Uppsala Center for Fiscal Studies

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1 Uppsala Center for Fscal Studes Department of Economcs Workng Paper 2010:11 he Standard Devaton of Lfe-Length, Retrement Incentves, and Optmal Penson Desgn homas Aronsson and Sören Blomqust

2 Uppsala Center for Fscal Studes Workng paper 2010:11 Department of Economcs September 2010 Uppsala Unversty P.O. Box 513 SE Uppsala Sweden Fax: he Standard Devaton of Lfe-Length, Retrement Incentves, and Optmal Penson Desgn homas Aronsson and Sören Blomqust Papers n the Workng Paper Seres are publshed on nternet n PDF formats. Download from

3 1 he Standard Devaton of Lfe-Length, Retrement Incentves, and Optmal Penson Desgn * homas Aronsson and Sören Blomqust September 2010 Abstract In ths paper, we consder how the retrement age as well as a tax fnanced penson system ought to respond to a change n the standard devaton of the length of lfe. In a frst best framework, where a benevolent government exercses perfect control over the ndvduals labor supply and retrement-decsons, the results show that a decrease n the standard devaton of lfe-length leads to an ncrease n the optmal retrement age and vce versa, f the preferences for the number of years spent n retrement are characterzed by constant or decreasng absolute rsk averson. A smlar result follows n a second best settng, where the government rases revenue va a proportonal tax (or penson fee) to fnance a lump-sum beneft per year spent n retrement. We consder two versons of ths model, one wth a mandatory retrement age decded upon by the government and the other where the retrement age s a prvate decson-varable. JEL Classfcaton: D61, D80, H21, H55. Keywords: Uncertan lfetme, retrement, penson system. * he authors would lke to thank Henry Ohlsson and omas Sjögren for helpful comments and suggestons. Research grants from the Bank of Sweden ercentenary Foundaton, the Swedsh Councl for Workng Lfe and Socal Research, and the Natonal ax Board are gratefully acknowledged. Address: Department of Economcs, Umeå Unversty, SE Umeå, Sweden. E-mal: homas.aronsson@umu.se. Address: Department of Economcs, Uppsala Unversty, SE Uppsala, Sweden. E-mal: Soren.Blomqust@nek.uu.se.

4 2 1. Introducton In 1913, the Swedsh parlament decded upon a general publc penson. he retrement age was set to 67. At that tme, the expected lfe-length was 55 years, mplyng that the expected length of lfe fell short of the retrement age by 12 years. Accordng to these rules, and the average length of lfe at the tme, most people would never beneft from the publc penson system. Snce then, the expected length of lfe has ncreased consderably and was 81 years n 2008 for Sweden. Smlar ncreases n the expected lfe-length have taken place n most other countres. here s a large lterature dealng wth the ageng of the populaton and the consequences ths ought to have for the desgn of penson schemes. 1 It s well known that the average length of lfe has ncreased consderably n the western world durng the last 150 years and more recently also n the less developed countres. However, another mportant change n the mortalty process s that the standard devaton of the length of lfe has decreased. Sweden has unusually good hstorcal demographc statstcs, so t s possble to follow ths development over tme. As the mortalty among nfants has a large nfluence both on the expected length of lfe and on the standard devaton, t s customary to calculate the standard devaton (or varance) n the length of lfe for those who have survved untl at least age 10. hs measure s usually denoted s 10. For Sweden, s 10 was 21 around 1750 and had decreased to around 12.5 n year here s also a large crosscountry varaton n the standard devaton of lfe-length. hs s partcularly so f one compares developed and less developed countres, although there s varaton also between developed countres. For nstance, whle Sweden has the most equal dstrbuton of lfe-length wth s 10 beng around 12.5 years, the U.S. has one of the most unequal dstrbutons among developed countres wth a standard devaton, s 10, of 15 years. 3 How would the optmal retrement age as well as the ndvdual contrbuton to and beneft from the penson system respond to changes n the standard devaton of the length of lfe? 1 hs lterature focuses on a varety of ssues such as how the penson system affects retrement ncentves at the ndvdual level and labor supply behavor among the elderly (e.g., Gruber and Wse, 1999; Cole and Gruber, 2007), the optmal legal retrement age (Lacomba and Lagos, 2006) as well as the drecton of reforms of socal securty (Damond, 2005; Damond and Orszag, 2005). 2 See Edwards and uljapurkar (2005, p. 654). 3 See Edwards and uljapurkar (2005, p. 653).

