How Family Status and Social Security Claiming Options Shape Optimal Life Cycle Portfolios

Transcription

1 Workng Paper WP How Famly Status and Socal Securty Clamng Optons Shape Optmal Lfe Cycle Portfolos Andreas Hubener, Ramond Maurer, and Olva S. Mtchell M R R C Project #: UM13-12

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3 How Famly Status and Socal Securty Clamng Optons Shape Optmal Lfe Cycle Portfolos Andreas Hubener Goethe Unversty of Frankfurt Ramond Maurer Goethe Unversty of Frankfurt Olva S. Mtchell The Wharton School, Unversty of Pennsylvana October 2013 Mchgan Retrement Research Center Unversty of Mchgan P.O. Box 1248 Ann Arbor, MI (734) Acknowledgements Ths work was supported by a grant from the Socal Securty Admnstraton through the Mchgan Retrement Research Center (Grant # 5 RRC ). The fndngs and conclusons expressed are solely those of the author and do not represent the vews of the Socal Securty Admnstraton, any agency of the Federal government, or the Mchgan Retrement Research Center. Regents of the Unversty of Mchgan Mark J. Bernsten, Ann Arbor; Jula Donovan Darlow, Ann Arbor; Laurence B. Detch, Bloomfeld Hlls; Shauna Ryder Dggs, Grosse Ponte; Dense Iltch, Bngham Farms; Andrea Fscher Newman, Ann Arbor; Andrew C. Rchner, Grosse Ponte Park ; Katherne E. Whte, Ann Arbor; Mary Sue Coleman, ex offco

4 How Famly Status and Socal Securty Clamng Optons Shape Optmal Lfe Cycle Portfolos Abstract Household decsons are profoundly shaped by a complex set of fnancal optons due to Socal Securty rules determnng retrement, spousal, and survvor benefts, along wth beneft adjustments that vary wth the age at whch these are clamed. These rules nfluence optmal household asset allocaton, nsurance, and work decsons, gven lfe cycle demographc shocks such as marrage, dvorce, and chldren. Our model generates a wealth profle and a low and stable equty fracton consstent wth emprcal evdence. We also confrm predctons that wves wll clam retrement benefts earler than husbands, whle lfe nsurance s manly purchased by younger men. Our polcy smulatons mply that elmnatng survvor benefts would sharply reduce clamng dfferences by sex whle dramatcally ncreasng men s lfe nsurance purchases. Ctaton Hubener, Andreas, Ramond Maurer, and Olva S. Mtchell (2013). How Famly Status and Socal Securty Clamng Optons Shape Optmal Lfe Cycle Portfolos. Ann Arbor MI: Unversty of Mchgan Retrement Research Center (MRRC) Workng Paper, WP Authors Acknowledgements The research reported heren was performed pursuant to a grant from the US Socal Securty Admnstraton to the Mchgan Retrement Research Center as part of the Retrement Research Consortum. Addtonal research support was provded by the Deutsche Forschungsgemenschaft, the German Investment and Asset Management Assocaton (BVI), the Penson Research Councl/Boettner center at The Wharton School of the Unversty of Pennsylvana, the Metzler Exchange Professor program, and the Eunce Shrver Kennedy Natonal Insttute of Chld Health and Development Populaton Research Infrastructure Program R24 HD at the Unversty of Pennsylvana. The authors thank Davd Love, Sta Slavov, Karen Smth, and John Shoven for generously sharng data and computer code; Yong Yu for excellent programmng assstance; and James Anderson for research assstance. Opnons and errors are solely those of the authors and not of the nsttutons wth whom the authors are afflated Hubener, Maurer, and Mtchell. All rghts reserved.

5 How Famly Status and Socal Securty Clamng Optons Shape Optmal Lfe Cycle Portfolos 1. Introducton Two crucal factors drve households optmal lfe cycle savng and nvestment decsons: labor market work and famly status. Ths s because decsons about hours of work as well as retrement ages shape labor earnngs, whch n turn nfluence how people spend, save, nvest, and buld up retrement benefts through the Socal Securty system. Not only are wages uncertan, but so too s famly status due to marrage/dvorce, the arrval/departure of chldren, and spousal death. Each of these poses fundamentally mportant rsks to the household s fnancal poston: for nstance, the arrval of chldren shape household spendng and savng patterns due to chld support n the case of martal splts and college costs. Not only do chldren nfluence fnances drectly; they also change the amount of tme that household members, especally mothers, can earn ncome essental to buld up fnancal assets (Kmmel and Connelly, 2007). Also key to lfe cycle decsons s the role of the Socal Securty system. In the Unted States, ths s a natonal mandatory deferred lfe annuty scheme wth complex clamng optons and cash-flow patterns that depend on age, work hstory, and famly status. Socal Securty s especally crucal because t represents such a large component of household assets. For example, the medan Baby Boomer household on the verge of retrement has accumulated $600,000, of whch 40 percent s Socal Securty wealth; and the remander s dvded evenly between home equty, non-penson fnancal assets, and penson wealth. 1 The rsk and return profle of ths mportant asset should therefore have profound consequences for how households manage ther fnancal wealth, both durng the work lfe and n retrement. And t s ncreasngly becomng clear that when to exercse the opton to clam Socal Securty benefts s one of the most crucal and complex fnancal decsons facng workers. For example, clamng benefts at age 70 nstead of age 62 boosts lfelong payments by 76 percent (Myers, 1985). Addtonally, the fnancal decson of when to clam Socal Securty benefts s dfferent from, but also related to, the decson about when to leave the labor force (c.f., Cole et al., 2002). For example, workers can retre early at age 62, delay clamng untl age 70 to boost benefts, and draw down fnancal assets to mantan 1 Ths measure (n $2010) ncludes fnancal assets, home equty, busness and penson assets, and Socal Securty benefts, and t nets out fnancal and mortgage debt (see Gustman et al., 2010). 1

