A Continuous Labour Supply Model in Microsimulation: A Life-Cycle Modelling Approach with Heterogeneity and Uncertainty Extension

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1 D I S C U S S I O N P A P E R S E R I E S IZA DP No. 698 A Contnuous Labour Supply Model n Mcrosmulaton: A Lfe-Cycle Modellng Approach wth Heterogenety and Uncertanty Extenson Jnjng L Densa Mara Sologon October 211 Forschungsnsttut zur Zukunft der Arbet Insttute for the Study of Labor

2 A Contnuous Labour Supply Model n Mcrosmulaton: A Lfe-Cycle Modellng Approach wth Heterogenety and Uncertanty Extenson Jnjng L Maastrcht Unversty / UNU-MERIT Densa Mara Sologon CEPS/INSTEAD, Maastrcht Unversty and IZA Dscusson Paper No. 698 October 211 IZA P.O. Box Bonn Germany Phone: Fax: E-mal: za@za.org Any opnons expressed here are those of the author(s) and not those of IZA. Research publshed n ths seres may nclude vews on polcy, but the nsttute tself takes no nsttutonal polcy postons. The Insttute for the Study of Labor (IZA) n Bonn s a local and vrtual nternatonal research center and a place of communcaton between scence, poltcs and busness. IZA s an ndependent nonproft organzaton supported by Deutsche Post Foundaton. The center s assocated wth the Unversty of Bonn and offers a stmulatng research envronment through ts nternatonal network, workshops and conferences, data servce, project support, research vsts and doctoral program. IZA engages n () orgnal and nternatonally compettve research n all felds of labor economcs, () development of polcy concepts, and () dssemnaton of research results and concepts to the nterested publc. IZA Dscusson Papers often represent prelmnary work and are crculated to encourage dscusson. Ctaton of such a paper should account for ts provsonal character. A revsed verson may be avalable drectly from the author.

3 IZA Dscusson Paper No. 698 October 211 ABSTRACT A Contnuous Labour Supply Model n Mcrosmulaton: A Lfe-Cycle Modellng Approach wth Heterogenety and Uncertanty Extenson * Ths paper advances a structural nter-temporal model of labour supply that s able to smulate the dynamcs of labour supply n a contnuous settng and to crcumvent two man drawbacks of most of the exstng models. The frst lmtaton s the nablty to ncorporate ndvdual heterogenety as every agent s sharng the same parameters of the utlty functon. The second one s the strong assumpton that ndvduals make decsons n a world of perfect certanty. Essentally, ths paper offers an extenson of margnal-utlty-of-wealthconstant labour supply functons known as Frsch functons under certanty and uncertanty wth homogenous and heterogeneous preferences. Two alternatve models are proposed for capturng ndvdual heterogenety. Frst, a fxed effect vector decomposton model, whch allows the ndvdual specfc effects to be correlated wth the explanatory varables ncluded n the labour supply model, and second, a mxed fxed and random coeffcent model, whch ncorporates a hgher degree of ndvdual heterogenety by specfyng ndvdual coeffcents. Uncertanty s controlled for by ntroducng an expectaton correcton nto the model. The valdaton of each smulaton model s realzed n comparson wth the standard Heckman model. The lfetme models based on the fxed effect vector decomposton yeld the most stable and unbased smulaton results, both under certanty and uncertanty. Due to ts mproved accuracy and stablty, ths lfetme labour supply model s partcularly sutable for enhancng the performance of the penson models, thus provdng a better reference for polcymakng. JEL Classfcaton: C2, D9, J22 Keywords: lfetme labour supply, dynamc mcrosmulaton Correspondng author: Densa Mara Sologon CEPS/INSTEAD 3, avenue de la Fonte 4364 Esch-sur-Alzette Luxembourg E-mal: Densa.Sologon@ceps.lu * Ths research s part of the Earnngs Dynamcs and Mcrosmulaton project supported by the Luxembourg Fonds Natonal de la Recherche through an AFR grant (PDR no ) under the Mare Cure Actons of the European Commsson (FP7-COFUND). The authors would lke to thank Cathal O Donoghue for hs helpful comments and advce.

4 I. Introducton The emprcal lterature on labour supply has ganed an ncreasng nterest over the past decades. The contnuous approach n the tradton of Burtles and Hausman (1978) has been complemented by an approach whch focuses on a dscrete choce specfcaton, manly nspred by Van Soest (1995). Most studes focus on estmatng and smulatng labour supply n a statc context, whereas only a few focus on the dynamcs of labour supply (Blundell and MaCurdy, 1999). Some of the exstng models for estmatng and/or smulatng labour supply n a statc or dynamc settng lack the robustness of economc theory. Other models are based on specfyng utlty functons used to derve consstent labour supply functons, whch s fundamental for projectng the labour supply behavour. The latter models, however, suffer from two major shortcomngs. Frst, as utlty s not observed, many assumptons are assumed n order to estmate the parameters of the utlty functon. Ths becomes less robust when more tems/behavours are ntroduced nto the utlty functon. Secondly, ths approach does not ncorporate ndvdual heterogenety as every agent shares the same parameters. Some models deal wth the heterogenety problem by capturng the ndvdual specfc effects usng error components models or by estmatng random-coeffcents models. The ndvdual specfc effect s used n the context of the fxed or random effects estmatons, where the man ssue concerns the choce between fxed and random effects and whether the ndvdual specfc effects can be assumed to be ndependent of the explanatory varables ncluded n the model. Assumng that the ndvdual component s correlated wth the explanatory varables trggers many problems when undertakng a smulaton. The estmated coeffcents cannot be used to generate a condtonal predcton of ndvdual earnngs wthout specfyng the jont process determnng the ndvdual specfc effects and the explanatory varables (Pudney, 1992). The mpractcalty of ths opton, together wth the fact that the fxed-effect specfcaton cannot accommodate covarates that are constant over tme, constraned most studes to mantan the assumpton of a zero correlaton between the ndvdual specfc effects and the other covarates, a rather strong and mprobable assumpton. The man drawback of the error components models s that they provde less stable smulatons due to the stochastc components, whch affects the reproducblty of the results. One way to ncorporate the heterogenety effects s to use random coeffcent models. Provded that heterogenety s present n the emprcal models of labour supply, the applcaton of random coeffcents models s necessary to avod based estmates. The man drawback of these models, however, s ther hgh computatonal cost. Gven ths lmtaton, exstng studes suggest that f heterogenety s non-exstent or the bas s nsgnfcant, the standard fxed coeffcent models represent the optmal choce (Haan, 24). Several studes estmatng dscrete choce labour supply models compare the fxed and random specfcatons and fnd no sgnfcant dfferences between the results (van Soest, 1995; Duncan and MacCrae, 1999; Haan, 24) The estmaton of contnuous labour supply models usng the random coeffcent specfcaton s lmted. Ths chapter proposes a structural nter-temporal model of labour supply, whch estmates and predcts the dynamcs of labour supply n a contnuous settng. It ams to capture the ndvdual heterogenety to a larger extent than the exstng labour supply models whle mantanng the consstency wth the lfetme economc theory. The model s estmated usng both a transformed fxed effect specfcaton that crcumvents the 2

