The merging of these two bodies ofthoughtmdashmacroeconomics and optimalbehavior of individualsmdashis among the great-est development of economics in the last halfcentury This advance has been recognizedby Nobel prizes to Lucas Kydland andPrescott and one should expect more prizesto be awarded to other important develop-ers Previous ldquoschools of thoughtrdquo have beenreplaced by groups differentiated by howthey settle the tradeoff between realism andstrict adherence to optimal economic behav-ior The specification of macroeconomicmodels the judgment of whether they aresensible and the understanding of theimpacts of economic policy is now more sys-tematic because of its embedding in therules of optimal individual behavior

The trouble is this embedding cannot beright without taking account of aggregationA one-person or five-person economy is justnot realistic You can simulate a model witha few actors and pretend that it is realisticbut there is nothing in casual observation orempirical data or economic theory that sug-gests that such a stance is valid There ismuch to be learned from rational individualbehavior but there must be a explicit bridgeto economic aggregates because real peopleand their situations are so very heteroge-neous Aggregation is essential because het-erogeneity is a pervasive and indisputablefact of life

In this paper we have covered recent workon aggregation problems in a style that wehope is useful to empirical economists Ourorientation has been to highlight the impor-tance of different types of individual hetero-geneity in particular heterogeneity in tastesand reaction heterogeneity in market partici-pation and heterogeneity in uninsurablerisks Our approach has been practical wehave covered recent advances in econometricmodeling that address issues in aggregation byconsidering explicit models at the individuallevel and among economic aggregates

We have covered a wide range of ideasFirst we have detailed the main approach for

incorporating distributional information intoaggregate relationships namely exact aggre-gation models in the context of how thatapproach has been applied to the analysis ofconsumer demands Second we have shownhow one can incorporate basic nonlinearityinsurance and dynamic elements in our cov-erage of aggregate consumption based onCRRA preferences Third we have shownhow to account for compositional hetero-geneity in our coverage of labor participationand wages The latter two topics requiredexplicit assumptions on the distribution ofindividual heterogeneity and we have basedour solutions on normal and lognormalassumptions on individual heterogeneityWhile these distributional restrictions arespecific they do permit explicit formulationsof the aggregate relationships of interest tobe derived and those formulations captureboth location and spread (mean and vari-ance) of the underlying elements of individ-ual heterogeneity We view our solutions inthese cases as representative and clear andgood starting points for empirical modelingin the respective areas

Whether one dates the beginning of thestudy of aggregation problems from the1940s 1930s or perhaps earlier one can atbest describe progress toward solutions asslow Aggregation problems are among themost difficult problems faced in either thetheoretical or empirical study of economicsHeterogeneity across individuals is extreme-ly extensive and its impact is not obviouslysimplified or lessened by the existence ofeconomic interaction via markets or otherinstitutions The conditions under whichone can ignore a great deal of the evidenceof individual heterogeneity are so severe asto make them patently unrealistic With ref-erence to our introduction as annoyancesgo aggregation problems are particularlycloying There is no quick easy or obviousfix to dealing with aggregation problems ingeneral

Yet we see the situation as hopeful andchanging and offer the solutions discussed



in this paper as evidence of that change Thesources of this change are two-fold and it isworth pointing them out as well as pointingout how both are necessary

The first source of change is the increas-ing availability of data on individualsobserved over sequential time periods Toaddress questions of what kinds of individualheterogeneity are important for aggregaterelationships one must assess what kinds ofheterogeneity are relevant to individualbehavior for the problem at hand and assesshow much the distributions of the relevantheterogeneity vary over time To the extentthat this heterogeneity reflects differences inunexpected shocks to individual agents themechanisms that are available to individualsto insure against such shocks will have astrong bearing on the form of the aggregaterelationship

While we have advanced the idea of usingaggregation factors (derived from time seriesof individual data) to summarize the impactsof aggregation the specific method one usesis less important than the ability to use allavailable types of information to study eco-nomic relationships That is it is importantto study any relationship among economicaggregates with individual data as well asaggregate data to get as complete a pictureas possible of the underlying structure Eventhough modeling assumptions will always benecessary to develop explicit formulations ofaggregate relationships testing thoseassumptions is extremely important and isnot possible without extensive individualdata over sequential time periods Our viewis that the prospects for meaningful advancecontinue to brighten as the data situationwith regard to individual behavior andaggregate economic variables will continueto improve

The second source of change in studyingaggregation problems is the recent rapidrise in computing power Realistic accom-modation of individual heterogeneity typi-cally requires extensive behavioral modelslet alone combinations of individual models

with aggregate relationships Within the lasttwenty five years (or the professional lives ofboth authors) there has been dramaticchanges in the ability to implement realisticmodels Before this it was extremely diffi-cult to implement models that are necessaryfor understanding impacts of individual heterogeneity in aggregation

Aggregation problems remain among themost vexing in all of applied economicsWhile they have not become less difficult inthe past decade it has become possible tostudy aggregation problems in a meaningfulway As such there are many reasons to beoptimistic about the prospects for steadyprogress on aggregation problems in thefuture The practice of ignoring or closetingaggregation problems as ldquojust too hardrdquo is nolonger appropriate

REFERENCES

Aigner Dennis J and Stephen M Goldfeld 1974ldquoEstimation and Prediction from Aggregate Datawhen Aggregates are Measured More Accurately thanTheir Componentsrdquo Econometrica 42(1) 113ndash34

Aiyagari S Rao 1994 ldquoUninsured Idiosyncratic Riskand Aggregate Savingrdquo Quarterly Journal ofEconomics 109(3) 659ndash84

Aiyagari S Rao 1994 ldquoCo-Existence of aRepresentative Agent Type Equilibrium with a Non-Representative Agent Type Equilibriumrdquo Journal ofEconomic Theory 57(1) 230ndash36

Alessie Rob Michael P Devereux and GuglielmoWeber 1997 ldquoIntertemporal ConsumptionDurables and Liquidity Constraints A CohortAnalysisrdquo European Economic Review 41(1) 37ndash59

Alessie Rob Bertrand Melenberg and GuglielmoWeber 1988 ldquoConsumption Leisure and Earnings-Related Liquidity Constraints A Noterdquo EconomicsLetters 27(1) 101ndash04

Allenby Greg M and Peter E Rossi 1991 ldquoThere IsNo Aggregate Bias Why Macro Logit ModelsWorkrdquo Journal of Business and Economic Statistics9(1) 1ndash14

Anderson Simon P Andre de Palma and Jacques-Franccedilois Thisse 1989 ldquoDemand for DifferentiatedProducts Discrete Choice Models and theCharacteristics Approachrdquo Review of EconomicStudies 56(1) 21ndash35

Atkeson Andrew and Masao Ogaki 1996 ldquoWealth-Varying Intertemporal Elasticities of SubstitutionEvidence from Panel and Aggregate Datardquo Journalof Monetary Economics 38(3) 507ndash34

Atkinson Anthony B Joanna Gomulka and NicholasH Stern 1990 ldquoSpending on Alcohol Evidencefrom the Family Expenditure Survey 1970ndash1983rdquoEconomic Journal 100(402) 808ndash27



Attanasio Orazio P 1999 ldquoConsumptionrdquo in Handbookof Macroeconomics John B Taylor and MichaelWoodford eds Amsterdam North Holland 741ndash812

Attanasio Orazio P and Martin Browning 1995ldquoConsumption over the Life Cycle and over theBusiness Cyclerdquo American Economic Review 85(5)1118ndash37

Attanasio Orazio P and Steven J Davis 1996ldquoRelative Wage Movements and the Distribution ofConsumptionrdquo Journal of Political Economy 104(6)1227ndash62

Attanasio Orazio P Pinelopi K Goldberg andEkaterini Kyriazidou In progress ldquoOn the Relevanceof Borrowing Constraints Evidence from Car Loansrdquo

Attanasio Orazio P and Guglielmo Weber 1993ldquoConsumption Growth the Interest Rate andAggregationrdquo Review of Economic Studies 60(3)631ndash49

Attanasio Orazio P and Guglielmo Weber 1995 ldquoIsConsumption Growth Consistent with IntertemporalOptimization Evidence from the ConsumerExpenditure Surveyrdquo Journal of Political Economy103(6) 1121ndash57

Banks James Richard Blundell and Agar Brugiavini2001 ldquoRisk Pooling Precautionary Saving andConsumption Growthrdquo Review of Economic Studies68(4) 757ndash79

Banks James Richard Blundell and Arthur Lewbel1997 ldquoQuadratic Engel Curves and ConsumerDemandrdquo Review of Economics and Statistics 79(4)527ndash39

Banks James Richard Blundell and Sarah Tanner1998 ldquoIs There a Retirement-Savings PuzzlerdquoAmerican Economic Review 88(4) 769ndash88