5 3 hs, of course, depends on what we see as the reason for havng a publc penson scheme. Many alternatve motves for publc penson schemes have been dscussed over tme. One s that the market for annutes s not well functonng. It s, therefore, dffcult for ndvduals to handle the uncertanty of the length of lfe. How much should be saved for the old age? How should ndvduals plan ther consumpton path n the old age? One role for a publc penson scheme s to mtgate the problems assocated wth an uncertan length of lfe. It s ths property that we focus on n the present paper. Our analyss shows that the standard devaton of the length of lfe has mportant mplcatons for the optmal retrement age. he nature of ths nfluence s, n turn, qute complex and depends crucally on ndvduals rsk averson wth respect to the number of years they plan to spend n retrement. An nterestng polcy mplcaton of our analyss s that Sweden and the US should have dfferent retrement ages due to dfferences n the standard devaton of lfe-length, even f we were to dsregard all other dfferences between the two countres. As far as we know, the connecton between the standard devaton of lfe-length and the optmal retrement age has not been examned before. 4 o obtan a tractable model, we make several smplfyng assumptons, the detals of whch wll be lad out n secton 2 below. One such smplfcaton s that the ndvduals have no better knowledge about ther own mortalty than the polcy maker (.e. we abstract from asymmetrc nformaton); another s that the ndvduals lfe-tme utlty functons exhbt temporal rsk neutralty. 5 In our model, the objectve of the polcy maker s to desgn the penson scheme n such a way that the expected lfetme utlty faced by the representatve consumer s mzed. Part of the soluton to ths problem s obtaned by provdng a certan consumpton stream durng the old age ndependently of how old an ndvdual becomes. We shall both consder a frst best framework where the government exercses full control over the resource allocaton by decdng upon ndvdual consumpton, labor supply and retrement ages, as well as two dfferent second best models where the government rases revenue by usng a tax (or penson fee) proportonal to labor ncome to fnance a penson-beneft per year spent n retrement. One of the second best models contans a mandatory retrement age 4 here are studes focusng on other aspects of the varaton n the length of lfe between ndvduals. For example, Bommer et al. (2007) consder redstrbuton between ndvduals wth dfferent lfe-lengths. A crucal assumpton n ther work s that the ndvdual lfe-tme utltes exhbt temporal rsk averson. 5 We also abstract from other sources of asymmetrc nformaton, such as unobserved dfferences n the ablty to work durng old age. Cremer et al. (2004) consder a model where the productvty and health status are prvate nformaton and vary between consumers. hey show that the second best optmal polcy may mply a dstorton of the retrement behavor.

6 4 decded upon by the government, whle the retrement age s a prvate decson-varable n the other. Real world penson systems often contan elements of both mandatory retrement (typcally n the form of a lowest age when penson benefts become avalable) and ndvdual retrement choces. o smplfy the analyss, and capture how uncertanty wth respect to the length of lfe affects the retrement age, we consder mandatory retrement and prvate retrement choces as two separate regmes. he rest of ths paper s organzed as follows. In secton 2, we present the model of ndvdual behavor. Secton 3 deals wth the frst best decson-problem and soluton; n partcular, we examne how the resource allocaton responds to a change n the standard devaton of the length of lfe (as well as a change n the average length of lfe). In Sectons 4 and 5, we consder a smple penson scheme n combnaton wth a choce of retrement age, whch s ether made by the government (Secton 4) or the ndvdual consumer (Secton 5). he penson system s such that the ndvdual pays a tax (or penson fee) proportonal to the labor ncome when young and receves a penson beneft when old. We use both versons of the model to analyze how the optmal penson fee, old-age beneft and retrement age depend on the standard devaton of the length of lfe. Secton 6 concludes. 2. he Model Consder a representatve ndvdual wth preferences over the lfe-tme consumpton, the number of years spent n retrement, and the tme spent on lesure whle beng part of the labor force, respectvely. he assumpton that the ndvdual derves utlty from the number of years spent n retrement s justfed by the observaton that the step from beng a worker to beng retred opens up a whole new spectrum of opportuntes regardng tme-use (e.g., developng new, and tme-consumng, hobbes) as well as mples fewer restrctons on the resdental choce. As a consequence, the utlty assocated wth retrement may dffer n a fundamental way from the utlty of lesure durng the workng-lfe, whch motvates that these two aspects of non-workng tme are treated separately. We assume that the utlty functon takes a quas-lnear form C P( R) U( H ht), (1) t t1 t1 R