5 standard problems mentoned above, and a random coeffcent specfcaton. Addtonally, the model ncorporates uncertantes regardng future wages to further explore the heterogenetes. II. Theoretcal Background Most of the emprcal work on labour supply s based on a statc, wthn-perod framework, where the labour supply decson rule refers to one perod. Typcally, the annual hours of work are regressed on the current hourly wage rate and some measure of property ncome. Labour supply responses are estmated usng the standard labour supply models, whch gnore the mportance of future wages on current hours suppled. Ths yelds a wage coeffcent that confuses the labour supply response to three types of wage changes: evolutonary wage changes arsng from movements along the lfetme wage profle, parametrc wage changes arsng from shfts n the wage profle, and those arsng from changes n the profle slopes. As a result, the wage coeffcents reported n many emprcal studes have no economc nterpretaton and are not useful for polcy evaluatons (MaCurdy, 198). The elastcty derved from the statc specfcaton can be placed n an nter-temporal settng, but s economcally meanngful only under the strong assumpton of ether complete myopa or perfectly constraned captal markets. These assumptons mply that t s mpossble to transfer captal across perods. In ths stuaton, the coeffcent on wage represents the uncompensated substtuton elastcty gven ncome, equvalent to the Marshallan wage elastcty n the statc model (Blundell and MaCurdy, 1998). There are three types of substtuton elastcses relevant for predctng the response of hours of work to changes n the wage rate: the nter-temporal elastcty, whch determnes the labour supply wage changes resultng from lfe-cycle wage growth and movements over a perfectly foreseen busness cycle, the uncompensated elastcty, whch determnes the labour supply response to shfts n wage profles whlst holdng the margnal utlty of wealth constant, and the compensated elastcty, whch can be used to predct the dfferences n hours of work across consumers wth dfferent wage profles but an dentcal lfetme utlty constant. In order to estmate meanngful behavoural parameters for the nter-temporal and the uncompensated substtuton effects, t s crucal to recognze that ndvduals make ther labour supply decsons wthn a lfe-cycle framework. Moreover, formulatng a model that reles on the economcs of lfetme behavour leads to a better understandng of consumer behavour (Blundell and MaCurdy, 1998; MaCurdy, 198). The acknowledgement that labour supply s part of a lfetme decson process s realzed by the mult-perod models of labour supply. In the context of ths chapter, specal attenton s devoted to the Frsch labour supply functons, whch estmate the effect of labour supply whlst holdng the margnal utlty of wealth constant. These functons are useful when analysng lfe-cycle maxmzaton problems. The Frsch labour supply functons represent a thrd type of labour supply functons together wth the Marshallng and Hcksan functons. The orgnal nterest n the lfe-cycle labour supply s motvated by the need to nvestgate the varous dmensons of labour supply, such as the determnants of the shape of the lfe-cycle hours profle, the labour supply response to the aggregate wage, the changes and the source of the dosyncratc year-to-year changes n labour supply. The exstng lterature, however, manages to shed lttle lght on the orgnal questons focusng 3

6 manly on one aspect of the nter-temporal hours varaton - the labour supply response to the wage growth along a known lfe-cycle trajectory, whlst gnorng other aspects. One gnored aspect s the labour supply response to wage changes under uncertanty, meanng wage changes that determne ndvduals to revse ther expectatons of ther future wages (Card, 1991). MaCurdy (1983, 1985) makes the most sgnfcant contrbuton n ncorporatng uncertanty. The lfe-cycle framework s proposed as an explanaton for all the components of the ndvdual labour supply (Card, 1991). Lucas and Rappng (197) consder that the lfe-cycle model can be used to explan the aggregate year-to-year movements n labour supply tme effects. Heckman (1974, 1975), Ghez and Becker (1975) clam that the lfe-cycle model s able to explan the systematc age effects n hours of work ( age effects ) and the dfferences across people wth respect to ther hours of work over the lfe-cycle personspecfc effects. MaCurdy (198) and Altonj (1986) formulate lfe-cycle models of labour supply whch explan the person-and-year specfc changes n hours of work through changes n wages. In the context of estmatng and smulatng lfe-cycle labour supply, the choce of the labour supply elastcty to be smulated depends on the scope of the exercse. If the nterest s n comparng the mpact of wage varatons across consumers on labour supply, the varaton n the entre wage profle must be examned. Because the varaton of the wage profle affects the value of the margnal utlty of wealth, Frsch elastcty cannot be used to measure the effect of ths varaton. The estmaton of the full mpact on wages requres the estmaton of the effect of the shfts of the wage profle on the hours of work besdes the estmaton of the nter-temporal elastcty. The estmaton of the parametrc shft requres the specfcaton of the mpact of the wage profle on the margnal utlty of wealth. The estmaton of the uncompensated substtuton elastcty s undertaken n a lmted number of studes as t reles on specfyng the functonal form of the margnal-utlty-of-wealth constant parameter, proven to be dffcult n practce. Some studes (MaCurdy 198, 1985) gnore the functonal form of ths parameter because of ts complcated functonal form n ntal assets, lfetme wages, the nterest rate, the rate of tme preference and the taste parameters. They assume that the approxmaton of ts lfe-cycle specfcaton s a lnear functon of measured characterstcs, the natural log of wages at each age, ntal wealth and an unobserved random varable representng unmeasured characterstcs. Others derve an expresson of the margnal utlty of wealth by takng nto account the restrctons of the optmzaton process (e.g. Bover, 1989). The estmaton of the full mpact of wage changes, both evolutonary and parametrc, are of core mportance for polcy evaluaton. Assumng that the tax and beneft reforms represent unantcpated shfts n net real wages today and n the future, the elastcty measurng the cumulated response to evolutonary changes and parametrc shfts n the lfe-cycle wage profle represent the most approprate means for descrbng the response to these reforms (Blundell and MaCurdy, 1981). III. A Lfe-Cycle Model of Labour Supply The model n ths chapter follows the theoretcal specfcaton ntroduced by MaCurdy (198, 1985) and Medoff and Abraham (1981), and the unfyng labour supply framework ntroduced by Blundell and 4

