I. Labour Supply. B. Intertemporal/Life-cycle Behaviour. Plan 1. Recent Trends: Age-LFP Profiles. 2. Life-Cycle/Intertemporal Labour Supply Model

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1 Forin Econ 560 Lecure 1B I. Labour Supply B. Ineremporal/Life-cycle Behaviour Plan 1. Recen Trends: Age-LFP Profiles 2. Life-Cycle/Ineremporal Labour Supply Model 3. Esimaing he life-cycle model

2 Forin Econ 560 Lecure 1B 1) Recen Trends: Age-LFP Profiles and Cohor Effecs Men Women A. Changes over Time A. Changes over Time Year B. Changes over he Life-Cycle Year B. Changes over he Life-Cycle Age <= All Age <= All Source: US-GSS, Forin (2009)

3 Forin Econ 560 Lecure 1B Figure 3: Employmen rae by age ( ) A. Male 1977 B. Male 2007 C. Female 1977 D. Female 2007 Source: Blundell, Bozio, and Laroque (2011)

4 Forin Econ 560 Lecure 1B Figure 2: Toal Hours by Age ( ) A. Male 1977 B. Male 2007 C. Female 1977 D. Female 2007

5 Forin Econ 560 Lecure 1B Labour supply is clearly par of a life-ime decision-making process: individuals aend school early in life, accumulae wealh while in he labour force and make reiremen decisions laer in life. o As shown by men s life-cyle LFP profiles Women s life-cycle LFP may be complicaed by labour force wihdrawal when hey have children (generally in heir mid-20s o mid-30s) o Because feriliy decisions have changed across generaions, women s life-cycle or age-profile LFP profiles are complicaed by he presence of cohor effecs, which may i difficul o disenangle o he pure effec of age from a birh cohor effec (e.g. he fac ha women born beween waied for heir children o leave school before enering he labour marke, bu no women born beween ) o These wo effecs canno be disenangled using a single cross-secion of daa. Women s LFP has increased dramaically over he wenieh cenury Wealh and/or savings from labour earnings are used o susain individuals during periods when hey are ou of he labour marke. Life-cycle/Ineremporal Labour Supply (ILS) models seek o undersand variaions in labour supply as individuals aged as well as year-o-year variaions.

6 Forin Econ 560 Lecure 1B There are hree main "dimensions" of ILS: 1) Wages have a hump-shaped paern over he lifecycle. Do people respond o his? Wha deermines he shape of he life-cycle profile? 2) A person may have emporarily higher wages in some period. How does he/she respond? Wha is he source of idiosyncraic (i.e. person-specific) changes in yearo-year labour supply? 3) The "macro" dimension. Hours of work vary over he business cycle. Can we inerpre his as an endogenous response o wages? Wha is he effec of a ransiory change in wage by comparison wih a permanen modificaion o he life-cycle wage profile? The ineremporal model of labour supply makes he poin ha here are hree ypes of wage shocks: permanen, ransiory, and anicipaed, and he labour supply response (elasiciy) will depend on wha ype of shock occurs.

7 Forin Econ 560 Lecure 1B There are 3 ypes of subsiuions elasiciies relevan for predicing he response of hours of work o changes in he wage rae: The ineremporal elasiciy deermines he labour-supply response o wage changes resuling from life-cycle wage growh. The uncompensaed and compensaed elasiciies deermine he hours of work response o shif in wage profiles. An imporan assumpion of he ineremporal model of labour supply is ime separabiliy, his assumpion rules ou habi-formaion or habi persisence, which has been shown o exis empirically. Wha else does i rule ou?

8 Forin Econ 560 Lecure 1B 2. Ineremporal Labour Supply (ITLS) Model A consumer makes his or her choices of consumpion and leisure (or hours of work) over a life-cycle ha goes from an iniial ime period 0 o a ending ime period T. Sanding a ime, he full life-cycle model is characerized by a uiliy funcion of he form U U C H, X ; C, H, X ;... C, H, X, where C is consumpion,, H is labour and X are individual characerisics. Mos ofen he life-cycle uiliy is assumed separable over ime, we hus consider a uiliy funcion of he form U C, H, X, (1) where 1/(1 ) is he subjecive discoun rae. In models wih uncerainy, preferences are considered addiive over saes of he world and ime, so ha uncerainy is handled by aking he expecaion of (1). In his world, individuals have he opporuniy o save and herefore earn an ineres rae r beween period and -1.