7 5 where C s annual consumpton, the length of lfe, R the retrement age and h the annual hours of work. he term H s nterpretable as a fxed tme endowment per perod. he functon UH h t shows the utlty of lesure n perod t. he number of years spent n retrement s gven by R and the utlty derved from the number of years spent n retrement s represented by the functon P ( R). We assume that the functons P() and U () are ncreasng and strctly concave n ther respectve argument. he propertes of the functon P drve many of our results; n partcular, the sgn of the thrd dervatve wll determne the sgn of mportant comparatve statcs. We assume here that the length of lfe s a random varable n the sense that, where s the expected lfe-length, a random varable wth mean zero and unt varance, and 0 a parameter. hs means that s nterpretable as the standard devaton of the length of lfe. In the absence of any penson system, the lfe-tme budget constrant facng the ndvdual can be wrtten as R wh C 0 (2) t t t t1 t1 where w s the hourly gross wage rate. In the next three sectons, we use ths basc model to analyze relatonshps between, on the one hand, the optmal retrement age and penson desgn and, on the other, the standard devaton of the length of lfe. 3. A Frst Best Approach We start by consderng a frst best decson-problem, where the polcy maker decdes upon the consumpton, hours of work and retrement age for a large number of dentcal ndvduals. o smplfy the analyss, and be able to focus on the relatonshp between the retrement age and standard devaton of lfe-length, we assume that the consumpton s equal durng all years, and that the hours of work are constant durng the years spent workng. Snce the age of death s a stochastc varable, the budget constrant facng an ndvdual wll be stochastc as well. However, we assume that the number of ndvduals n the economy as a whole s large enough to mply that the resource constrant at the aggregate level can be

8 6 treated as determnstc. he resource constrant for the polcy maker can, therefore, be wrtten as C Rwh 0. (3) he objectve of the polcy maker s to mze the expected utlty of a typcal ndvdual,.e. E[ CP( R) RV( H h)] CE[ P( R)] RU( H h) (4) where E denotes the expectatons operator. Usng, where E( ) 0 and Var( ) 1, and then substtutng equaton (3) nto equaton (4), we can wrte the optmzaton problem of the polcy maker as Rh, Max Rwh E P( R) RU ( H h). he frst order condtons for R and h become wh E P '( R) U ( H h) 0 (5a) wu '( H h) 0. (5b) We can then derve the followng functons for the optmal retrement age and hours of work; R R( w,, ) (6) h h( w ) (7) n whch the sgn of the comparatve statcs dervatve (when the sgn s unambguous) s gven above each argument. he comparatve statcs dervatves are presented n the Appendx. Equatons (6) and (7) show that an ncrease n the wage rate leads to an ncrease n the retrement age and an ncrease n the hours of work per perod when partcpatng n the labor market: n other words, both labor supply dmensons are modfed n response to a

9 7 change n the wage rate. A change n average length of lfe, on the other hand, only leads to an ncrease n the retrement age, whle the hours of work per perod reman unaffected. hs s a consequence of quas-lnearty, whch means that a hgher does not affect the tradeoff between consumpton and lesure wthn a gven perod. In fact, as we show n the Appendx, our model mples R/ 1, so an ncrease n the lfe expectancy by one year means that the retrement s postponed by one year as well. he reason as to why does not affect the hours of work s analogous; t does not affect the tradeoff between consumpton and lesure. he qualtatve effect on the retrement age of an ncrease n the standard devaton of lfelength,, s ambguous n general. One can show that sgn R / sgn cov(, P ''). If cov(, P '') 0 ( 0 ), then R / 0 ( 0 ). A suffcent condton for cov(, P '') to be postve s that the sub-utlty functon capturng the preferences for the number of years spent n retrement, P(), s characterzed by constant or decreasng absolute rsk averson n the sense that P''( A ) P''( A) 2 1 P'( A2) P'( A1) for 2 1 A A. Wth ncreasng absolute rsk averson, cov(, P '') can be ether postve or negatve. In summary, we have derved the followng results; Proposton 1. In the frst best, an ncrease (decrease) n the lfe-expectancy,, leads to an ncrease (decrease) n the optmal retrement age. If the preferences for the number of years spent n retrement are characterzed by constant or decreasng absolute rsk averson, an ncrease (decrease) n the standard devaton of the length of lfe,, leads to a decrease (ncrease) n the optmal retrement age,.e. R / 0. he ntuton behnd the effect of a change n s as follows: f the preferences for the number of years spent n retrement are characterzed by constant or decreasng absolute rsk averson, there s no precautonary motve to delay the retrement age n response to ncreased uncertanty about the length of lfe. Instead, the optmal response wll be to decrease the retrement age, whch ncreases the lkelhood that any consumer wll be able to enjoy a perod of retrement. Conversely, reduced uncertanty about the length of lfe,.e. a decrease