Barker Terence S and M Hashem Pesaran 1990Disaggregation in Econometric Modeling LondonRoutledge

Barnett William A 1979 ldquoTheoretical Foundations forthe Rotterdam Modelrdquo Review of Economic Studies46(1) 109ndash30

Barten Anton P 1964 ldquoFamily Composition Prices andExpenditure Patternsrdquo in Econometric Analysis forNational Economic Planning P E Hart F Mills andJ K Whitaker eds London Butterworths 277ndash92

Bean Charles R 1986 ldquoThe Estimation of lsquoSurprisersquoModels and the lsquoSurprisersquo Consumption FunctionrdquoReview of Economic Studies 53(4) 497ndash516

Beaudry Paul and David Green 1996 ldquoCohortPatterns in Canadian Earnings Assessing the Role ofSkill Premia in Inequality Trendsrdquo University ofBritish Columbia Working Paper

Becker Gary S 1962 ldquoIrrational Behavior andEconomic Theoryrdquo Journal of Political Economy70(1) 1ndash13

Berndt Ernst R Masako N Darrough and W EDiewert 1977 ldquoFlexible Functional Forms andExpenditure Distributions An Application toCanadian Consumer Demand FunctionsrdquoInternational Economic Review 18(3) 651ndash75

Bewley Truman F 1977 ldquoThe Permanent IncomeHypothesis A Theoretical Formulationrdquo Journal ofEconomic Theory 16(2) 252ndash92

Bierens Herman J and Hettie A Pott-Buter 1990

ldquoSpecification of Household Engel Curves byNonparametric Regressionrdquo Econometric Reviews9(2) 123ndash84

Bils Mark J 1985 ldquoReal Wages over the BusinessCycle Evidence from Panel Datardquo Journal ofPolitical Economy 93(4) 666ndash89

Blackorby Charles Richard Boyce and R RobertRussell 1978 ldquoEstimation of Demand SystemsGenerated by the Gorman Polar Form AGeneralization of the S-Branch Utility TreerdquoEconometrica 46(2) 345ndash63

Blackorby Charles Daniel Primont and R RobertRussell 1978 Duality Separability and FunctionalStructure Theory and Economic ApplicationsAmsterdam North Holland

Blinder Alan S 1975 ldquoDistribution Effects and theAggregate Consumption Functionrdquo Journal ofPolitical Economy 83(3) 447ndash75

Blundell Richard 1988 ldquoConsumer BehaviourTheory and Empirical EvidencemdashA SurveyrdquoEconomic Journal 98(389) 16ndash65

Blundell Richard Martin Browning and Ian ACrawford 2003 ldquoNonparametric Engel Curves andRevealed Preferencerdquo Econometrica 71(1) 205ndash40

Blundell Richard Martin Browning and CostasMeghir 1994 ldquoConsumer Demand and Life-CycleAllocation of Household Expendituresrdquo Review ofEconomic Studies 61(1) 57ndash80

Blundell Richard Alan Duncan and KrishnaPendakur 1998 ldquoSemiparametric Estimation andConsumer Demandrdquo Journal of AppliedEconometrics 13(5) 435ndash61

Blundell Richard John Ham and Costas Meghir1987 ldquoUnemployment and Female Labour SupplyrdquoEconomic Journal 97 44ndash64

Blundell Richard and Thomas MaCurdy 1999ldquoLabor Supply A Review of AlternativeApproachesrdquo in Handbook of Labor Economics VolIII(a) O Ashenfelter and D Card eds AmsterdamNorth Holland 1560ndash95

Blundell Richard Panos Pashardes and GuglielmoWeber 1993 ldquoWhat Do We Learn About ConsumerDemand Patterns from Micro Datardquo AmericanEconomic Review 83(3) 570ndash97

Blundell Richard Luigi Pistaferri and Ian Preston2003 ldquoConsumption Inequality and PartialInsurancerdquo University College London DiscussionPaper 0306

Blundell Richard and Ian Preston 1998ldquoConsumption Inequality and Income UncertaintyrdquoQuarterly Journal of Economics 113(2) 603ndash40

Blundell Richard Howard Reed and Thomas MStoker 2003 ldquoInterpreting Aggregate Wage GrowthThe Role of Labor Market Participationrdquo AmericanEconomic Review 93(4) 1114ndash31

Blundell Richard and Jean-Marc Robin 2000ldquoLatent Separability Grouping Goods without WeakSeparabilityrdquo Econometrica 68 (1)53-84

Blundell Richard and Jean-Marc Robin 1999ldquoEstimation in Large and Disaggregated DemandSystems An Estimator for Conditionally LinearSystemsrdquo Journal of Applied Econometrics 14(3)209ndash32



Blundell Richard and Thomas M Stoker 1999ldquoConsumption and the Timing of Income RiskrdquoEuropean Economic Review 43(3) 475ndash507

Blundell Richard and Thomas M Stoker 2003ldquoModels of Aggregate Economic Relationships thatAccount for Heterogeneityrdquo Mimeo

Bourguignon Franccedilois and Pierre-Andreacute Chiappori1994 ldquoThe Collective Approach to HouseholdBehaviorrdquo in The Measurement of HouseholdWelfare R Blundell I Preston and I Walker edsCambridge Cambridge University Press 70ndash86

Brown Donald J and Rosa L Matzkin 1996ldquoTestable Restrictions on the Equilbrium ManifoldrdquoEconometrica 64(6) 1249ndash62

Browning Martin 1992 ldquoChildren and HouseholdEconomic Behaviorrdquo Journal of EconomicLiterature 30(3) 1434ndash75

Browning Martin 1993 ldquoEstimating Micro Parametersfrom Macro Data Alone Some PessimisticEvidencerdquo Ricerche Economiche 47(3) 253ndash67

Browning Martin Lars P Hansen and James JHeckman 1999 ldquoMicro Data and GeneralEquilibrium Modelsrdquo in Handbook ofMacroeconomics J Taylor and M Woodford edsAmsterdam Elsevier 543ndash633

Browning Martin and Annamaria Lusardi 1996ldquoHousehold Saving Micro Theories and MicroFactsrdquo Journal of Economic Literature 34(4)1797ndash1855

Browning Martin and Costas Meghir 1991 ldquoTheEffects of Male and Female Labor Supply onCommodity Demandsrdquo Econometrica 59(4) 925ndash51

Buse Adolph 1992 ldquoAggregation Distribution andDynamics in the Linear and Quadratic ExpenditureSystemsrdquo Review of Economics and Statistics 74(1)45ndash53

Caballero Ricardo J 1990 ldquoConsumption Puzzles andPrecautionary Savingsrdquo Journal of MonetaryEconomics 25(1) 113ndash36

Caballero Ricardo J and Eduardo M R A Engel1991 ldquoDynamic (Ss) Economiesrdquo Econometrica59(6) 1659ndash86

Caballero Ricardo J and Eduardo M R A Engel1992 ldquoMicroeconomic Adjustment Hazards andAggregate Dynamicsrdquo NBER Working Paper No4090

Cameron A Colin 1990 ldquoAggregation in DiscreteChoice Models An Illustration of NonlinearAggregationrdquo in Disaggregation in EconometricModeling T S Barker and M H Pesaran edsLondon Routledge 206ndash34

Caplin Andrew and Barry Nalebuff 1991ldquoAggregation and Imperfect Competition On theExistence of Equilibriumrdquo Econometrica 59(1)25ndash59

Caplin Andrew S and Daniel F Spulber 1987ldquoMenu Costs and the Neutrality of MoneyrdquoQuarterly Journal of Economics 102(4) 703ndash25

Carroll Christopher D 2000 ldquoRequiem for theRepresentative Consumer Aggregate Implicationsof Microeconomic Consumption BehaviorrdquoAmerican Economic Review 90(2) 110ndash15

Chetty V K and James J Heckman 1986 ldquoA

Dynamic Model of Aggregate Output Supply FactorDemand and Entry and Exit for a CompetitiveIndustry with Heterogeneous Plantsrdquo Journal ofEconometrics 33(1-2) 237ndash62

Chiappori Pierre-Andre 1988 ldquoRational HouseholdLabor Supplyrdquo Econometrica 56(1) 63ndash90

Cochrane John H 1991 ldquoA Simple Test ofConsumption Insurancerdquo Journal of PoliticalEconomy 99(5) 957ndash76

Cogan John F 1981 ldquoFixed Costs and Labor SupplyrdquoEconometrica 49(4) 945ndash63

Cowing Thomas G and Daniel L McFadden 1984Microeconomic Modeling and Policy Analysis NewYork Academic Press

Deaton Angus S 1985 ldquoPanel Data from Time Seriesof Cross-Sectionsrdquo Journal of Econometrics 30(1-2)109ndash26

Deaton Angus S 1991 ldquoSaving and LiquidityConstraintsrdquo Econometrica 59(5) 1221ndash48