7 MaCurdy (1998). The model ams to estmate the effect of evolutonary wage changes assumng no parametrc shft n the wage profles. The labour supply responses are estmated and smulated under two scenaros: frst, assumng that ndvduals make labour supply decsons n a world of perfect certanty, and second, assumng that ndvduals make decsons n a world n whch they are uncertan about ther future wages. 1.III.1 A Lfe-Cycle Model of Labour Supply under Certanty Under the assumpton of certanty, the effect of evolutonary wage changes on hours of work represents the conventonal margnal-utlty-of-wealth constant nter-temporal elastcty of labour supply obtaned from the Frsch labour supply equatons and the Euler condton. The margnal-utlty-of-wealth parameter serves as a suffcent statstc that captures the nformaton from the other perods needed to solve the maxmzaton process n the current perod (Blundell and MaCurdy, 1998). The nter-temporal elastcty estmated from the Frsch specfcatons s relevant for predctng the ndvdual labour supply nto the future assumng a steady state, manly due to the presence of the margnal utlty of wealth, whch s ndvdualzed, constant over tme, and accounts for the worker s future plans (MaCurdy, 198). Followng MaCurdy (198, 1983), the theory underlyng the model of lfetme hours of work used n ths chapter represents an extenson of Fredman s (1957) permanent ncome theory to a stuaton where the relatve prce of consumpton and lesure vares over the lfe-cycle. The permanent ncome hypothess can be extended to the lfetme labour supply to assume that ndvduals/households look nto the future when decdng the current number of hours suppled on the labour market. Ths theory allows us to make the dstncton between a consumer s dynamc behavour and the factors determnng the dfferences n hours of work between consumers. Ths separaton leads to a manageable emprcal model that accommodates the dfferentaton between the labour supply responses to evolutonary changes and those to parametrc changes n the wage profles. 1.III.1.1 An economc model of labour supply under certanty wth homogenous preferences and heterogeneous ndvdual effects Ths secton presents an economc model of lfe-cycle labour supply decsons assumng that a worker takes hs/her decsons n an envronment of perfect certanty wth respect to hs/her future ncome. The worker s assumed to choose consumpton and hours of work at each age to maxmze a lfetme preference functon, strongly separable over tme, subject to wealth constrants. The model descrbed n ths chapter s desgned for sngle decson-makers, but an extenson to jont decson makers s straghtforward. In the applcaton below, male s labour supply behavour s consdered as ndependent whle women s labour supply s condtoned on other household ncomes besdes ther own earnngs. Assumng that the consumer at the age of t has a utlty gven by the concave functon Ut Ct, Lt, Xt, where Ct () s the consumpton at age t, Lt () s the number of hours of lesure at age t and X() t s a vector of taste shfters varables at age t, the vector X() t can nclude both observed and unobserved varables. 5

8 Due to the assumpton of separatng utlty, the lfetme preference functon can be formally presented as the sum of dscounted future utltes at the moment t a, equvalent to the begnnng of the actve lfe, where t represents age and a the age of entrance nto the labour market: T 1 U t Ct, Lt, Xt (1) ta 1 ρ t The lfetme (actve lfe) s assumed to consst of T a 1 perods, where T represents the age of retrement. The rate of tme preference used for dscountng the value of future utlty s represented by ρ. Formally, the consumer has to choose C(t) and L(t) at each age to maxmze ther lfetme preference functon (1) subject to a lfetme wealth constrant: T ( ) ( ) (2) ta T ta A R t H t W t R t C t Where A() s the level of assets at the begnnng of the actve lfe of each consumer, H(t) the number of labour market hours at age t, W(t) the exogenous wage rate at age t, then Rt1/ 1r1 * * 1 rt s the dscount rate whch s used to convert the real ncome at age t nto ts equvalent for age a and j refers to the sample perod. In perod j, the consumer can borrow and lend at a rate of nterest equal to r(j). It s assumed that the rate of nterest stays constant over tme, so that the dscount rate at age Rt 1/ 1 rt. To create the Frsch labour supply functons, t s assumed that the contemporaneous utlty functon for each ndvdual at age t takes the form: Ψ t t t t t σ U =G C,X - H (3) Where G s a monotoncally ncreasng functon of C t, σ s a tme-nvarant preference parameter common across consumers and Ψ t s an age-specfc parameter of tastes, whch depends on the consumer characterstcs expected to nfluence hs/her utlty at age t (age, educaton, number of chldren, etc.). The partcpaton decson s ncluded n the preference parameter. The analyss assumes Ψ t s related to the * * * worker s characterstcs by the functon Ψt exp{ X t ψ vt } where vt represents the contrbuton of * unobserved characterstcs and ψ s a vector of preference parameters. Assumng an nteror optmum, the mpled Frsch labour supply functon or margnal utlty of wealth constant labour supply functon s obtaned from maxmzng the utlty n perod t subject to the lfetme wealth constrant. The Lagrange functon then takes the form: t T T U t λ A Rt HtWt RC t t t t. The notaton s smplfed by consderng the age of entrance nto the labour market equal to zero. The frst order condtons result n the Frsch or λ constant consumpton and labour supply functons: C C( λ, W, X ) ; H H( λ, W, X ) t t t t t t 6