9 Forin Econ 560 Lecure 1B The second elemen of he model is he wealh consrain. Leing he change in he value of asses be A A ( 1 r w H C 1 0 1) 2 A1 1 r2 ) A ( w2h 2 C2 A 1 A ( 1 r 1) w 1H 1 C 1 (2) where he price of consumpion is normalized o 1 so ha w is he real wage rae, r is he real ineres rae in period and A is he real value of asses, or equivalenly T 1 A0 w H C (1 r ), (2 ) so ha here are no bequess. s 0 Bu borrowing and lending is permied, and so he budge consrain essenially summarizes he ime pah of wealh (or asses). Wha would if consumers were no allowed o have a negaive asse posiion A 0, 1,... T? For example, hold sudens loans borrowed agains fuure human capial, raher han collaeral. (Hin: saic labour supply model, see Deaon (Economerica Sep 1991)) s 1 1 1

10 Forin Econ 560 Lecure 1B There are many ways o solve his problem, bu given he ime separabiliy, he simples one is simply o se up a Lagrangian T 0 C, H, X L ( C, H, ) U [ A T 0 A 1 (1 r ) w H C ] The FOC condiions for an inerior soluion o he maximizaion of (1) subjec o (2) are U C ( C, H, X ) 0 0,... T (3a) U H ( C, H, X ) w 0 0,... T (3b) and he Euler equaion ( 1 r ) 1 0 (3c) where is he marginal uiliy of wealh in period. Equaion (3c) deermines he rule for he allocaion of wealh across periods. I says ha he consumer chooses savings so ha he marginal uiliy of wealh in period equals he discouned value of he marginal uiliy of wealh in period 1, where he marginal rae of discoun is ( 1 r ) : (1 r ) (1 r ) (1 ) (3c ) 1

11 Forin Econ 560 Lecure 1B Equaion (3c) defines he opimal sraegy and he lifeime problem decomposes ino wo levels. A he firs level, an individual allocaes his wealh over his life such ha his marginal uiliy of wealh evolves as he ages according o equaion (3c). A he second level, condiional upon wealh allocaed o a given period, he wihin period allocaion problem is addressed. The firs wo condiions (3a) and (3b) are he same as in he saic model and require ha he U c / U h MRS HC w in every period as a consequence of ime separabiliy. Time-separabiliy ransforms he ineremporal allocaion problem ino a wo-sage budgeing problem The FOC condiions imply consumpion and hours-of-work supply funcions of he form * C C, w, X * H H, w, X 0 These are commonly referred o as Frisch demand funcions or consan-marginaluiliy-of-wealh labour supply funcions.

12 Forin Econ 560 Lecure 1B Variables such as fuure wealh, wages, or personal characerisics affec labour supply and consumpion only by changing he value of, which serves he role of sufficien saisic capuring all informaion from oher periods ha is relevan. Whereas Marshallian funcions held income consan, Hicksian funcions held uiliy consan, Frisch funcions hold marginal uiliy of wealh consan. As shown in MaCurdy (1981) and Browning e al. (1985), he Frisch elasiciies are he larges of he corresponding hree wage elasiciies, when an inerior soluion is choosen M H F H H H. w w w The Euler equaion implies a ime pah for of he form ln b ln 1, where b ln[ (1 r )]. Repeaed subsiuion yields ln ln b j 0 (4) j 1

13 Forin Econ 560 Lecure 1B Hence he erm in he Frisch supply funcions can be capured as an individual fixed effec 0 plus a funcion of age which is common across consumers. In he world of perfec cerainy, he pah of hrough ime is deermined solely by he known pah of ineres raes and discoun facor. Then apar from ase changes and a geomeric rend, he life-cycle profile of labour supply is compleely deermined by he profile of wages. * * A priori, 0, which is obained by subsiuing C and H ino he wealh consrain (2 ), is affeced by all fuure values of wages received by he consumer over his/her lifeime. Thus a ransiory change in a single w is unlikely o have a large effec on 0, bu he effecs of a permanen change in he wage profile is likely o have an effec on 0.