Deaton Angus S 1993 Understanding ConsumptionOxford Oxford University Press

Deaton Angus S and John Muellbauer 1980a ldquoAnAlmost Ideal Demand Systemrdquo American EconomicReview 70(3) 312ndash26

Deaton Angus S and John Muellbauer 1980bEconomics and Consumer Behavior CambridgeCambridge University Press

Deaton Angus S and Christina Paxson 1994ldquoIntertemporal Choice and Inequalityrdquo Journal ofPolitical Economy 102(3) 437ndash67

Debreu Gerard 1974 ldquoExcess Demand FunctionsrdquoJournal of Mathematical Economics 1(1) 15ndash23

Dickens Richard 1996 ldquoThe Evolution of IndividualMale Earnings in Great Britain 1974ndash1994rdquo CEPDiscussion Paper No 306

Diewert W E 1977 ldquoGeneralized Slutsky Conditionsfor Aggregate Consumer Demand FunctionsrdquoJournal of Economic Theory 15(2) 353ndash62

Dumas Bernard 1989 ldquoTwo-Person DynamicEquilibrium in the Capital Marketrdquo Review ofFinancial Studies 2(2) 157ndash88

Dynan Karen E 1993 ldquoHow Prudent AreConsumersrdquo Journal of Political Economy 101(6)1104ndash13

Epstein Larry G and Stanley E Zin 1989ldquoSubstitution Risk Aversion and the TemporalBehavior of Consumption and Asset Returns ATheoretical Frameworkrdquo Econometrica 57(4)937ndash69

Epstein Larry G and Stanley E Zin 1991ldquoSubstitution Risk Aversion and the TemporalBehavior of Consumption and Asset Returns AnEmpirical Analysisrdquo Journal of Political Economy99(2) 263ndash86

Fair Ray C and Kathryn M Dominguez 1991ldquoEffects of the Changing US Age Distribution onMacroeconomic Equationsrdquo American EconomicReview 81(5) 1276ndash94

Feenberg Daniel R and Harvey S Rosen 1983ldquoAlternative Tax Treatments of the FamilySimulation Methodology and Resultsrdquo in BehavioralSimulation Methods in Tax Policy Analysis M SFeldstein ed Chicago University of Chicago Press



Forni Mario and Marco Lippi 1997 Aggregation andthe Microfoundations of Dynamic MacroeconomicsOxford Clarendon Press

Freixas Xavier and Andreu Mas-Colell 1987 ldquoEngelCurves Leading to the Weak Axiom in theAggregaterdquo Econometrica 55(3) 515ndash31

Garcia Rene Annamaria Lusardi and Serena Ng1997 ldquoExcess Sensitivity and Asymmetries inConsumptionrdquo Journal of Money Credit andBanking 29(2) 154ndash76

Goodfriend Marvin 1992 ldquoInformation-AggregationBiasrdquo American Economic Review 82(3) 508ndash19

Gorman William M 1953 ldquoCommunity PreferenceFieldsrdquo Econometrica 21(1) 63ndash80

Gorman William M 1981 ldquoSome Engel Curvesrdquo inEssays in the Theory and Measurement of ConsumerBehaviour A S Deaton ed Cambridge CambridgeUniversity Press 7ndash29

Gosling Amanda Stephen Machin and CostasMeghir 2000 ldquoThe Changing Distribution of MaleWages in the UK 1966ndash93rdquo Review of EconomicStudies 67(4) 635ndash66

Grandmont Jean-Michel 1992 ldquoTransformations ofthe Commodity Space Behavioral Heterogeneityand the Aggregation Problemrdquo Journal of EconomicTheory 57(1) 1ndash35

Granger Clive W J 1980 ldquoLong MemoryRelationships and the Aggregation of DynamicModelsrdquo Journal of Econometrics 14(2) 227ndash38

Granger Clive W J 1987 ldquoImplications of Aggregationwith Common Factorsrdquo Econometric Theory 3208ndash22

Granger Clive W J 1990 ldquoAggregation of Time-SeriesVariables A Surveyrdquo in Disaggregation inEconometric Modelling T Barker and M H Pesaraneds London and New York Routledge 17ndash34

Grunfeld Yehuda and Zvi Griliches 1960 ldquoIsAggregation Necessarily Badrdquo Review of Economicsand Statistics 42(1) 1ndash13

Hall Robert E 1978 ldquoStochastic Implications of theLife Cycle-Permanent Income Hypothesis Theoryand Evidencerdquo Journal of Political Economy 86(6)971ndash87

Hall Robert E 1988 ldquoIntertemporal Substitution inConsumptionrdquo Journal of Political Economy 96(2)339ndash57

Hansen Gary D 1985 ldquoIndivisible Labor and theBusiness Cyclerdquo Journal of Monetary Economics16(3) 309ndash27

Hansen Lars Peter and Kenneth J Singleton 1982ldquoGeneralized Instrumental Variables Estimation ofNonlinear Rational Expectations ModelsrdquoEconometrica 50(5) 1269ndash86

Hansen Lars Peter and Kenneth J Singleton 1983ldquoStochastic Consumption Risk Aversion and theTemporal Behavior of Asset Returnsrdquo Journal ofPolitical Economy 91(2) 249ndash65

Haumlrdle Wolfgang Werner Hildenbrand and MichaelJerison 1991 ldquoEmpirical Evidence on the Law ofDemandrdquo Econometrica 59(6) 1525ndash49

Hausman Jerry A Whitney K Newey HidehikoIchimura and James L Powell 1991ldquoIdentification and Estimation of Polynomial

Errors-in-Variables Modelsrdquo Journal ofEconometrics 50(3) 273ndash95

Hausman Jerry A Whitney K Newey and James LPowell 1994 ldquoNonlinear Errors-In-VariablesEstimation of Some Engel Curvesrdquo Journal ofEconometrics 65(1) 205ndash33

Hayashi Fumio 1985 ldquoThe Effect of LiquidityConstraints on Consumption A Cross-sectionalAnalysisrdquo Quarterly Journal of Economics 100(1)183ndash206

Hayashi Fumio 1987 ldquoTests for Liquidity ConstraintsA Critical Surveyrdquo in Advances in EconometricsFifth World Congress Vol 2 T Bewley edCambridge Cambridge University Press 91ndash120

Hayashi Fumio Joseph Altonji and LaurenceKotlikoff 1996 ldquoRisk-Sharing between and withinFamiliesrdquo Econometrica 64(2) 261ndash94

Heaton John and Deborah J Lucas 1996ldquoEvaluating the Effects of Incomplete Markets onRisk Sharing and Asset Pricingrdquo Journal of PoliticalEconomy 104(3) 443ndash87

Heckman James J 1974 ldquoEffects of Child-CarePrograms on Womenrsquos Work Effortrdquo Journal ofPolitical Economy 82(2) S136ndash163

Heckman James J 1979 ldquoSample Selection Bias as aSpecification Errorrdquo Econometrica 47(1) 153ndash61

Heckman James J 1990 ldquoVarieties of Selection BiasrdquoAmerican Economic Review 80(2) 313ndash18

Heckman James J and Bo E Honore 1990 ldquoTheEmpirical Content of the Roy ModelrdquoEconometrica 58(5) 1121ndash49

Heckman James J and Guilherme Sedlacek 1985ldquoHeterogeneity Aggregation and Market WageFunctions An Empirical Model of Self-Selection inthe Labor Marketrdquo Journal of Political Economy93(6) 1077ndash1125

Heckman James J and Guilherme L Sedlacek 1990ldquoSelf-Selection and the Distribution of HourlyWagesrdquo Journal of Labor Economics 8(1) S329ndash63

Heckman James J and James R Walker 1989ldquoForecasting Aggregate Period-Specific Birth RatesThe Time Series Properties of a MicrodynamicNeoclassical Model of Fertilityrdquo Journal of theAmerican Statistical Association 84(408) 958ndash65

Heckman James J and James R Walker 1990 ldquoTheRelationship between Wages and Income and theTiming and Spacing of Births Evidence fromSwedish Longitudinal Datardquo Econometrica 58(6)1411ndash41

Heineke J M and H M Shefrin 1990 ldquoAggregationand Identification in Consumer Demand SystemsrdquoJournal of Econometrics 44(3) 377ndash90

Hicks J R 1956 A Revision of Demand TheoryLondon Oxford University Press

Hildenbrand Werner 1981 ldquoShort-Run ProductionFunctions Based on Microdatardquo Econometrica49(5) 1095ndash1125

Hildenbrand Werner 1983 ldquoOn the lsquoLaw ofDemandrsquordquo Econometrica 51(4) 997ndash1019

Hildenbrand Werner 1992 ldquoMarket Demand Theoryand Empirical Evidencerdquo Universitaumlt BonnDiscussion Paper A-359