10 8 n, gves an ncentve to postpone retrement, because the future benefts assocated wth retrement are less uncertan than before. Wth ncreasng absolute rsk averson, on the other hand, there s a precautonary motve for labor market partcpaton (.e. to earn more ncome) n response to ncreased uncertanty, whch has a counteractng nfluence and renders the total effect ambguous. 4. A Second Best Model wth Mandatory Retrement We begn ths secton by characterzng and solvng the decson-problem faced by the ndvdual, whch s here lmted to a choce of consumpton and work hours (for a gven retrement age). he fnal part concerns the polcy-problem facng the government and ts soluton; n partcular, how the (mandatory) retrement age, penson fee and old-age beneft respond to a change n the standard devaton of the length of lfe. Indvdual Behavor In realty, the consumpton path of an ndvdual may vary over the lfe-cycle; for nstance, the planned consumpton mght be smaller for years where the probablty of lvng s low than for years where the probablty of lvng s hgh. Lkewse the hours of work mght vary between perods of the workng lfe. o be able to focus attenton on retrement behavor and penson-polcy wthn a smple model, we abstract from these aspects by assumng (as we dd n sectons 2 and 3 above) that the consumpton remans constant over the lfe-cycle, and that the hours of work per perod are constant when the consumer partcpates n the labor market. o be more specfc, suppose that the ndvdual lves for at most years, and that the subutlty functon for consumpton and lesure, respectvely, has the property mn Ct and t1 R mn U( H ht ). t1 hs formulaton wll mply that the consumpton s equalzed over the lfe-cycle, and that the hours of work are constant over all years of work (as n the prevous secton). As a

11 9 consequence, we elmnate the tme-ndex, t, below. As before, the ndvdual also receves utlty from the number of years spent n retrement, P ( R). he ndvdual plans for the eventualty of lvng untl the age of years, although the probablty of ths event mght be qute small. 6 herefore, the budget constrant takes the form Rwh (1 ) ( R ) B C 0. (8) In equaton (8), s the contrbuton rate to the penson system, whch s proportonal to the labor ncome and pad durng the tme-perod spent n the labor force, and B a lump-sum penson per year spent n retrement. In ths secton, we consder a mandatory retrement age decded upon by the government, whch s treated as exogenous by the ndvdual. herefore, the ndvdual chooses consumpton and work hours to mze the expected utlty subject to the budget constrant,.e. Max E[ C P( R )] R U ( H h ) C, h (9) subject to equaton (8). Snce s stochastc, n all events except when the ndvdual wll leave a bequest. We assume ths bequest gves no utlty to the ndvdual. It s a consequence of the uncertan lfe-tme and the absence of a market for annutes. As before, we have, where E( ) 0 and Var( ) 1. By substtutng the budget constrant nto the objectve functon, the decson-problem facng ndvdual can be wrtten as Max R wh (1 ) ( R ) BE P( R ) R U ( H h ) h. he frst order condton for h becomes 6 hs assumpton s, of course, arbtrary. Our argument s that, f the ndvdual does not consder the possblty that he/she may lve to the age of, he/she may end up wth too lttle resources durng old age. As long as the ndvdual plans for the possblty that hs/her lfe may be longer than the average lfe-length, the fnal year assumed n the budget s not mportant.

12 10 w (1 ) U '( H h ) 0. (10) As the ndvduals are dentcal ex-ante, they wll choose the same number of work hours. Equaton (10) s a slght modfcaton of the standard labor supply condton sayng that the margnal utlty of consumpton tmes the margnal wage rate per perod spent workng must be equal to the margnal utlty of lesure. he weght / attached to the margnal wage rate appears as a consequence of expected utlty mzaton n combnaton wth the assumpton that the consumer recognzes that he/she may reach the age. We can use equaton (10) to derve the labor suppled when partcpatng n the labor market as functon of and (suppressng the before-tax wage rate) h h h(, ) (11) n whch we have ndcated the sgn of each comparatve statc dervatve above the correspondng argument. he ntuton s straght forward: an ncrease n the contrbuton rate to the penson system reduces the margnal wage rate and, therefore, hours of work, whle an ncrease n the lfe-expectancy provdes an ncentve to accumulate more ncome when partcpatng n the labor market. Note also that B and do not affect the hours of work. he Government he government mzes the expected utlty for a typcal cohort wth ex-ante dentcal ndvduals by choosng the retrement age, R, contrbuton rate to the penson system,, and penson beneft per year spent n retrement, B. We can wrte the objectve functon of the government n terms of the expected ndrect utlty functon facng ths typcal ndvdual V B R Rwh R B E P R RU( Hh(, )) (,,,, ) { (, )(1 ) ( ) } [ ( )]. (12) he government s budget constrant s wrtten as

13 11 Rwh(, ) [ R] B 0. (13) he budget constrant (13) bulds on the assumpton that all (unntended) bequests are returned lump-sum to the government. By solvng equaton (13) for B and substtutng nto equaton (12), we can wrte the decsonproblem faced by the government as an unconstraned optmzaton problem n and R,.e. Rwh Max Rwh R E P R RU H h, R (1 ) ( ) [ ( )] ( ) R where, whle the dependence of h on and has been suppressed for notatonal convenence. he frst order condtons can then be wrtten as : R h wh wh w 0 R (14a) R : wh(1 ) Rwh whe[ P '( R)] U ( H h) 0. (14b) R 2 ( R) R Equatons (14a) and (14b) mplctly defne the optmal contrbuton rate and retrement age, respectvely. herefore, by solvng for the optmal contrbuton rate and retrement age, and then substtutng nto the publc budget constrant n equaton (13), we can derve the optmal penson beneft per year spent n retrement. Our man concern here s how a change n the standard devaton of lfe-length affects the optmal retrement age, penson fee and old age beneft, respectvely. Wth equatons (13) and (14) at our dsposal, we have derved the followng result; Proposton 2. If the preferences for the number of years spent n retrement are characterzed by constant or decreasng absolute rsk averson, an ncrease (decrease), n the standard devaton of lfe-length leads to () a decrease (ncrease) n the retrement age, () a decrease (ncrease) n the contrbuton rate to the penson system and () a decrease