14 Forin Econ 560 Lecure 1B

15 Forin Econ 560 Lecure 1B Wih concaviy in preferences, as shown in Heckman (1974, 1976), he FOC condiions imply: H o an 0 0 cons w, effec of a movemen along anicipaed wage profile, also called evoluionary wage changes H o 0, effec of a change in he marginal uiliy of wealh 0 o 0 A, marginal uiliy of wealh is declining 0 o 0 w, wages generae wealh effecs The las wo proposiions say ha he marginal uiliy of wealh (based on iniial asses) diminishes wih wages and iniial asses. o The underlying concep of dynamic equilibrium is key o undersand he model: i yields a se of equilibrium ime profiles for labour supply and consumpion deermined for a given value of iniial wealh A 0 and for a given profile of wages, w for all periods.

16 Forin Econ 560 Lecure 1B o Noe ha anicipaed effecs are jus movemens along a given labour supply profile, and should be analyzed as an issue of dynamic equilibrium, no comparaive dynamics. o I is naural o hink ha simple efficiency consideraions would dicae ha he individual will work he mos during periods when he wage (he opporuniy cos of leisure) is highes. o Wha would you do if you were old ha your wage was raised o $50,000 per hour for jus one day? o Bu suppose your wage was raised o $50,000 per hour, effecive for he res of your life. Comparing your remaining oal lifeime labor supply under ha scenario o wha you are currenly planning, would i be less or more? o So, wha happens if w If his increase was anicipaed, he profile of work and consumpion is no affeced. Bu if his increase is unanicipaed hen we will observe a wealh effec and a subsiuion effec.

17 Forin Econ 560 Lecure 1B The former is for sure going o reduce labour supply, he second will increase labour supply in he surprise period, and depending on he cross-subsiuion effec will increase or reduce leisure a imes differen from ha. o Temporary ransfer has a small effec on labour supply in all periods. Bu in realiy, emporary ransfers seem o have large effecs on labor supply [e.g., severance paymens, Card-Chey-Weber (2008)] o The inuiion for hese resuls has o do wih he fac ha a one-period wage increase generaes lile or no wealh effecs when income is pooled across all periods of he lifeime. In conras o he saic, or liquidiy-consrained case where you migh decide o ake i easy when wages rise, you should always make hay when he sun shines if you have access o capial markes.

18 Forin Econ 560 Lecure 1B Adding uncerainy Under uncerainy, FOC (3a) and (3b) remain he same, whereas becomes a random variable ha follows he sochasic process described by (1 r ) E 1 1 Assuming raional expecaions 1 E 1, where 1 is an innovaion o he marginal uiliy of wealh, and deerminisic ineres rae, he expression can be reduced o (1 r 1 ) 1 ln 1 ln r 1 (1 ) The implicaion of he presence of uncerainy is ha we need a measure of anicipaed wage change beween -1 and

19 Forin Econ 560 Lecure 1B 3. Esimaing he life-cycle model a. Srucural Approach Earlier aemps a evaluaing he elasiciy of labour supply in Frisch s sense used a srucural approach by posiing a specific funcional form for he uiliy. For example, MaCurdy (1981), used he addilog uiliy funcion, which is addiive in log consumpion and hours: ~ ~ i 1 U i Ci, H i ; X i, i bi ( Ci ) a i ( H i ), ~ where 1 1, 0, , b 0, a exp ( ). i i X i The Frisch hours of work equaion for individual i a age is hen log-linear in hours and wages: ln Hi ln 0 i lnwi X i (ln ) i, (4) where ( 1 /(1 r). Here, he parameer is he ineremporal subsiuion elasiciy: i measures he proporional change in hours of work induced by a proporional increase in wage raes i