Hildenbrand Werner 1994 Market Demand Theory



and Empirical Evidence Princeton PrincetonUniversity Press

Hildenbrand Werner 1998 ldquoHow Relevant areSpecifications of Behavioral Relations on the Micro-Level for Modelling the Time Path of PopulationAggregatesrdquo European Economic Review 42(3-5)437ndash58

Hildenbrand Werner and Alois Kneip 1993 ldquoFamilyExpenditure Data Heteroscedasticity and the Lawof Demandrdquo Ricerche Economiche 47(2) 137ndash65

Houthakker Hendrick S 1955 ldquoThe ParetoDistribution and the Cobb-Douglas ProductionFunction in Activity Analysisrdquo Review of EconomicStudies 23(1) 27ndash31

Jappelli Tullio 1990 ldquoWho Is Credit Constrained inthe US Economyrdquo Quarterly Journal ofEconomics 105(1) 219ndash34

Jappelli Tullio and Marco Pagano 1994 ldquoSavingGrowth and Liquidity Constraintsrdquo QuarterlyJournal of Economics 109(1) 83ndash109

Jappelli Tullio Jorn-Steffen Pischke and Nicholas SSouleles 1998 ldquoTesting for Liquidity Constraints inEuler Equations with Complementary Data SourcesrdquoReview of Economics and Statistics 80(2) 251ndash62

Johansen Leif 1972 Production FunctionsAmsterdam North Holland

Joint Committee on Taxation 1992 ldquoDiscussion ofRevenue Estimation Methodology and Processrdquo JCS14-92 August 13 1992 Washington DC USGovernment Printing Office

Jorgenson Dale W Lawrence J Lau and Thomas MStoker 1980 ldquoWelfare Comparison and ExactAggregationrdquo American Economic Review 70(2)268ndash72

Jorgenson Dale W Lawrence J Lau and Thomas MStoker 1982 ldquoThe Transcendental LogarithmicModel of Aggregate Consumer Behaviorrdquo inAdvances in Econometrics Vol 1 R L Basmannand G Rhodes eds Greenwich JAI Press 97ndash238

Jorgenson Dale W and Daniel T Slesnick 1984ldquoAggregate Consumer Behavior and theMeasurement of Inequalityrdquo Review of EconomicStudies 51(3) 369ndash92

Jorgenson Dale W Daniel T Slesnick and ThomasM Stoker 1988 ldquoTwo-Stage Budgeting and ExactAggregationrdquo Journal of Business and EconomicStatistics 6(3) 313ndash25

Jorgenson Dale W and Thomas M Stoker 1997ldquoNonlinear Three-Stage Least Squares Pooling ofTime Series and Cross-Section Datardquo in Advances inStatistical Analysis and Statistical Computing Vol 1D W Jorgenson ed Greenwich JAI Press 87ndash116

Katz Lawrence F and Kevin M Murphy 1992ldquoChanges in Relative Wages 1963ndash1987 Supply andDemand Factorsrdquo Quarterly Journal of Economics107(1) 35ndash78

Kelejian Harry H 1980 ldquoAggregation andDisaggregation of Nonlinear Equationsrdquo inEvaluation of Econometric Models J Kmenta and JB Ramsey eds New York Academic Press

Kirman Alan P 1992 ldquoWhom or What Does theRepresentative Individual Representrdquo Journal ofEconomic Perspectives 6(2) 117ndash36

Krusell Per and Anthony A Smith Jr 1998 ldquoIncomeand Wealth Heterogeneity in the MacroeconomyrdquoJournal of Political Economy 106(5) 867ndash96

Lam Pok-Sang 1991 ldquoPermanent Income Liquidityand Adjustments of Automobile Stocks Evidencefrom Panel Datardquo Quarterly Journal of Economics106(1) 203ndash30

Lau Lawrence J 1977 ldquoExistence Conditions forAggregate Demand Functionsrdquo Institute forMathematical Studies in the Social SciencesStanford University Technical Report No 248

Lau Lawrence J 1982 ldquoA Note on the FundamentalTheorem of Exact Aggregationrdquo Economics Letters9(1) 119ndash26

Lee Kevin C M Hashem Pesaran and Richard GPierse 1990 ldquoTesting for Aggregation Bias in LinearModelsrdquo Economic Journal 100(400) 137ndash50

Lee Lung-Fei and Robert H Porter 1984 ldquoSwitchingRegression Models with Imperfect SampleSeparation Information With an Application onCartel Stabilityrdquo Econometrica 52(2) 391ndash418

Leser Conrad E V 1963 ldquoForms of Engel FunctionsrdquoEconometrica 31(4) 694ndash703

Lewbel Arthur 1989a ldquoA Demand System RankTheoremrdquo Econometrica 57(3) 701ndash05

Lewbel Arthur 1989b ldquoExact Aggregation and aRepresentative Consumerrdquo Quarterly Journal ofEconomics 104(3) 621ndash33

Lewbel Arthur 1990 ldquoIncome DistributionMovements and Aggregate Money Illusionrdquo Journalof Econometrics 43(1-2) 35ndash42

Lewbel Arthur 1991 ldquoThe Rank of DemandSystems Theory and Nonparametric EstimationrdquoEconometrica 59(3) 711ndash30

Lewbel Arthur 1992 ldquoAggregation with Log-LinearModelsrdquo Review of Economic Studies 59(3) 635ndash42

Lewbel Arthur 1993 ldquoDistribution MovementsMacroeconomic Regularities and the RepresentativeConsumerrdquo Ricerche Economiche 47(2) 189ndash99

Lewbel Arthur 1994 ldquoAggregation and SimpleDynamicsrdquo American Economic Review 84(4)905ndash18

Lippi Marco 1988 ldquoOn the Dynamic Shape ofAggregated Error Correction Modelsrdquo Journal ofEconomic Dynamics and Control 12(2-3) 561ndash85

Lusardi Annamaria 1996 ldquoPermanent IncomeCurrent Income and Consumption Evidence fromTwo Panel Data Setsrdquo Journal of Business andEconomic Statistics 14(1) 81ndash90

Mace Barbara J 1991 ldquoFull Insurance in the Presenceof Aggregate Uncertaintyrdquo Journal of PoliticalEconomy 99(5) 928ndash56

MaCurdy Thomas E 1982 ldquoThe Use of Time SeriesProcesses to Model the Error Structure of Earningsin a Longitudinal Data Analysisrdquo Journal ofEconometrics 18(1) 83ndash114

MaCurdy Thomas E 1987 ldquoA Framework forRelating Microeconomic and MacroeconomicEvidence on Intertemporal Substitutionrdquo inAdvances in Econometrics Fifth World CongressVol 2 T F Bewley ed Cambridge CambridgeUniversity Press 149ndash76

Mankiw N Gregory and David N Weil 1989 ldquoThe



Baby Boom the Baby Bust and the HousingMarketrdquo Regional Science and Urban Economics19(2) 235ndash58

Marshak Jacob 1939 ldquoPersonal and Collective BudgetFunctionsrdquo Review of Economic Statistics 21161ndash70

McFadden Daniel and F Reid 1975 ldquoAggregateTravel Demand Forecasting from DisaggregatedBehavioral Modelsrdquo Transportation ResearchRecord No 534

Meghir Costas and Luigi Pistaferri 2004 ldquoIncomeVariance Dynamics and HeterogeneityrdquoEconometrica 72(1) 1ndash32

Meghir Costas and Guglielmo Weber 1996ldquoIntertemporal Nonseparability or BorrowingRestrictions A Disaggregate Analysis Using a USConsumption Panelrdquo Econometrica 64(5)1151ndash81

Meghir Costas and Edward Whitehouse 1996 ldquoTheEvolution of Wages in the United KingdomEvidence from Micro Datardquo Journal of LaborEconomics 14(1) 1ndash25

Modigliani Franco 1970 ldquoThe Life Cycle Hypothesisof Saving and Intercountry Differences in the SavingRatiordquo in Induction Growth and Trade W Eltis MScott and I Wolfe eds Oxford Clarendon Press197ndash225

Moffitt Robert 1991 ldquoIdentification and Estimationof Dynamic Models with a Time Series of RepeatedCross Sectionsrdquo Brown University Draft

Muellbauer John 1975 ldquoAggregation IncomeDistribution and Consumer Demandrdquo Review ofEconomic Studies 42(4) 525ndash43

Muellbauer John 1976 ldquoCommunity Preferences andthe Representative Consumerrdquo Econometrica 44(5)979ndash99

Muellbauer John 1981 ldquoLinear Aggregation inNeoclassical Labour Supplyrdquo Review of EconomicStudies 48(1) 21ndash36

Pesaran M Hashem Richard G Pierse and Mohan SKumar 1989 ldquoEconometric Analysis of Aggregationin the Context of Linear Prediction ModelsrdquoEconometrica 57(4) 861ndash88