14 12 (ncrease) n the penson beneft per year spent n retrement,.e. R / 0, / 0 and B / 0 at the second best optmum. he calculatons behnd Proposton 2 are presented n the Appendx. he ntuton s straght forward: wth constant or decreasng absolute rsk averson, ncreased uncertanty about the length of lfe provdes an ncentve for the government to decrease the retrement age whch, n turn, ncreases the lkelhood that the consumer wll be able to enjoy the benefts assocated wth retrement. Equvalently, reduced uncertanty about the length of lfe gves an ncentve to postpone retrement. hs s analogous to the correspondng result derved n a frst best framework n Secton 3. urnng to the responses n the penson varables, note that a decrease n the retrement age (followng an ncrease n the standard devaton of lfe-length) leads to a decrease n the planned consumpton over the whole lfe-cycle, ceters parbus, at the second best optmum wth the contrbuton rate held constant. hs can be seen by substtutng B Rwh /( R) from the publc budget constrant nto equaton (8) and then dfferentatng wth respect to R, n whch case we obtan C [( R){( Rwh) /( R)}] wh(1 ) 0. R R A decrease n the contrbuton rate to the penson system counteracts ths effect, as t leads to ncreased consumpton,.e. C Rw(1 ) h 0 n whch we have used that equatons (14) are fulflled at the second best optmum. As a consequence, there s an ncentve for the polcy maker to offset the lost consumpton assocated wth early retrement by reducng the contrbuton rate or vce versa. hs explans why the optmal retrement age and contrbuton rate to the penson system move n the same drecton n response to ncreased or decreased uncertanty wth respect to the length of lfe. hen, by usng the publc budget constrant, t follows that the penson beneft per year spent n retrement must move n the same drecton as well. Gven the assumptons on whch Proposton 2 s based, an nterestng mplcaton s that a country wth a relatvely low standard devaton of the length of lfe can be expected to have a

15 13 hgher retrement age and a more generous penson system (n terms of the levels of contrbuton rates and penson benefts) than a country wth a relatvely hgh standard devaton of the length of lfe, everythng else held constant. For nstance, our model predcts that Sweden ought to have a hgher retrement age, hgher contrbuton rates to the penson system and hgher penson benefts per year spent n retrement than the U.S., because the standard devaton of lfe-length s hgher n the U.S. than n Sweden. Note also that f the preferences for the number of years spent n retrement are characterzed by ncreasng absolute rsk averson (nstead of constant or decreasng absolute rsk averson as n the precedng dscusson), none of the three effects dscussed n the proposton can be sgned unambguously. In other words, we can nether sgn the effect on the optmal retrement age, nor the effects on the contrbuton rate and penson beneft, followng a change n the standard devaton of the length of lfe. As before, ths ambguty s due to a precautonary motve to retre later (earler) n response to ncreased (decreased) uncertanty. he optmal polcy-responses to an ncrease n the average length of lfe,, are ambguous. In fact, and somewhat surprsngly, each of the two partal dervatves / and R / can be ether postve or negatve n the second best model. here are two counteractng forces here (each of whch s comprsed of several mechansms). Frst, the concavty of the functon P() and the tax base effect of (va ncreased work hours) consttute a jont ncentve for the government to ncrease both and R n response to a hgher. he ntuton s that () the expected margnal utlty of an addtonal year spent on retrement decreases, and () the government can collect more revenue wthout lowerng the total lfetme consumpton, C (wth held constant, ths varable decreases n response to a hgher contrbuton rate). Second, a hgher lowers the penson beneft per year spent n retrement va the publc budget constrant as well as reduces the budgetary gan of an ncrease n the penson fee. hese effects work to decrease the optmal penson fee and retrement age. Note also that and R need not necessarly move n the same drecton, as the strengths of the mechansms descrbed above may vary between the polcy-varables. he effect of an ncrease n on the optmal penson beneft per year spent n retrement s, of course, also ambguous. 5. A Second Best Model wth a Prvate Retrement Decson