9 The frst order condton wth respect to hours of work at age t mples that: ln H α(lnλlnσ) ψx βtαlnw v (4) t t t t 1 * * Where α ; ψ αψ, vt αvt, and β α ρ r. At ths stage of the analyss the assumpton s that α, 1 σ ψ and β are constant across consumers and tme. The expresson (4) represents the Frsch or margnal utlty of wealth constant labour hours of work functon. Its functonal form depends on the form of the utlty functon and on whether a corner soluton s chosen for hours of work at age t (Blundell and MaCurdy, 1999). If an ndvdual chooses to partcpate n the labour market, then model (4) apples and an nteror soluton s assumed. If however, the decson s not to partcpate, the hours of work are set to zero. The Frsch labour supply functon decomposes the labour supply decson wth respect to the hours of work nto personal and professonal characterstcs ( Xt and age) observed at tme t, the wage rate at tme t ( W t ) and the λ component, whch represents the suffcent statstc summarzng the relevant nformaton for each consumer from the other perods. The optmal value of λ s obtaned by substtutng the λ constant consumpton and labour supply functons nto the budget constrant gven by equaton (3). λ s expressed as a functon of ntal assets, lfetme wages, nterest rates, rates of tme preference and tastes. In other words, λ summarzes the lfetme nformaton that the consumer requres when choosng hs/her optmal level of current consumpton and labour supply. λ represents the correspondent statstc to the permanent ncome from Fredman s (1957) permanent ncome theory and represents a permanent component, whch together wth the current wage, determnes the consumer s current consumpton and labour supply. λ also represents a statstc capable of charactersng hstorc and future nformaton for lfetme wages and assets that are relevant for the current choces of consumpton and labour supply. The general concluson s that, assumng perfect certanty, the λ constant consumpton and labour supply functons fully characterze a consumer s dynamc behavour (Blundell and MaCurdy, 1998; MaCurdy, 198, 1985). As shown by Blundell and MaCurdy (1999), n a world of perfect certanty, λ can be captured as an ndvdual specfc effect, whch s constant over tme. For ths reason, for each ndvdual, changes n wages have no mpact on λ. In concluson, the Frsch elastcty s the correct elastcty for assessng the mpact of the evolutonary changes along the lfe-cycle wage profle on labour supply. The man characterstcs of the λ component, the fact that t s specfc to each ndvdual, constant over tme and ncorporates the future plans of each worker n a smple way, allows the use of the λ constant functon (4) to forecast ndvdual labour supply outsde the sample perod. Gven that λ s tme-nvarant and unque for each worker, the ntercept term from the labour supply equaton (4) s: F α(lnλ ln σ) Ths represents a tme-nvarant component, unque to each ndvdual and can therefore be treated as a fxed effect n the estmaton. The argument n favour of ths choce steams from the content of the λ component 7

10 that depends on the past and future nformaton of the varables relevant for determnng the ndvdual labour supply. Because λ s correlated wth the exogenous varables n the model, the natural concluson s that F s also correlated wth the exogenous varables n the model, and therefore cannot be treated as a random effect. The parameters n model (4) are estmated by fxed effects. In order to crcumvent the problem nduced by the fxed effects specfcaton n a smulaton context, ths chapter apples a three-stage procedure to decompose the unt fxed effects. The emprcal specfcaton of the ndvdual effects requres a formal specfcaton of the λ component. The one obtaned from the optmsaton process s too complcated to estmate t emprcally. Ths chapter adopts the suggestons made by MaCurdy (1981), Blundell and MaCurdy (1999). For the emprcal specfcaton, ln λ can be approxmated by: T * * * * tln t t lnλ Dφ γ W A θ a D s a vector of observed tme-nvarant characterstcs or rarely changng varables, φ * *, γ, t * θ are * parameters assumed constant across consumers, and a s an error term. Ths emprcal specfcaton mposes strong restrctons, as t s assumed that the worker knows he/she wll work T years, and ther total lfetme ncome. It also ncorporates the effect of nterest rates and rates of tme preference n the ntercepts and the other parameters. Under ths assumpton, the tme-nvarant ntercept of equaton (4) can be approxmated by: F α Dφ γ W A θ u σ Dφ g W a T T * * * * t ln t t ln t ln t t t (5) * * * φ αφ, γ αγ t, θ αθ and the ntercept nclude the term ( αlnσ ), and A represents the ntal ncome, assumed to be zero for all ndvduals. The present value of lfetme ncome s constant for each ndvdual and can be approxmated by functons of all the personal and professonal characterstcs nfluencng a lfetme wage path as: T t t γlnw = f X,X,, X t 1 2 T Where X k, k=1,2,t represents a vector of personal and professonal career characterstcs whch can be expressed as functons of two components: a tme nvarant part D1 and an unobserved part D 2 : T γlnw t t f X D +D,X D +D,, X D +D (6) T 1 2 t Assumng that ths s an addtve functon for D1 and D 2, the present value of the lfetme wage can be approxmated by: T γlnw t t f D 1 + f D2 t By substtutng equaton (6) nto (5), the ndvdual effect can be presented as follows: 8