Pischke Jorn-Steffen 1995 ldquoIndividual IncomeIncomplete Information and AggregateConsumptionrdquo Econometrica 63(4) 805ndash40

Pollak Robert A 1985 ldquoA Transaction Cost Approachto Families and Householdsrdquo Journal of EconomicLiterature 23(2) 581ndash608

Pollak Robert A and Terence J Wales 1981ldquoDemographic Variables in Demand AnalysisrdquoEconometrica 49(6) 1533ndash51

Powell James L and Thomas M Stoker 1985 ldquoTheEstimation of Complete Aggregation StructuresrdquoJournal of Econometrics 30(1ndash2) 317ndash44

Powell James L 1994 ldquoEstimation of Semi-Parametric Modelsrdquo in Handbook of EconometricsVol 4 R F Engel and D L McFadden edsAmsterdam Elsevier 2443ndash2521

Prescott Edward C 1986 ldquoTheory Ahead of BusinessCycle Measurementrdquo Federal Reserve Bank ofMinneapolis Quarterly Review 10(4) 9ndash22

Ray Ranjan 1983 ldquoMeasuring the Costs of Children

An Alternative Approachrdquo Journal of PublicEconomics 22(1) 89ndash102

Rubinstein Mark 1974 ldquoAn Aggregation Theorem forSecurities Marketsrdquo Journal of Financial Economics1(3) 225ndash44

Runkle David E 1991 ldquoLiquidity Constraints and thePermanent-Income Hypothesis Evidence fromPanel Datardquo Journal of Monetary Economics 27(1)73ndash98

Roy Andrew D 1951 ldquoSome Thoughts on theDistribution of Earningsrdquo Oxford Economic Papers3 135ndash46

Samuelson Paul A 1956 ldquoSocial Indifference CurvesrdquoQuarterly Journal of Economics 70(1) 1ndash22

Sato Kazuo 1975 Production Functions andAggregation Amsterdam North Holland

Scheinkman Jose A and Laurence Weiss 1986ldquoBorrowing Constraints and Aggregate EconomicActivityrdquo Econometrica 54(1) 23ndash45

Shafer Wayne and Hugh Sonnenschein 1982 ldquoMarketDemand and Excess Demand Functionsrdquo inHandbook of Mathematical Economics Vol 2 K JArrow and M D Intriligator eds New York NorthHolland 339ndash53

Skinner Jonathan 1988 ldquoRisky Income Life CycleConsumption and Precautionary Savingsrdquo Journalof Monetary Economics 22(2) 237ndash55

Solon Gary Robert Barsky and Jonathan A Parker1994 ldquoMeasuring the Cyclicality of Real WagesHow Important Is Composition Biasrdquo QuarterlyJournal of Economics 109(1) 1ndash25

Sonnenschein Hugo 1972 ldquoMarket Excess DemandFunctionsrdquo Econometrica 40(3) 549ndash63

Stoker Thomas M 1978 ldquoThe Pooling of CrossSection and Average Time Series Datardquo DraftHarvard University

Stoker Thomas M 1982 ldquoThe Use of Cross-SectionData to Characterize Macro Functionsrdquo Journal ofthe American Statistical Association 77(378)369ndash80

Stoker Thomas M 1984a ldquoCompletenessDistribution Restrictions and the Form ofAggregate Functionsrdquo Econometrica 52(4)887ndash907

Stoker Thomas M 1984b ldquoExact Aggregation andGeneralized Slutsky Conditionsrdquo Journal ofEconomic Theory 33(2) 368ndash77

Stoker Thomas M 1985 ldquoAggregation StructuralChange and Cross-Section Estimationrdquo Journal ofthe American Statistical Association 80(391)720ndash29

Stoker Thomas M 1986a ldquoAggregation Efficiencyand Cross-Section Regressionrdquo Econometrica 54(1)171ndash88

Stoker Thomas M 1986b ldquoConsistent Estimation ofScaled Coefficientsrdquo Econometrica 54(6) 1461ndash81

Stoker Thomas M 1986c ldquoThe Distributional WelfareEffects of Rising Prices in the United States The1970rsquos Experiencerdquo American Economic Review76(3) 335ndash49

Stoker Thomas M 1986d ldquoSimple Tests ofDistributional Effects on MacroeconomicEquationsrdquo Journal of Political Economy 94(4)


763ndash95Stoker Thomas M 1992 Lectures on Semiparametric

Econometrics CORE Lecture Series Louvain-la-Neuve CORE Foundation

Stoker Thomas M 1993 ldquoEmpirical Approaches to theProblem of Aggregation Over Individualsrdquo Journalof Economic Literature 31(4) 1827ndash74

Theil Henri 1954 Linear Aggregation of EconomicRelations Amsterdam North Holland

Theil Henri 1975 Theory and Measurement ofConsumer Demand Vol 1 Amsterdam NorthHolland

Townsend Robert M 1994 ldquoRisk and Insurance inVillage Indiardquo Econometrica 62(3) 539ndash91

Van Daal Jan and Arnold H Q M Merkies 1984Aggregation in Economic Research Dordrecht DReidel

Weber Guglielmo 1993 ldquoEarnings-RelatedBorrowing Restrictions Empirical Evidence from aPseudo Panel for the UKrdquo Annales drsquoEconomie et

de Statistique 29 157ndash73de Wolff Pieter 1941 ldquoIncome Elasticity of Demand

a Micro-Economic and a Macro-EconomicInterpretationrdquo Economic Journal 51 140ndash45

Working Harold 1943 ldquoStatistical Laws of FamilyExpenditurerdquo Journal of the American StatisticalAssociation 38 43ndash56

Zeldes Stephen P 1989a ldquoConsumption and LiquidityConstraints An Empirical Investigationrdquo Journal ofPolitical Economy 97(2) 305ndash46

Zeldes Stephen P 1989b ldquoOptimal Consumption withStochastic Income Deviations from CertaintyEquivalencerdquo Quarterly Journal of Economics104(2) 275ndash98

Zellner Arnold 1969 ldquoOn the Aggregation Problem aNew Approach to a Troublesome Problemrdquo inEconomic Models Estimation and RiskProgramming Essays in Honor of Gerhard TintnerK A Fox G V L Narasimhan and J K Senguptaeds Berlin Springer 365ndash74




1 J R Hicks (1956) p 552 This effort is underway primarily in work on psycho-

logical tendencies in economics finance and marketingThis work examines what is ldquoconsistent economic behav-iorrdquo and whether departures from rational behavior aresystematic or idiosyncratic

3 For instance to study inequality a relevant aggregatewould be the Gini coefficient The choice of aggregate mayeven be informed by empirical regularities in individualdata For example if an individual model is best specifiedwith the logarithm of observed income the geometricmean of income might be a more natural aggregate thantotal income or average income

ldquoreasonably homogeneousrdquo groups of house-holds firms or other types of economicagents or to Hicksrsquos Mr Brown or Mr Jonesas well as any of our own relatives or neigh-bors 1 To assert that there is a ldquocorrectrdquo indi-vidual level at which to apply a mathematicalmodel that is in line with rational behavior isto take a stand on those issues a stand whichcould only be properly validated by experi-mentation or much more extensive empiricalresearch than has been performed to date2

At the other extreme are questions per-taining to what the appropriate ldquoaggregaterdquois One typically considers sums or averagesas reported in national income accounts asthe relevant aggregates They are usually themost interpretable numbers and the mostrelevant for pricing or policy analysis Butwith large populations one could considermany other kinds of aggregates or statisticsfrom the population3

Once these issues are settledmdashwhat is therelevant ldquoindividual levelrdquo and what is therelevant ldquoaggregaterdquomdashthen aggregationproblems become purely practical For anyapplication a model must be specifiedwhich captures all important economiceffects allows for relevant individual hetero-geneity and bridges the gap between indi-vidual and aggregate facilitating analysis atboth levels

This survey covers specific solutions andrelated work in primarily three applicationareas consumer demand analysis consump-tion and saving analysis and analysis ofwages and labor market participation A keyissue is to identify what kinds of individual

4 This roughly coincides with the categorization of het-erogeneity discussed in Martin Browning Lars P Hansenand James J Heckman (1999 chapter 8)

differences or heterogeneity are relevantfor each application area As an organizingprinciple we consider (1) heterogeneity inindividual tastes and incomes (2) hetero-geneity in wealth and income risks faced byindividuals and (3) heterogeneity in marketparticipation4 There is a generic tensionbetween the degree of individual hetero-geneity accounted for and the ease withwhich one can draw implications for econom-ic aggregates We point out how differenttypes of heterogeneity are accommodated inthe different application areas