16 14 In ths secton, we relax the assumpton that the retrement age s decded upon by the government and assume, nstead, that the retrement age s a prvate decson varable. he government only decdes upon the penson fee and penson beneft per year spent n retrement. Except for ths dfference, the structure of the model wll be the same as above. Indvdual Behavor he ndvdual chooses consumpton, work hours and retrement age to mze the expected utlty subject to the budget constrant presented n equaton (8). By substtutng the budget constrant nto the objectve functon, ths decson-problem can be wrtten as Max R wh R BE P R R U H h R, h (1 ) ( ) ( ) ( ) where (as before) wth E( ) 0 and Var( ) 1. he frst order condtons for h and R become w (1 ) U '( h ) 0 (15a) wh (1 ) BE P '( R ) U ( H h ) 0. (15b) Equaton (15a) takes the same form as ts counterpart n Secton 4, whereas equaton (15b) s novel here and reflects the ndvdual consumer s choce of retrement age. We can use equatons (15a) and (15b) to derve the retrement age and the labor suppled when partcpatng n the labor market as functons of, B, and. hs gves R R R(, B,, ) (16a) h h h(, ). (16b) he comparatve statcs are derved n the Appendx. We summarze the results as follows;

17 15 Proposton 3. An ncrease n the contrbuton rate to the penson system,, or penson beneft per year spent n retrement, B, wll reduce the retrement age, and an ncrease n the contrbuton rate also leads to a decrease n the hours of work, ceters parbus. Wth the contrbuton rate and penson beneft per year spent n retrement held constant, t follows that () an ncrease n the average length of lfe leads to ncreased hours of work as well as an ncrease (decrease) n the retrement age f wh (1 ) B 0 ( 0 ), and () an ncrease n the standard devaton of the length of lfe leads to a decrease n the retrement age, f the preferences for the number of years spent n retrement, P(), are characterzed by constant or decreasng absolute rsk averson. We now turn to the decson-problem faced by the government and, n partcular, the optmal polcy-responses to a change n the standard devaton of lfe-length. he Government As before, the government mzes the expected utlty for a typcal cohort wth ex-ante dentcal ndvduals. We wrte the objectve functon of the government as the expected ndrect utlty functon of ths typcal ndvdual V(, B,, ) R(, B,, ) wh(, )(1 ) ( R(, B,, )) B EP [ ( R(, B,, ))] R(, B,, ) U( Hh(, )). (17) he budget constrant of the government s now gven by G(, B,, ) R(, B,, ) wh(, ) [ R(, B,, )] B 0, (18) snce the government must consder how the hours of work and retrement age (decsons made by the ndvdual consumer) are affected by the polcy-varables. herefore, the government behaves as f t chooses and B to mze the Lagrangean LV(, B,, ) G(, B,, ) (19)

18 16 where s the Lagrange multpler assocated wth the government s budget constrant. o avod notatonal clutter, we wrte the frst order condtons for and B on the followng compact form; V G 0 (20a) V B G 0 B (20b) where the subscrpts attached to the functons V () and G() denote partal dervatves. We assume that the second order condton 0 G G G V G V G 0 B B G V G V G B B B BB BB B (21) s fulflled, where a double subscrpt denotes second order partal dervatve. he formulas for V and V B are presented n the Appendx. Snce V 0 and VB 0, t follows that G 0 and G 0 at the second best optmum. In addton, f the preferences for the number of years B spent n retrement are characterzed by constant or decreasng absolute rsk averson n the sense descrbed above, f t straght forward to show that G 0. In general, and contrary to the results derved n the prevous secton, the changes n the optmal contrbuton rate and penson beneft followng a change n the standard devaton of the length of lfe are ambguous here. hs s not surprsng, snce the government has fewer nstruments at ts dsposal here,.e. the contrbuton rate and penson beneft must, n ths case, also be used to nfluence the retrement age (as well as nfluence the hours of work and consumpton, as n the prevous secton). o be able to derve an unambguous result, we add the followng assumpton: A1. () G 0 and G 0, and () V G V G 0. B B B B B he frst part of assumpton A1 restrcts the effects that a change n the standard devaton may have on the slopes of the publc budget constrant n - and B -space. It means that an