11 F Dφ f D + f D +a (7) 1 2 The tme nvarant characterstcs D, can be decomposed nto the observed part D 1 and the unobserved part D 2 : F= D 1+D2 φ+f D 1+fD 2+a The observed part can then be approxmated by a lnear form of D 1, whereas the unobserved part, assumed to follow a normal dstrbuton, s ncorporated n the error term as: F D φ f ( D ) D φ f ( D ) a (8) * F= Dφ 1 c (9) Therefore, the ndvdual unt effect can be approxmated by a lnear functon of the tme-nvarant characterstcs plus a normally dstrbuted error that accounts for the unexplaned part of the ndvdual effect. Substtutng equaton (9) nto (4), the hours of work process can be approxmated as follows: ln H D φ ψx βt αlnw c v (1) * t t t t The smulaton of labour supply based on the economc model gven n (4) requres also a mechansm for predctng lfetme wage profles. It s assumed that the lfetme wage profle s: 2 lnw t L π, k P t π1t π2t ε t (11) L s the ndvdual specfc effect, P t s a vector of personal and professonal career characterstcs, π,k s a vector of coeffcents whch are constant across tme and ndvduals, k s the number of personal and professonal career characterstcs n the model and ε t the error term. Experence and educaton are assumed to be exogenously determned. 1.III.1.2 An expanson of the economc labour supply model under certanty wth heterogeneous preferences In ths secton, the model s extended by allowng for ncreased heterogenety amongst the ndvduals, who are assumed to have dfferent preferences on ther labour supply. The preference parameter σ ~ N,Γ s ndvdual specfc but tme nvarant. α, β, α(lnλ ln σ) are assumed to be ndvdual specfc and follow normal dstrbutons across ndvduals. Gven these assumptons, the lfetme labour supply functon s: lnh α lnλ lnσ ψx β tα lnw v, t t t t, 1 1, 2 2, lnh ζ ς ψx ζ ς t ζ ς lnw v (12) ln ln ζ, t t t t α λ σ ς s the ndvdual effect constant over tme, β ζ ς the random coeffcent of age 1, and α ζ 2 ς2, the random coeffcent of wage. ζ s the mean ntercept of hours of work, β the mean slope of age, α the mean slope of the natural logarthm of wage and ς k,, k[,2] the devaton of ndvdual ntercept 9

12 and slopes from the mean values of ζ,ζ 1,ζ 2. ψ s a vector of fxed coeffcents, constant across ndvduals and tme. In addton, n order to allow for ncreased heterogenety also n wage, the lfetme wage path has the followng specfcaton: ln W ( π ξ ) R ( π ξ ) ( π ξ ) t π P π t ε (13) 2 t, t 1 1, 2 2, 3 t 4 t R t and age represents the varables wth random coeffcents, π represents the mean ntercept of wage, πk, k1,2 s a vector of coeffcents llustratng the mean slopes of R t and age, and ξ, k, k,1,2 devaton of the ndvdual ntercepts and slopes from the mean values of π, π 1 and π 2. 1.III.2 An expanson of the economc labour supply model under uncertanty represents the The above economc models of lfe-cycle consumpton and labour supply behavours are assumed to be certan at the tme perod, meanng that agents have perfect nformaton and perfect predctons n the begnnng. The followng model extenson relaxes the assumpton of certanty as ndvduals are assumed to act n a world of uncertanty, where they adjust ther expectatons every perod based on current and past nformaton. Ths uncertanty s accounted for n the model by ncludng a forecastng error whch ncorporates the past and future dfferences between realzed values and ther expectatons. It s assumed that agents form an expectaton of wage adaptvely based on what has happened n the past. Due to the mperfecton of forecastng, the wealth constrant needs to be updated every tme perod. Therefore the lfetme wealth constrant at tme perod k n tme perod wth a value of WC k, can be expressed as the sum between the assets n perod, the dscounted sum of the total earnngs between perod and perod k, and the sum of the expected total earnngs, wth dscount factor d( t ) between perod k untl retrement (perod T): k () () WC k A d t H t W t d t E W t H t t tk1 T (14) Alternatvely, t can be expressed as the sum between the assets n perod, the expected dscounted total earnngs over the actve lfe formed before enterng the labour market and an adjustment error. The adjustment error equals the sum between the dfference between the realsed earnngs and ther expectatons made before enterng the labour market over the perod to k (part A), and the expectatons adjustment of the total earnngs made n the perod t+1 to T (Part B) t k T T WC k A d(t) E H t I E W t I A d(t)h t W t - d(t) E H(t)W t I-1 t d(t) E H(t)W t I k - d(t) E H(t)W t I-1 t k 1 WC k B (15) 1

13 I t represents all the nformaton avalable to an agent at tme perod t, whch ncludes personal characterstcs and labour supply actvtes, and k represents the error adjustment term. Ths suggests that current expectatons of lfe tme wealth reflects past expectatons, and an error adjustment term that could ether lower or rase the total constrant. To further smplfy the equaton, t s assumed that the nterest rate certan. Therefore, the error adjustment term s expressed as: t f vtw, vth, vtx f vt 1W, vt 1H, vtx, Wt, HtWt (16) vt W s a vector of all wage nformaton untl the tme perod t, t v (H) s a vector of workng hour nformaton untl tme perod t, and v t (X) s a vector of personal characterstcs nformaton untl tme perod t. As the workng hour s modelled usng the wage rate and the personal characterstcs, the new expectaton adjusted lfetme wealth constrant can be approxmated as: k d( t) WtEWt Ik 1 X t t WCkWC k WC() pe (17) p s an approxmaton parameter, X t s a vector of personal characterstcs whch affect the form of the current expectaton, and s the vector of coeffcents of personal characterstcs. lnwck lnwc can be formulated as: lnwck lnwc μa k X t A k k t k j 1 1 r j W t E W t I k1 Combnng the antcpated paths for wages and ncome wth the approxmated emprcal Lagrange functon modfes the lfetme constrant. Therefore, the λ component takes the form: T * * * * t t ln t k t t ln λ Dφ γ W μa X A θ a (18) μ s an adjustment rato, whch shows the margnal change of workng hours caused by the error term H. explans the role of personal characterstcs n the formulaton of expectaton. It s not possble, A k however, to dstngush from the personal character coeffcent vector ψ n equaton (12). IV. Emprcal Implementaton Ths secton ntroduces the econometrc and the smulaton technques for estmatng and smulatng lfetme labour supply, both under certanty and uncertanty. The model s structured n three parts: a selecton model, a model for lfetme wage profles and a model for lfetme labour supply profles. 11