Our approach is practical and we hope toaddress many of the concerns faced byempirical research regarding aggregationWe take a ldquomicro-econometricrdquo view of theindividual modelmdashnamely an econometricmodel (obeying restrictions of economic the-ory) is applicable to individuals or house-holds We consider model specifications thatare typically used in empirical analysis ofindividual data in each application area Wetake the relevant ldquoaggregatesrdquo to be eithertotaled or per-capita (averaged) values of theindividual variables of interest coincidingwith aggregates as typically reported forregions or whole economies Whether suchaggregates are easy to model or not they arethe most interpretable and the most usefultypes of aggregates for interpretation policyanalysis or other uses of empirical results

We are concerned with models that strike abalance between realism (flexibility) adher-ence to restrictions from economic theoryand connections between individual behaviorand aggregate statistics We consider severalsettings where individual models are intrinsi-cally nonlinear and for those we must makespecific assumptions on the distributions ofrelevant heterogeneous characteristics Wepresent results that can be used to explore theimpact of heterogeneity in empirical applica-tions that assume reasonable (and hopefully






























(1)


8


(2)


(3)



sum=

sumi ijt

t

i it

t

q

nj J

mn

1 hellip and







(4)


(5)


(6)


(7)

times












(8)

times

times

where

(9)



π 2 titE m z

= iit

it itE m E z( )

( ) ( )

π1tit it

it it

E m mE m E m

=( )

( ) ( )ln

lnand











(10)


(11)

times









(12)


(13)

times




E q a p b pt ijt j t j t( ) = ( ) + ( )0 0










(14)

times


(15)

times



E p q

E mE w

b p b

t jt ijt

t itt it ijt

j t j

( )( ) ( )

= ( ) +

micro

0 1 ppt( )


wp q

mb p b pijt

jt ijt

itj t j t= = ( ) + ( )0 1



(16)

times


(17)





π ktt it k it

k t it

E h mh E m

k M=( )( )

( )( ) =micro

1 hellip



b p b pt t= ( ) +0 1 (( )π1t

E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(18)



(19)


distribution

(20)




(21)


πmicro σ1 2

11

2 1tmt mt

= + ( ) +

πmicro

1tt it it

t it

t it it

t it

E mE m

E m mE m

=( )

( ) =( )ln

lnln

(( ) ( )ln E mt it


E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(22)


(23)


(24)



(25)


(26)




p E mj t t it+ ( )b1 ln (( )

E p q

E mpt jt ijt

t itj t

( )( ) = ( )b0

b p c c Ej t t+ ( ) +1 1 2 ln mmit( )( )

E p q

E mb pt jt ijt

t itj t

( )( ) = ( )0








(27)




(28)




t it itE m m=

(ln ))( )( ) ( )( )

2

2E m E mt it t itln

πmicro

2

2

2t

t it it

t it

E m

E m=

( )( )( )( )ln

ln


17


(29)




(30)





t itE m z= iit it

t it t it t it

mE m E z E m

lnln

( )( ) ( ) ( )

πmicro

1tz t it it it

t it t it

E z mE z E m

=( )

( ) ( )ln

ln

πmicro

tz t it it

t it

t it it

t it t i

E zE z

E m zE m E z

=( )

( ) =( )

( ) tt( )




3 4 5 6 70

01

02

03

04

05

06

07

Foo

d Sh

are

Log Expenditure






(31)




=prime

prime

γ

γ

1

N



ln

mj

it+minus

λln lnna p

c put

tijt

( )( )( ) +

2





35 4 45 5 55 6 65 7


Alc

ohol

Sha

re

Log Expenditure












(32)


(33)


ˆˆ

πmicro

tz i it it t

i it t

z nz n

=sum

sum

ˆˆ

πmicro

1ti it it t

i it t

m nm n

=sum ( )

sum( )ln

ln




1

105

11

115

12

125

13

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000












102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999









(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )






101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES









































































































































































































































jn05_Article 1 61005 137 PM 49




















(1)


8


(2)


(3)



sum=

sumi ijt

t

i it

t

q

nj J

mn

1 hellip and







(4)


(5)


(6)


(7)

times












(8)

times

times

where

(9)



π 2 titE m z

= iit

it itE m E z( )

( ) ( )

π1tit it

it it

E m mE m E m

=( )

( ) ( )ln

lnand











(10)


(11)

times









(12)


(13)

times




E q a p b pt ijt j t j t( ) = ( ) + ( )0 0










(14)

times


(15)

times



E p q

E mE w

b p b

t jt ijt

t itt it ijt

j t j

( )( ) ( )

= ( ) +

micro

0 1 ppt( )


wp q

mb p b pijt

jt ijt

itj t j t= = ( ) + ( )0 1



(16)

times


(17)





π ktt it k it

k t it

E h mh E m

k M=( )( )

( )( ) =micro

1 hellip



b p b pt t= ( ) +0 1 (( )π1t

E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(18)



(19)


distribution

(20)




(21)


πmicro σ1 2

11

2 1tmt mt

= + ( ) +

πmicro

1tt it it

t it

t it it

t it

E mE m

E m mE m

=( )

( ) =( )ln

lnln

(( ) ( )ln E mt it


E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(22)


(23)


(24)



(25)


(26)




p E mj t t it+ ( )b1 ln (( )

E p q

E mpt jt ijt

t itj t

( )( ) = ( )b0

b p c c Ej t t+ ( ) +1 1 2 ln mmit( )( )

E p q

E mb pt jt ijt

t itj t

( )( ) = ( )0








(27)




(28)




t it itE m m=

(ln ))( )( ) ( )( )

2

2E m E mt it t itln

πmicro

2

2

2t

t it it

t it

E m

E m=

( )( )( )( )ln

ln


17


(29)




(30)





t itE m z= iit it

t it t it t it

mE m E z E m

lnln

( )( ) ( ) ( )

πmicro

1tz t it it it

t it t it

E z mE z E m

=( )

( ) ( )ln

ln

πmicro

tz t it it

t it

t it it

t it t i

E zE z

E m zE m E z

=( )

( ) =( )

( ) tt( )




3 4 5 6 70

01

02

03

04

05

06

07

Foo

d Sh

are

Log Expenditure






(31)




=prime

prime

γ

γ

1

N



ln

mj

it+minus

λln lnna p

c put

tijt

( )( )( ) +

2





35 4 45 5 55 6 65 7


Alc

ohol

Sha

re

Log Expenditure












(32)


(33)


ˆˆ

πmicro

tz i it it t

i it t

z nz n

=sum

sum

ˆˆ

πmicro

1ti it it t

i it t

m nm n

=sum ( )

sum( )ln

ln




1

105

11

115

12

125

13

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000












102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999









(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )






101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES




















































































































































































































































(1)


8


(2)


(3)



sum=

sumi ijt

t

i it

t

q

nj J

mn

1 hellip and







(4)


(5)


(6)


(7)

times












(8)

times

times

where

(9)



π 2 titE m z

= iit

it itE m E z( )

( ) ( )

π1tit it

it it

E m mE m E m

=( )

( ) ( )ln

lnand











(10)


(11)

times









(12)


(13)

times




E q a p b pt ijt j t j t( ) = ( ) + ( )0 0










(14)

times


(15)

times



E p q

E mE w

b p b

t jt ijt

t itt it ijt

j t j

( )( ) ( )

= ( ) +

micro

0 1 ppt( )


wp q

mb p b pijt

jt ijt

itj t j t= = ( ) + ( )0 1



(16)

times


(17)





π ktt it k it

k t it

E h mh E m

k M=( )( )

( )( ) =micro

1 hellip



b p b pt t= ( ) +0 1 (( )π1t

E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(18)



(19)


distribution

(20)




(21)


πmicro σ1 2

11

2 1tmt mt

= + ( ) +

πmicro

1tt it it

t it

t it it

t it

E mE m

E m mE m

=( )

( ) =( )ln

lnln

(( ) ( )ln E mt it


E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(22)


(23)


(24)



(25)


(26)




p E mj t t it+ ( )b1 ln (( )

E p q

E mpt jt ijt

t itj t

( )( ) = ( )b0

b p c c Ej t t+ ( ) +1 1 2 ln mmit( )( )

E p q

E mb pt jt ijt

t itj t

( )( ) = ( )0








(27)




(28)




t it itE m m=

(ln ))( )( ) ( )( )

2

2E m E mt it t itln

πmicro

2

2

2t

t it it

t it

E m

E m=

( )( )( )( )ln

ln


17


(29)




(30)





t itE m z= iit it

t it t it t it

mE m E z E m

lnln

( )( ) ( ) ( )

πmicro

1tz t it it it

t it t it

E z mE z E m

=( )

( ) ( )ln

ln

πmicro

tz t it it

t it

t it it

t it t i

E zE z

E m zE m E z

=( )

( ) =( )

( ) tt( )




3 4 5 6 70

01

02

03

04

05

06

07

Foo

d Sh

are

Log Expenditure






(31)




=prime

prime

γ

γ

1

N



ln

mj

it+minus

λln lnna p

c put

tijt

( )( )( ) +

2





35 4 45 5 55 6 65 7


Alc

ohol

Sha

re

Log Expenditure












(32)