19 17 ncrease n must not decrease the margnal budgetary gan of an ncrease n the contrbuton rate, and t must not ncrease the margnal budgetary loss of an ncrease n the penson beneft. hs s also nterpretable to mean that the budgetary ncentve to tax and spend must not be deterorated. he second part of assumpton A1 places restrant on the cross dervatves: t ensures that the second order condton summarzed by (21) s fulflled. We can now derve the followng result; Proposton 4. If the preferences for the number of years spent n retrement are characterzed by constant or decreasng absolute rsk averson, and under assumpton A1, an ncrease n the standard devaton of lfe-length leads to an ncrease n the optmal contrbuton rate,.e. / 0 at the second best optmum. he calculatons underlyng Proposton 4 are presented n the Appendx. As long as the preferences for the number of years spent n retrement are characterzed by constant or decreasng absolute rsk averson, assumpton A1 provdes a suffcent, not necessary, condton for the optmal contrbuton rate to ncrease n response to an ncrease n the standard devaton of the length of lfe. he ntuton behnd Proposton 4 s as follows: wth constant or decreasng absolute rsk averson, an ncrease n unambguously reduces the retrement age wth and B held constant and, therefore, the tax base. o compensate the lost revenue, the government ncreases the contrbuton rate to the penson system. However, t remans unclear whether the lower tax revenue due to a hgher exceeds, or falls short of, the revenue gan followng the adjustment n the contrbuton rate. As a consequence, the penson beneft per year spent n retrement, B, may ether ncrease or decrease. hs also mples that the equlbrum effect of an ncrease or decrease n on the retrement age remans ambguous. 7 he optmal polcy-responses to an ncrease n the average length of lfe,, reman ambguous here as well. As we saw above, an ncrease n the average lfe-length typcally leads to an ncrease n the tax base both va ncreased retrement age and ncreased hours of 7 Note that dr R R R B d B. Although the frst two terms on the rght hand sde are negatve, the thrd s ambguous. As a consequence, we were not able to sgn the sum on the rght hand sde; let be that a postve value runs counter to ntuton.

20 18 work. hs means that the tax revenue ncreases for a gven contrbuton rate, and the government may respond ether by an ncrease or decrease n the contrbuton rate and stll obtan hgher revenue than before. In addton, t s not clear whether the average number of years spent n retrement wll ncrease or decrease. hs also mples that the effect on the optmal penson beneft per year spent n retrement may change n ether drecton. 4. Concluson Contrary to earler lterature on the relatonshps between lfe-length, retrement age and publc polcy, whch typcally concentrates on effects of changes n the average length of lfe, the present paper focuses much attenton on how the retrement age as well as the penson system (n terms of the contrbuton rate and penson beneft) ought to change n response to a change n the standard devaton of the length of lfe. Such a study has clear practcal relevance both because ths standard devaton has decreased substantally n most countres durng the latest centures, and because sgnfcant dfferences between countres stll reman. As a consequence, t s mportant to understand how changes n the standard devaton of lfelength affects the optmal choce of retrement age as well as the optmal contrbuton to and beneft from the penson system. he present paper serves ths purpose. Our results show that the atttude towards rsk consttutes a major determnant for how ndvdual behavor and publc polcy may respond to a change n the standard devaton of lfe-length. Although ths nsght s not n tself very surprsng, our relatvely smple model allows us to derve several qute strong results. In a frst best framework, where a benevolent government exercses perfect control over the ndvduals consumpton, labor supply and retrement decsons, the results show that a decrease n the standard devaton of lfe-length leads to an ncrease n the optmal retrement age, f the preferences for the number of years spent n retrement are characterzed by constant or decreasng absolute rsk averson. he ntuton s that reduced uncertanty about the length of lfe gves an ncentve to postpone retrement, because the future benefts assocated wth retrement are less uncertan than before. In a second best framework, where the government rases revenue va a contrbuton to the penson system (or penson fee) attached to the labor ncome to fnance a unform lump-sum beneft per year spent n retrement, we consder two possble scenaros; () the retrement age

21 19 s mandatory and decded upon by the government, and () the retrement age s a prvate decson-varable. In the frst scenaro, we are able to derve a strong result: f the preferences for the number of years spent n retrement are characterzed by constant or decreasng absolute rsk averson, then the optmal retrement age, contrbuton rate and beneft per year spent n retrement all ncrease n response to a decrease n the standard devaton of lfelength. All else equal, ths suggests that a country wth a lower standard devaton of lfelength ought to have a hgher retrement age and more generous penson system (n terms of contrbuton rates and old age benefts) than a country wth a hgher standard devaton of lfelength. In the second scenaro, where the retrement age s a prvate decson-varable, the results are less clear cut. We are able to show that a decrease n the standard devaton of lfelength leads to an ncrease n the retrement age, wth the contrbuton rate and penson beneft held constant, and n a specal case that the optmal contrbuton rate decreases n response to a decrease n the standard devaton. Agan, ths presupposes constant or decreasng absolute rsk averson. he ambguty n the second scenaro s not surprsng, snce the government has fewer polcy nstruments at ts dsposal n that case. As a consequence, t must use the contrbuton rate and penson beneft per year spent n retrement to exercse control over the ndvdual choces of consumpton, work hours and retrement age. In the frst scenaro, on the other hand, the retrement age s decded upon by the government, whch means that the government may use the contrbuton rate and penson beneft solely to affect the labor supply and consumpton behavor. One nterestng drecton for future research would be to relax the (unrealstc) assumpton that the government s as well nformed about ndvdual mortalty as the ndvduals themselves. Instead, f we were to ntroduce asymmetrc nformaton, the second best resource allocaton would be related to these asymmetres, and not to arbtrary restrctons on the set of polcy nstruments. As such, ths would enable us to examne how the optmal use of a nonlnear tax and penson system responds to ncreased varaton n the length of lfe, whch s arguably a realstc extenson. Another nterestng ssue refers to the optmal combnaton of fully-funded and pay-as-you-go pensons, when the length of lfe s uncertan. We leave these and other extensons for future research. Appendx