14 1.IV.1 Data Ths study uses the German Soco Economc Panel (GSOEP) data from 1996 to 23 for estmaton and data for the smulaton valdaton exercse. Only adults respondent who have not reached the retrement age are ncluded n the study. Some basc employment varable nformaton s presented n Table 1. Table 1 Basc Descrptve of Some Employment Varables Varable Mean Std. Dev. Age Number of chldren Employed n formal sector.5.5 Fulltme experence Part-tme experence Unemployment perod Health stuaton.8.28 Hours of workng Educaton (Years) Total number of observatons The dependent varable s the logarthmc value of actual workng hours per week. The wage rate n the estmaton and the smulaton s calculated usng the actual workng hours. Partcpaton s defned as employment n the formal labour market, except for vocatonal tranng, zero workng hours, mltary servce and communty servce, whch are all modelled as non-partcpaton n the selecton model. 1.IV.2 The Emprcal Model of Lfetme Labour Supply under Certanty Ths subsecton develops the econometrc model for the labour supply under certanty. The emprcal model follows the structure of the economc model: n the frst stage t assumes a homogenous preference parameter σ, and n the second t allows for a hgher degree of heterogenety amongst ndvduals. 1.IV.2.1 An Econometrc Model of Labour Supply under Certanty wth Homogenous Preferences For estmatng a lfetme labour supply whch ncorporates a lower degree of heterogenety ths chapter proposes a model smlar to Plumper and Troeger (27) and Hsao (23). The model extends the standard fxed effects specfcaton to an estmaton procedure whch enables an effcent estmaton of tme-nvarant and rarely changng varables by applyng a fxed effects vector decomposton. The lfetme labour supply model follows the structure of the economc model n (4): K y ζ X D u v (19) t k kt c c t k1 c1 C y t s the natural logarthm of hours worked per week, ζ s the ntercept of the base unt, X are the tmevaryng varables (the natural logarthm of wage, age, age squared, chldren, health dummes, household type dummes, cumulated experence untl last year (full tme, part-tme, unemployment), other household 12

15 ncome (for women only), sector dummes), D are the tme-nvarant varables (educaton, educaton squared, cohort dummes), u are the unobserved ndvdual specfc effects, v t s an..d. error term, and J and are parameters common to all ndvduals and constant over tme. The lfetme labour supply model s estmated usng a three-stage procedure. To start, the model (19) s estmated usng the standard fxed effects estmator and t s possble to extract the ndvdual fxed effects or unt effects as: K FE u y k xk v (2) k1 The estmated ndvdual specfc effect u captures the unobserved ndvdual-specfc effect, the observed ndvdual -specfc effects D, the ndvdual means of the resduals v and the ndvdual means of the tmevaryng varables. In the second step, the estmated ndvdual specfc effect s regressed aganst the observed tme-nvarant characterstcs and the rarely changng varables to obtan the unexplaned part of the ndvdual-specfc effects. The ndvdual specfc effects are decomposed as follows: C u cdc h (21) c1 C z s the explaned part. h s the resdual from equaton (21) and captures the unexplaned part of the c c c1 ndvdual-specfc effect: C OLS h u D c c c1 (22) In the thrd stage, the ndvdual effect from the model (19) s substtuted wth the unexplaned part of the decomposed ndvdual fxed effect vector obtaned n the prevous stage, resultng n an error that s no longer correlated wth the tme varyng covarates ncluded n the model. Therefore, the model (23) can be estmated consstently by pooled OLS: K y ζ X D ρh v (23) t k kt c c t k1 c1 C The Monte Carlo smulatons conducted by Plumper and Troeger (27) reveal the crcumstances under whch the FEVD s nferor to the pooled OLS, random effects (RE) and fxed effects (FE). OLS s more approprate when there are no ndvdual effects, RE when the ndvdual effects are uncorrelated wth the other explanatory varables, and FE when the assumptons of the RE are volated and the wthn-varance of the varables of nterest s suffcently large compared wth the between-varance. If ths condton does not occur, the vector decomposton technque has better fnte sample propertes n estmatng models that have tme-nvarant or rarely changng varables correlated wth ndvdual-specfc effects. The lfetme wage process s estmated based on model (24) usng the same procedure: w π X π D π ξ ε (24) t t 1 2, t 13

16 wt s the natural logarthm of gross hourly wage s, X t s a vector of tme-varyng varables, and D a vector of tme nvarant characterstcs. ξ, s the ndvdual-specfc effect and ε t the error term. The tme-varyng covarates n the wage equatons are age, age squared, chldren, health dummes, household type dummes, cumulated unemployment experence untl last year, and sector dummes. The tme-nvarant varables are educaton, educaton squared and cohort dummes. When analysng the wage and the labour supply processes usng panel data, the natural queston that arses s whether the non-response or the mssng observatons are endogenously determned. One source of sample selectvty s the unbalanced nature of the panel. If the panel attrton s endogenous to the wage and the hours processes, sample selecton would be nformatve for wage and hours, and therefore the estmates from the wage and hours equatons would be based. The panel attrton bas s dsregarded from the analyss because prevous studes usng the GSOEP have ndcated that the sample selecton bas s not sgnfcant (Galler, 1996; Rendel and Buechel, 1994). Another sample selectvty bas comes from the fact that wages and workng hours are observed only for the ndvduals n the labour market and selecton bas s determned by the dfferences between workers and non-workers. If the sample under analyss s randomly selected, t s assumed that both workers and nonworkers have smlar observed and unobserved characterstcs and the selecton process does not bas the estmates obtaned usng the workng sample. In contrast, f the decson to work s no longer random and people select themselves nto the labour market based on certan characterstcs, then t s reasonable to assume that workers and non-workers have dfferent observed and unobserved characterstcs. A selecton bas arses when some component of the partcpaton decson s relevant to the wage and hours processes. Dsregardng these relatonshps, the estmates of wages and hours of work from the subsample of workng ndvduals wll be based. If the relatonshps between the partcpaton decson and the wage and hours processes occur through the observables, the secton bas can be controlled by ntroducng the approprate condtonng varables n the wage and hours equatons. If the relatonshps between the partcpaton decson and the wage and hours processes occur through the unobservable, meanng that the unobserved characterstcs affectng the partcpaton decson are correlated wth the unobservable from the wages and hours equatons, smply controllng for the observables s not enough to obtan unbased estmates. If the observables are correlated wth the unobservable, n order to get unbased estmates, the wage and hours equatons should nclude an estmate for the unobservable (Vella, 1998). The selecton model s defned usng the latent varable model: s * X J D u v (25) t t t The selecton ndcator s t defnes the observed employment status y t : y t ìï * 1 f st > = ï í ï * f st ïî (26) For fxed u the probablty of observng y = 1 s gven by: t 14