(33)


ˆˆ

πmicro

tz i it it t

i it t

z nz n

=sum

sum

ˆˆ

πmicro

1ti it it t

i it t

m nm n

=sum ( )

sum( )ln

ln




1

105

11

115

12

125

13

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000












102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999









(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )






101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES





































































































































































































































(4)


(5)


(6)


(7)

times












(8)

times

times

where

(9)



π 2 titE m z

= iit

it itE m E z( )

( ) ( )

π1tit it

it it

E m mE m E m

=( )

( ) ( )ln

lnand











(10)


(11)

times









(12)


(13)

times




E q a p b pt ijt j t j t( ) = ( ) + ( )0 0










(14)

times


(15)

times



E p q

E mE w

b p b

t jt ijt

t itt it ijt

j t j

( )( ) ( )

= ( ) +

micro

0 1 ppt( )


wp q

mb p b pijt

jt ijt

itj t j t= = ( ) + ( )0 1



(16)

times


(17)





π ktt it k it

k t it

E h mh E m

k M=( )( )

( )( ) =micro

1 hellip



b p b pt t= ( ) +0 1 (( )π1t

E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(18)



(19)


distribution

(20)




(21)


πmicro σ1 2

11

2 1tmt mt

= + ( ) +

πmicro

1tt it it

t it

t it it

t it

E mE m

E m mE m

=( )

( ) =( )ln

lnln

(( ) ( )ln E mt it


E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(22)


(23)


(24)



(25)


(26)




p E mj t t it+ ( )b1 ln (( )

E p q

E mpt jt ijt

t itj t

( )( ) = ( )b0

b p c c Ej t t+ ( ) +1 1 2 ln mmit( )( )

E p q

E mb pt jt ijt

t itj t

( )( ) = ( )0








(27)




(28)




t it itE m m=

(ln ))( )( ) ( )( )

2

2E m E mt it t itln

πmicro

2

2

2t

t it it

t it

E m

E m=

( )( )( )( )ln

ln


17


(29)




(30)





t itE m z= iit it

t it t it t it

mE m E z E m

lnln

( )( ) ( ) ( )

πmicro

1tz t it it it

t it t it

E z mE z E m

=( )

( ) ( )ln

ln

πmicro

tz t it it

t it

t it it

t it t i

E zE z

E m zE m E z

=( )

( ) =( )

( ) tt( )




3 4 5 6 70

01

02

03

04

05

06

07

Foo

d Sh

are

Log Expenditure






(31)




=prime

prime

γ

γ

1

N



ln

mj

it+minus

λln lnna p

c put

tijt

( )( )( ) +

2





35 4 45 5 55 6 65 7


Alc

ohol

Sha

re

Log Expenditure












(32)


(33)


ˆˆ

πmicro

tz i it it t

i it t

z nz n

=sum

sum

ˆˆ

πmicro

1ti it it t

i it t

m nm n

=sum ( )

sum( )ln

ln




1

105

11

115

12

125

13

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000












102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999









(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )






101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES







































































































































































































































(10)


(11)

times









(12)


(13)

times




E q a p b pt ijt j t j t( ) = ( ) + ( )0 0










(14)

times


(15)

times



E p q

E mE w

b p b

t jt ijt

t itt it ijt

j t j

( )( ) ( )

= ( ) +

micro

0 1 ppt( )


wp q

mb p b pijt

jt ijt

itj t j t= = ( ) + ( )0 1



(16)

times


(17)





π ktt it k it

k t it

E h mh E m

k M=( )( )

( )( ) =micro

1 hellip



b p b pt t= ( ) +0 1 (( )π1t

E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(18)



(19)


distribution

(20)




(21)


πmicro σ1 2

11

2 1tmt mt

= + ( ) +

πmicro

1tt it it

t it

t it it

t it

E mE m

E m mE m

=( )

( ) =( )ln

lnln

(( ) ( )ln E mt it


E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(22)


(23)


(24)



(25)


(26)




p E mj t t it+ ( )b1 ln (( )

E p q

E mpt jt ijt

t itj t

( )( ) = ( )b0

b p c c Ej t t+ ( ) +1 1 2 ln mmit( )( )

E p q

E mb pt jt ijt

t itj t

( )( ) = ( )0








(27)




(28)




t it itE m m=

(ln ))( )( ) ( )( )

2

2E m E mt it t itln

πmicro

2

2

2t

t it it

t it

E m

E m=

( )( )( )( )ln

ln


17


(29)




(30)





t itE m z= iit it

t it t it t it

mE m E z E m

lnln

( )( ) ( ) ( )

πmicro

1tz t it it it

t it t it

E z mE z E m

=( )

( ) ( )ln

ln

πmicro

tz t it it

t it

t it it

t it t i

E zE z

E m zE m E z

=( )

( ) =( )

( ) tt( )




3 4 5 6 70

01

02

03

04

05

06

07

Foo

d Sh

are

Log Expenditure






(31)




=prime

prime

γ

γ

1

N



ln

mj

it+minus

λln lnna p

c put

tijt

( )( )( ) +

2





35 4 45 5 55 6 65 7


Alc

ohol

Sha

re

Log Expenditure












(32)


(33)


ˆˆ

πmicro

tz i it it t

i it t

z nz n

=sum

sum

ˆˆ

πmicro

1ti it it t

i it t

m nm n

=sum ( )

sum( )ln

ln




1

105

11

115

12

125

13

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000












102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999









(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )






101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES







































































































































































































































(14)

times


(15)

times



E p q

E mE w

b p b

t jt ijt

t itt it ijt

j t j

( )( ) ( )

= ( ) +

micro

0 1 ppt( )


wp q

mb p b pijt

jt ijt

itj t j t= = ( ) + ( )0 1



(16)

times


(17)





π ktt it k it

k t it

E h mh E m

k M=( )( )

( )( ) =micro

1 hellip



b p b pt t= ( ) +0 1 (( )π1t

E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(18)



(19)


distribution

(20)




(21)


πmicro σ1 2

11

2 1tmt mt

= + ( ) +

πmicro

1tt it it

t it

t it it

t it

E mE m

E m mE m

=( )

( ) =( )ln

lnln

(( ) ( )ln E mt it


E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(22)


(23)


(24)



(25)


(26)




p E mj t t it+ ( )b1 ln (( )

E p q

E mpt jt ijt

t itj t

( )( ) = ( )b0

b p c c Ej t t+ ( ) +1 1 2 ln mmit( )( )

E p q

E mb pt jt ijt

t itj t

( )( ) = ( )0








(27)




(28)




t it itE m m=

(ln ))( )( ) ( )( )

2

2E m E mt it t itln

πmicro

2

2

2t

t it it

t it

E m

E m=

( )( )( )( )ln

ln


17


(29)




(30)





t itE m z= iit it

t it t it t it

mE m E z E m

lnln

( )( ) ( ) ( )

πmicro

1tz t it it it

t it t it

E z mE z E m

=( )

( ) ( )ln

ln

πmicro

tz t it it

t it

t it it

t it t i

E zE z

E m zE m E z

=( )

( ) =( )

( ) tt( )




3 4 5 6 70

01

02

03

04

05

06

07

Foo

d Sh

are

Log Expenditure






(31)




=prime

prime

γ

γ

1

N



ln

mj

it+minus

λln lnna p

c put

tijt

( )( )( ) +

2





35 4 45 5 55 6 65 7


Alc

ohol

Sha

re

Log Expenditure












(32)


(33)


ˆˆ

πmicro

tz i it it t

i it t

z nz n

=sum

sum

ˆˆ

πmicro

1ti it it t

i it t

m nm n

=sum ( )

sum( )ln

ln




1

105

11

115

12

125

13

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000












102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999









(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )






101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES



































































































































































































































(18)



(19)


distribution

(20)




(21)


πmicro σ1 2

11

2 1tmt mt

= + ( ) +

πmicro

1tt it it

t it

t it it

t it

E mE m

E m mE m

=( )

( ) =( )ln

lnln

(( ) ( )ln E mt it


E p q

E mE wt jt ijt

t itt it ijt

( )( ) = ( )micro





(22)


(23)


(24)



(25)


(26)




p E mj t t it+ ( )b1 ln (( )

E p q

E mpt jt ijt

t itj t

( )( ) = ( )b0

b p c c Ej t t+ ( ) +1 1 2 ln mmit( )( )

E p q

E mb pt jt ijt

t itj t

( )( ) = ( )0








(27)




(28)




t it itE m m=

(ln ))( )( ) ( )( )

2

2E m E mt it t itln

πmicro

2

2

2t

t it it

t it

E m

E m=

( )( )( )( )ln

ln


17


(29)




(30)





t itE m z= iit it

t it t it t it

mE m E z E m

lnln

( )( ) ( ) ( )