22 20 Frst Best Resource Allocaton Let EP [ ''( R)] U''( Hh) 0. he comparatve statcs become R hu''( H h) 0 w R E[ P''( R)] U''( H h ) 1 R E[ P''( R)] U''( H h) (A1) (A2) (A3) h E[ P''( R)] 0. (A4) w Second Best Model wth Mandatory Retrement Let us use as a short notaton for the objectve functon faced by the government s Secton 4. herefore, Rwh R Rwh(1 ) ( R) E[ P( R)] RU ( H h), (A5) where the dependence of h on and has been suppressed. he frst order condtons 0 and 0 are gven by equatons (14a) and (14b), respectvely. he second order R condtons,.e. 0, RR 0 and that 0, are assumed to be fulflled. Note RR 2 R h Rwh Rw 0 2 R ( R). (A6) We can then use equatons (14a) and (14b) to derve 1 E R [ P ''( R )] (A7)

23 21 R 1 E[ P''( R)]. (A8) If the preferences for the number of years spent n retrement are characterzed by constant or decreasng absolute rsk averson, we have E[ P''( R)] 0, so / 0 and R / 0. Fnally, the publc budget constrant can be wrtten as Rwh B. (A9) R If E[ P''( R)] 0, we have B wh R Rwh ( R) R 2 0. (A10) Second Best Model wth a Prvate Retrement Decson Let EP [ ''( R)] U''( Hh) 0 and can then derve the followng responses at the ndvdual level; ( wh (1 ) B)/ E[ P ''( R )]. We R ''( ) wh U H h 0 (A11) R B ''( ) U H h 0 (A12) R U''( H h ) R E[ P''( R )] U''( H h ) h E[ P''( R )] w 0 h E[ P''( R )] w(1 ) / (A13) (A14) (A15) 0. (A16)

24 22 urnng to the government, we can use the frst order condtons for the ndvduals to derve the followng propertes of the functons V () and G() ; V Rwh 0 (A17) [ V ] 0 B R (A18) V V B B R wh B (A19) R (A20) 1 R h V rwh [ whrw ] V B R wh 1 R [ ] V R V B (A21) (A22) (A23) R. (A24) R h R G Rwh whrw B 0 R GB [ whb] [ R] 0 B 2 R R R h R G B wh [ whb] wh w B B B R h G [ whb] Rw B R G [ whb] 2 R h R R h R h G whrw wh w w 2 2 h R Rw 2 2 R R R h R G wh wh w B (A25) (A26) (A27) (A28) (A29) (A30) (A31)

25 R R h R R G wh w B1 B B B (A32) 2 R R GB [ whb] B. (A33) Dfferentatng equaton system (20a) and (20b) wth respect to, B, and, whle usng 0 G G V G V G 0 G B B G V G V G B B B BB BB gves the comparatve statcs dervatves B G[ GLBB GBLB] GB[ GBL G L ] B B G [ GBL G L B] G [ G L GBL ] B G [ GLBB GBLB ] GB[ GBL G LB ] B G [ GBL G L B] G [ G LB GBL ] (A34a) (A34b) (A34c) (A34d) where L V G and correspondngly for the other L-dervatves. References Bommer, A., Leroux, M-L. and Lozachmeur, J-M. (2007) Uncertan Lfetme, Redstrbuton and Nonlnear Prcng of Annutes. Mmeo. Cole, C. and Gruber, J. (2007) Future Socal Securty Enttlements and the Retrement Decson. Revew of Economcs and Statstcs 89, Cremer, H., Lozachmeur, J-M. and Pesteau, P. (2004) Socal Securty, Retrement Age and Optmal Income axaton. Journal of Publc Economcs 88, Damond, P. (2005) Pensons for an Ageng Populaton. NBER Workng Papers: Damond, P. and Orszag, R. (2005) Savng Socal Securty. Journal of Economc Perspectves 19,

26 24 Edwards, R.D and uljapurkar, S. (2005) Inequalty n Lfe Spans and a New Perspectve on Mortalty Convergence Across Industralzed Countres. Populaton and Development Revew 31, Gruber, J. and Wse, D. (1999) Socal Securty and Retrement around the World. Chcago: he Unversty of Chcago Press. Lacomba, J. and Lagos, F. (2006) Populaton Ageng and Legal Retrement Age. Journal of Populaton Economcs 19,

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