17 P y 1 X, D Φ( u X J D) (27) t t t Φ(.) s the cumulatve dstrbuton functon and v t s assumed to be normally dstrbuted. Wth..d. error terms, the log lkelhood functon for a fxed effect probt model s gven by: log L(,, u1, u2,, un) yt log ( u XtD) t, yt u Xt J D t, (1 ) log ( ) For a fxed T and N, the maxmum lkelhood s nconsstent because the number of unknown parameters grows wth the number of ndvduals n the sample. Ths s the so called ncdental parameter problem whch mpedes the fxed effects probt model to be estmated consstently for a fxed number of perods. To crcumvent ths problem and to estmate the selecton model accountng for the unobserved heterogenety, the model decomposes the unobserved effect u, followng the approach ntroduced by Mundlak (1978). The assumpton s that the unobserved effect can be modelled as: u ηx τ a (28) Ths equaton assumes that the correlaton between u and x t acts only through the tme averages of the exogenous varables, whereas a represents the remanng part of the unobserved effect whch s ndependent of the tme-varyng varables. Equaton (28) can be substtuted nto the selecton model as follows: y t f η X tj Z X τ a vt 1 f η X tj Z X τ a vt (29) v * X, Z ~ N,1, t 1,, T (3) t To summarze, the selecton model ncludes tme-nvarant characterstcs (educaton, educaton squared, cohort dummes, natonalty dummes), tme-varyng varables (age, age squared, nteracton between educaton and age, cumulated experence (full tme, part-tme, unemployment), health dummes, household type dummes, chldren and other household ncome (for women only)) and the means for the tmenvarant varables (except for age). To test for selecton bas n both hours and wage equatons for men and women, the study appled the approach ntroduced by Wooldrdge (1995) for the FE specfcaton. A standard probt was run for each tme perod based on the model (3). For people partcpatng n the labour market, the nverse Mlls ratos were computed and then ntroduced nto the ntal models for hours and wages. The fnal step nvolved estmatng the augmented models by applyng the FE specfcaton and testng the coeffcent of the nverse Mlls ratos. To correct for the sample selecton n the wage and hour equaton, the nverse Mlls ratos for each year were ncluded as tme varyng varables n the FEVD estmaton of the wage and hours models. For smulatng both labour supply and wage processes, the man requrement mposed on the estmaton method s to provde consstent and unbased estmates. Allowng for unobserved heterogenety and for a 15

18 correlaton between the unobserved ndvdual effects and the other explanatory varables requres the use of FE estmaton, however ths estmaton technque s rather useless n a smulaton context. A good alternatve s the FEVD technque, whch crcumvents the problems of a standard FE model (.e. assumes a correlaton structure between the unobserved ndvdual effects and the other explanatory varables) by decomposng the ndvdual effect and estmatng the last stage as a pooled OLS. 1.IV.2.2 An Econometrc Model of Labour Supply under Certanty wth Heterogeneous Preferences For the estmaton of the lfetme labour supply model whch ncorporates a hgh degree of ndvdual heterogenety, the study proposes a mxed fxed and random coeffcent model smlar to Rabe-Hesketh and Skrondal (25). The fxed coeffcents are consdered constant across consumers and tme, and the random coeffcents are unque for each ndvdual, but constant over tme. The assumpton s that the labour supply model can be approxmated by: y ζ J ( ς v ), 1,, N; t 1,, T (31) t t, t J t s a 1 K M vector of explanatory varables, s the correspondng vector of coeffcents, ς, s the ndvdual-specfc effect, and vt the error term. As the ς, component nduces a wthn-ndvdual dependence, the composte error term ( ς, vt) s correlated over tme. Serally correlated errors mply then the OLS or maxmum lkelhood standard errors are no longer vald. Ths dependence can be taken nto account ether by usng the sandwch estmator for the standard errors, whch does not make any assumptons about the dstrbuton of wthn-dependence of the resduals, or by modellng the dependence explctly (Rabe-Hesketh and Skrondal, 25). One way to model dependence s to decompose the error component nto a tme-constant or a permanent error component ς,, unque for each ndvdual, and a transtory error component v t, whch vares across ndvduals and tme. The permanent error component represents the combned effect of the omtted tmeconstant covarates and of the unobserved heterogenety. ς, and vt are assumed normally dstrbuted, ndependent of each other. v t s assumed ndependent across ndvduals and over tme. Model (31) can be rewrtten by movng ς, n the ntercept: y ( ζ ς ) J v (32) t, t t Model (32) represents a random-ntercept model, whch s a regresson model wth an ndvdual specfc ntercept. ς, can be nterpreted as a random parameter that s estmated together wth the varance of the v t. The parameters of the random-ntercept model are estmated by maxmum. To conclude, the lnear random-ntercept model allows the overall level of hours to vary across ndvduals after controllng for covarates. Addtonal heterogenety s ncorporated by ncludng addtonal random coeffcents besdes the random ntercept, meanng that the effects of some covarates are allowed to vary across ndvduals. The resultng model s a mxed fxed and random coeffcent model. By ntroducng ndvdual-specfc slopes, the assumpton of parallel ndvdual-specfc regresson lnes s relaxed and our model becomes: 16