πmicro

1tz t it it it

t it t it

E z mE z E m

=( )

( ) ( )ln

ln

πmicro

tz t it it

t it

t it it

t it t i

E zE z

E m zE m E z

=( )

( ) =( )

( ) tt( )




3 4 5 6 70

01

02

03

04

05

06

07

Foo

d Sh

are

Log Expenditure






(31)




=prime

prime

γ

γ

1

N



ln

mj

it+minus

λln lnna p

c put

tijt

( )( )( ) +

2





35 4 45 5 55 6 65 7


Alc

ohol

Sha

re

Log Expenditure












(32)


(33)


ˆˆ

πmicro

tz i it it t

i it t

z nz n

=sum

sum

ˆˆ

πmicro

1ti it it t

i it t

m nm n

=sum ( )

sum( )ln

ln




1

105

11

115

12

125

13

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000












102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999









(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )






101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES





































































































































































































































(27)




(28)




t it itE m m=

(ln ))( )( ) ( )( )

2

2E m E mt it t itln

πmicro

2

2

2t

t it it

t it

E m

E m=

( )( )( )( )ln

ln


17


(29)




(30)





t itE m z= iit it

t it t it t it

mE m E z E m

lnln

( )( ) ( ) ( )

πmicro

1tz t it it it

t it t it

E z mE z E m

=( )

( ) ( )ln

ln

πmicro

tz t it it

t it

t it it

t it t i

E zE z

E m zE m E z

=( )

( ) =( )

( ) tt( )




3 4 5 6 70

01

02

03

04

05

06

07

Foo

d Sh

are

Log Expenditure






(31)




=prime

prime

γ

γ

1

N



ln

mj

it+minus

λln lnna p

c put

tijt

( )( )( ) +

2





35 4 45 5 55 6 65 7


Alc

ohol

Sha

re

Log Expenditure












(32)


(33)


ˆˆ

πmicro

tz i it it t

i it t

z nz n

=sum

sum

ˆˆ

πmicro

1ti it it t

i it t

m nm n

=sum ( )

sum( )ln

ln




1

105

11

115

12

125

13

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000












102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999









(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )






101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES



































































































































































































































3 4 5 6 70

01

02

03

04

05

06

07

Foo

d Sh

are

Log Expenditure






(31)




=prime

prime

γ

γ

1

N



ln

mj

it+minus

λln lnna p

c put

tijt

( )( )( ) +

2





35 4 45 5 55 6 65 7


Alc

ohol

Sha

re

Log Expenditure












(32)


(33)


ˆˆ

πmicro

tz i it it t

i it t

z nz n

=sum

sum

ˆˆ

πmicro

1ti it it t

i it t

m nm n

=sum ( )

sum( )ln

ln




1

105

11

115

12

125

13

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000












102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999









(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )






101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES



































































































































































































































35 4 45 5 55 6 65 7


Alc

ohol

Sha

re

Log Expenditure












(32)


(33)


ˆˆ

πmicro

tz i it it t

i it t

z nz n

=sum

sum

ˆˆ

πmicro

1ti it it t

i it t

m nm n

=sum ( )

sum( )ln

ln




1

105

11

115

12

125

13

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000












102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999









(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )






101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES



































































































































































































































1

105

11

115

12

125

13

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000












102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999









(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )






101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES


































































































































































































































102

104

106

108

11

112

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999









(34)

d

p

= ggdp

dF m g p mdgdm

dF m

S

j comp

jint intminus

=

( ) ( ) ( )

minus A

dGdp

dgdp

g p mdgdm

dF mj j comp

j= minus

int ( ) ( )






101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES



































































































































































































































101

102

103

104

105

19791981

19831985

19871989

19911993

19951997

1999



(35)



d g p m

dmdF m

j=( ) ( )int

12

2

A g p mdgdm

























(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES





















































































































































































































































(36)



2

(37)


(38)


(39)






(40)


(41)




y yitP

itP

itP= +minus1 η

y y yit itP

itT= +








τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES




































































































































































































































τ tL tr r r= +( ) minus +( )( )



(42)

(43)



(44)


(45)




(46)





y u vitT

t it= +

η η εitP

t it= +



(47)


(48)



(49)


(50)









1 13 β





(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES




































































































































































































































(51)




(52)


(53)











(54)



(55)



(56)



k kitt


12

11εε κ ηit t+ 2



U c ec

s

it ita it

s

it

it

it

( ) =minus

minus11

11








Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES







































































































































































































































Andasht1



(57)

(58)



(59)




1 ++ +minusk Att

t21

2σ κ η


E Eit i tt


1 0

E Eit i tt


1 0









1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES


































































































































































































































1981ndash1985 1986ndash1990

1991ndash1995 1996ndash2001





(60)


(61)



2σ κ ηAtt

tminus +


k Att

t+ +minusσ κ η21

2


(62)





tρ σ κ η21

2

∆ =( )( )

minus minus

ln ( ) lnE cE c

E ct itt it

t it1 1


1 exp β ϕ

minus= + +( )21

2ρ σ κ ηt Att

tr kexp middot


2κ ηt










(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES







































































































































































































































(65)



(66)


σσθ

θ

θ θc t

t zc t

zc t t

t zz t t

minus

minus

minus

sumsum

sum

1

1

1

2

ln

c

zN

E zit

t it

c t

t t it

minus

( )

minus1 1

θmicroθ

~

lnE c

E ct it

t it

minus

minus minus

( )( )

=1

1 1

0

E ct+ ln iit

t itE c

minus

minus minus

( )( )

1

1 1

+minus

+ln

E ct it

β1

exp ϕϕr z

E c

t it

t it

( )( )

( )

prime

minus1

(67)

and

(68)


(69)

times






(70)


k kItt


12

11εε κ ηit t+ 2



21

β ϕ


12 β ϕ β ϕ


12 tt



c t tminus+

1


1






(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES




































































































































































































































(71)

where


εt









1 1 12 κ σε


ε


12 AAt

tt t

minus + + ( )11

12

κ η



















(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES












































































































































































































































(72)








-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES




































































































































































































































-002

-0015

-001

-0005

0

0005

001

0015

002

0025

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000













(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES








































































































































































































































(73)


(74)













(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES






































































































































































































































(75)


(76)



γ δ ζ+ +Ait it







(77)


(78)


(79)
































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES























































































































































































































































(80)


w r Hti t i=



Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES


































































































































































































































Et(1nHi) = js


(81)



it are lognormal

(82)


it or with

(83)




(84)








(85)



(86)


(87)


E I x zz

t it itit[ ] =

+

α α

σ0

2

ε ε

ε

εit

it

Nν

σσ

σσ

~

00

2

2










(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES




































































































































































































































(88)




(89)



(90)


xx xz

xz zz



β βα α

β βα α

0

0

0

0

+ prime+ prime

+ prime+ prime

x

zN

E x

E zit

it

it~( )( iit)

microitit

iti I

h

h=

isin =( )sum 1

wh w

hwt

i I it it

i I itit it

i I

=sum

sum=isin =( )


1 1

micro

xit+ prime ++ prime

00β β

σσ

λα αε

zzit

σ

2



(91)




(92)





(93)



22β β σε


ln ln[ ]

[ ]w

E h w IE h It

it it it

it it

==

=|

|1

1

prime +sumα α σ

2

zz

zz

+prime sum + 2

σ

α α σε

λλα α

α α σ0

2

+ prime

prime sum +

E zit

zz

( )


prime +sumα α σ

2

zz

E IE z

tit

zz

[ ] =+ prime ( )

prime +

sum

α α

α α σ0

2







(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES






































































































































































































































(95)


(96)




ln ln

ln

w E w I

E wt it it t

t it t t

= =[ ] += ( ) + + +

1

tt









γ α

σεprime sum +

prime sum

zz t

2 middot a

zzz tα σ λ+

2 middot a


it xzE z ln



+ prime

prime sum +

( )

α α

α α σ0

2

E zit

zz

tit xz

zz

E z=


prime sum +

ln

( ) ( )α α β α σ

α α σε0

2




log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES



































































































































































































































log

aver

age

wag

e

year

78 80 82 84 86 88 90 92 94 96

18

19

20

21

22









78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES



































































































































































































































78 80 82 84 86 88 90 92 94 96075

080

085

090

year









year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES


































































































































































































































year

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10

78 80 82 84 86 88 90 92 94 96

06

07

08

09

10









-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES



































































































































































































































-005

000

005

010

015

020

025

030

035

040

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Log aggregate wage

log

Wag

e













5 Conclusion





















REFERENCES





































































































































































































































5 Conclusion





















REFERENCES









































































































































































































































jn05_Article 1 61005 138 PM 4








REFERENCES








































































































































































































































REFERENCES
















































































































































































































jn05_Article 1 61005 138 PM 0

















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