20 Forin Econ 560 Lecure 1B as a worker age. I is necessarily posiive and greaer han he Hicksian elasiciy if leisure is a normal good. The parameer represens he elasiciy of hours wih respec o he marginal uiliy of wealh, i mus also be posiive if leisure is a normal good. MaCurdy (1981) used en annual observaions on 513 whie, coninuously married, men from he PSID who were aged in 1967 and observed in each of he en years beween 1967 and He finds esimaes of ranging from 0.14 o 0.35 wih sandard errors going from 0.07 o b. Basic empirical specificaion A ypical empirical specificaion for esimaing he ineremporal elasiciy of subsiuion begins wih he common log-linear specificaion, ln H i ai lnwi bx i ln i, In a sochasic environmen, he consumer maximizes her/his expeced uiliy and equaion (3c) becomes ( 1 r ) E[ 1] 0 o Leing i ln i E 1 ln i is he one-period-ahead forecas error in he logarihm of he marginal uiliy of wealh and i E 1[exp( i )].

21 Forin Econ 560 Lecure 1B The firs-differenced equaion becomes ln H i ai lnwi b X i ( r 1 ) i i, Thus, he change in labour supply consiss of a componen due o changes in ases ( ai ) and characerisics ( X i ) a componen due o variaion in wages (here no sure wha ype?), a componen dues o he difference beween he real ineres rae and he rae of ime preference, a componen due o any updaing in he logarihm of he marginal uiliy of income. Since here is no closed form soluion for he laer, i is ofen reaed as a nuisance parameer. How well does he lifecycle model explain he lifecycle of hours? o Well, here is much growh in boh hours and wages a he beginning of he lifecycle (beween ages 20-30), bu a midlife (ages 30-50) he quasi-absence of growh in hours does no mach he coninued growh in wages and owards reiremen (ages 50-60) hours fall more han wages.

22 Forin Econ 560 Lecure 1B Consider he following able (from Card, 1991) based on six cohors of male: group wage change hours change employmen rae change aged % +45% +10 % poins aged % 0% -5 % poins aged % -5-10% -20 % poins 0-8 years school (30-50) +40% 0% 12 years school (30-50) +20% 0% >= 16 years school (30-50) +10% 0% The able shows ha he age profile of wage change and he profile of hours change are hardly parallel, in conras o wha he lifecycle labor supply model would predic. Also, looking across educaion groups, over he life-cycle he wage growh of college graduaes in much higher han ha of high school graduaes bu he hours profile of differen educaion groups are very similar. A fairly elaborae se of ase parameers would be needed o explain hese daa. An imporan problem in evaluaing he validiy of he lifecycle model is ha very lile is know abou he evoluion of he marginal uiliy of income or he size of he wealh elasiciy.

23 Source: Card (1991) Figure 4 Lifecycle Profiles Men Age in of cps Wages C) 0 ci) C) E *5 0) B Age O-8YearsEduc l2yearseduc 16-i-YearsEduc

24 Source: Card (1991) Figure 5 Lifecycle Profiles of Hours A e in CPS, C 0IC3 C C) 0 -J Age a 0-8 Years Educ ' 12 Years Educ Years Educ

25 Forin Econ 560 Lecure 1B Pisaferri (2003) improves on his poin by using daa on expecaions of fuure wages colleced in he panel secion of he Bank of Ialy SHIW o isolae anicipaed and unanicipaed wage changes and o consruc a measure of wage risk. o He esimaes an ineremporal subsiuion elasiciy of around 0.7. Anoher problem is ha individual hours are influenced direcly by employer-specific demand condiions. Much recen research has focused on evaluaing he ineremporal elasiciy of labour supply in case where workers are free o choose heir hours of work.

26 Forin Econ 560 Lecure 1B Addiional readings: Cahuc and Zylberberg, Labor Economics, MIT Press, Chap. 1. Card, David. Ineremporal Labor Supply: An Assessmen, in C. Sims, ed., Advances in Economerics, Sixh World Congress, New York: Cambridge Universiy Press (also as Princeon Universiy IR Secion Working Paper no. 269, Sepember 1990) Pencavel, J. Labor Supply of Men, in Ashenfeler, O.C. and R. Layard, ediors, Handbook of Labor Economics, Norh-Holland, vol I, 1986, pp,