19 y ζ, 1, 2 ς, X ψ R ζ ς, v k (33) t t t k k t ζ represents the mean ntercept of hours of work, ζk, k 1,2 the mean slopes of the covarates chosen to have random coeffcents, and ς, k, k[,2] the devaton of ndvdual ntercept and slopes from the mean values of ζ k. ψ s a vector of fxed coeffcents, constant across ndvduals and tme X t represents a1 K vector of covarates wth fxed coeffcents, whereas R s a 1 M vector of covarates wth random coeffcents. t The wage model follows a smlar specfcaton; besdes the random ndvdual-specfc ntercept, the model s specfed wth two addtonal random coeffcents for age and educaton: w ( π ξ ) R ( π ξ ) P π ε (34) t, t 1 1, t 2 t R t s the vector of covarates wth random coeffcents. P t s the vector of covarates wth fxed coeffcents. π s the mean ntercept of wage, π 1 the vector of coeffcents llustratng the mean slopes of the R t varables and ξ, k,1 represents the devaton of ndvdual ntercept and slopes from the mean values of π k,, π 1. Gven the hgh computaton costs of estmatng the selecton model usng a random coeffcent specfcaton, only fxed coeffcent specfcaton s used n the study. The nverse Mlls ratos from the estmated probt regressons are ncluded n the random coeffcents wage equaton of both men and women, whereas for hours, the addtonal term s ncluded only for women. 1.IV.3 The Emprcal Model of Lfetme Labour Supply under Uncertanty The extenson of the emprcal model of lfetme labour supply to ncorporate uncertanty s straghtforward. The methodology s the same as the one under certanty, except for an addtonal regressor that captures the forecastng error. Ths term s ntroduced to ncorporate that ndvduals adapt ther expectatons regardng ther future wages each perod. 1.IV.3.1 An Econometrc Model of Labour Supply under Uncertanty wth Homogenous Preferences Under uncertanty, the labour supply model assumng homogenous preferences s expressed as: y ζ X D μln u v (35) t t t t 1 ln lnwk lnwt pk t (36) ka 1 r j ja ln represents the forecastng error equal to the cumulated dscounted dfference between the actual wage and the expected wage multpled by the probablty of beng employed, from the start of the actve lfe (a) untl the current year. The selecton and the wage models are mantaned as under certanty, assumng homogenous preferences. 1.IV.3.2 An Econometrc Model of Labour Supply under Uncertanty wth Heterogeneous Preferences Under uncertanty, the labour supply model assumng heterogeneous preferences s expressed as under certanty, wth the assumpton that the forecastng error n equaton (36) has a random coeffcent. Ths 17

20 change mples that the effects of the varables vary across ndvduals, but the selecton and wage models reman the same as under certanty. Table 2 summares the dfferences of all model varants proposed. Table 2 An Overvew of All Models Descrbed Models Estmaton Method for Labour Supply Uncertanty n Future Wages Standard Random Effect (Heckman extended) Model Random Effects Model No Proposed Labour supply model wth homogenous preferences FEVD No Proposed Labour supply model wth heterogeneous preferences Mxed Coeffcents Model No Proposed Labour supply model wth homogenous preferences wth uncertan extenson Proposed Labour supply model wth heterogeneous preferences wth uncertan extenson FEVD Mxed Coeffcents Model All varables ncluded n the estmatons are the same except when the adjustment varable under uncertanty cases Yes Yes 1.IV.4 Smulatons Ths secton descrbes how the models developed are appled to smulate the labour supply responses for 24 and 25. The accuraces of the projectons obtaned from these models are compared wth the smple extended Heckman model that s commonly used n smulatng contnuous labour supply. Snce the GSOEP dataset s a panel, the Heckman model s extended to ncorporate the unobserved heterogenety. Whle the estmaton of the probt model s dentcal wth one used by the selecton model, the second step of the Heckman procedure s estmated by a standard RE model. The smulatons are performed usng the estmates from the emprcal models presented n the prevous sectons. The smulaton follows the basc structure of a dynamc mcrosmulaton. To smplfy the exercse, ths smulaton only conssts of the demographc and labour market module. The demographc module updates some basc demographc varables lke age over tme. Besdes, t also updates the varables that may nteract wth the demographc varables. The labour market module s the core part of the smulaton, whch updates employment status, wage, and the hours of work. Fgure 1 llustrates the steps of smulaton. For the employment model (selecton model), the smulaton uses pooled probt as yearly probt s unfeasble n a smulaton exercse. Ths also crcumvents the problem that the nverse Mlls ratos cannot be updated durng smulaton. The number of formally employed ndvduals s algned wth the real data n 24 and 25. The wage and the hours smulatons are based on the updated personal characterstcs and are the result of the selecton smulaton ncludng the selecton correcton. The results of the wage smulaton are used n the workng hour smulaton. 18

21 Fgure 1 An Overvew of Smulaton Steps The smulaton follows the sequence as follows: It starts by determnng the value of demographc varables n the new tme perod and updates the related varables. For the personal characterstcs that are not nfluenced by employment, the smulaton uses the actual characterstcs observed n 24 and 25. The smulaton then moves on to the next step where labour market varables are smulated. It predcts the probablty of employed gven the employment selecton model. Afterwards, wage can be predcted usng prevously estmated equatons. Wth the wage and personal characterstcs nformaton, t s now possble to smulate the hours of work. In the case of models wth uncertanty extenson, correcton terms were calculated rght after the predcted wage becomes avalable. Lastly, the varables that reflect the labour trajectory are updated, ths nclude workng experence (full tme, part-tme, unemployed) etc. The projectons of the models are compared wth the actual observed hours of work for 24 and 25. V. Results Ths secton presents the estmaton and smulaton results. All models, ncludng wage and labour supply models are estmated separately for men and women. Smulaton s conducted under a smple framework as descrbed n earler secton. 1.V.1 Estmaton Results The parameter estmates for the probt models are largely skpped as the estmates are n lne wth prevous fndngs and all key varables are sgnfcant. Inverse Mll s ratos were ncluded n wage and labour supply model wth the excepton of male labour supply equaton, where the nverse Mll s rato s not sgnfcant usng any estmaton method. For the random coeffcents models, a test was performed to verfy whether the random ntercept model s suffcent to capture the heterogenety n the wage and hours estmaton. The lkelhood-rato test suggests that the random coeffcents model fts better than the smple random ntercept model, both for wage and labour supply. Table 3 and 5.4 show the parameter estmates for the wage equatons for women and men. Both model specfcatons are standard. The age effect s as expected for both men and women n all three models: a postve mpact wth decreasng margnal effects, showng that hourly wage has the standard humped-shaped age pattern. When estmated usng FEVD, the wage profle of women shows a stronger curvature than for men. The larger coeffcents of the lnear and quadratc age varables show that the growth rate of wages s hgher for women at younger ages, but the growth rate reduces more rapdly at later ages than for men. After the ntroducton of heterogenety, the mpact of age s reduced for women, whereas for men t s ncreased. At younger ages, however, both men and women have a smlar curvature of the wage profle. The estmated rate of return on educaton dffers qute a lot between men and women and between the three models. The model 19