6 consumpton. Or they can clam at the earlest possble age 62 by acceptng lower benefts, contnue to work, and concurrently receve ncome from work and Socal Securty benefts. The Socal Securty system offers a complex set of famly benefts whch also shapes optmal fnancal wealth and clamng patterns. Thus couples buld up an enttlement to ther own old age retrement benefts over ther workng lves, as well as spousal and wdow(er) benefts that depend on the partners work hstores. Moreover, the Socal Securty rules permt ndvduals to frst clam old age benefts on ther own work records, and later swtch to spousal/wdow benefts. In other words, the decson about when to clam benefts depends ntmately on famly status; n turn, the clamng age has a large effect on payouts to spouses and survvors. Thus these famly benefts can have a pronounced effect on savng and nvestment decsons ncludng the demand for rsky stocks and lfe nsurance products. For ths reason, theoretcal analyss of the clamng dynamcs and the nfluence of Socal Securty benefts on fnancal wealth management requres examnng a full household optmzaton framework over the complete lfe cycle whch jontly models the work, savng, nvestment, and clamng decsons. Untl now, such a model has not been avalable n the lterature. Ths paper ncorporates these key elements of the household lfe cycle Socal Securty benefts and famly dynamcs nto a realstcally-calbrated portfolo and consumpton choce lfe cycle model n dscrete tme wth forward-lookng ratonal multperson households. We allow for rsky asset returns as well as uncertanty n famly status, mortalty, labor ncome, and retrement ncome. Usng data from the Panel Study of Income Dynamcs (PSID), we calbrate wage rate dynamcs by age, sex, educaton, and famly status. In addton we calbrate the mpact of chld care tme on the households avalable budget for ncome generatng work hours usng the Amercan Tme Use Survey (ATUS). We track the ndvdual work hstory for each spouse separately and realstcally model Socal Securty old age benefts wth spousal and survvor benefts as well as delayed clamng optons. In ths envronment, an ndvdual makes decsons about savng, nvestment (stocks/bonds/lfe nsurance products), work hours, and the beneft clamng age. We show that famly status has enormous mpact on nvestment and clamng decsons. Couples wth chldren nvest less n rsky assets and purchase much more lfe nsurance than chldless couples or sngles. Also marred women clam ther own Socal Securty benefts much earler than sngle women, whle marred men clam much later. Interestngly, chldren have lttle mpact on clamng decsons. These predctons from our theoretcal model are consstent wth emprcal evdence n the Health and Retrement Study (HRS). We also show that Socal Securty benefts have a powerful mpact on how households manage fnancal wealth and work 2

7 patterns. Reducng survvor benefts would lead to hgher lfe nsurance demand on men, later clamng and more work hours for women, wth lttle mpact on allocatons to rsky stocks. Our research bulds on and extends the lterature ntated by Merton (1969) on lfe cycle consumpton and portfolo choce models. Recent researchers have sought to make these models more realstc by ntroducng new rsk sources, 2 mportant non-fnancal assets, 3 and endogenety of labor supply or retrement ages. 4 Whle most lfe cycle asset allocaton studes take the perspectve of an ndvdual representatve agent rather than examnng the possbly dfferng perspectves of households of varyng szes and compostons, Love s (2010) work s an mportant excepton. Hs was the frst model 5 to ncorporate the effect of famly and martal status rsk on portfolo and savng choce, drawng on PSID data and the Urban Insttute s Model of Income n the Near Term (MINT) to ft famly transton probabltes, housng cost processes, and labor ncome that depend on age, sex, martal status, and chldren. Hs man results are that, frst, chldren lead on average to less accumulaton of fnancal assets whle present n the household, but second, they also ncrease the household s share of rsky assets. In addton, he showed that households wth chldren have substantally hgher demand for term lfe nsurance than sngles. Yet ths mportant pror study s slent on the lkely mpact of endogenous labor supply and retrement age on optmal household patterns, takng account of Socal Securty rules. By contrast, our realstc formulaton of Socal Securty beneft optons departs rather dramatcally from pror studes whch assume that retrement benefts are smply a fxed fracton of labor earnngs as of a pre-specfed date. And our more general approach permts us to evaluate possble polcy reforms such as changes n Socal Securty rules. Other work related to ours ncludes that of Shoven and Slavov (2012) and Cole et al. (2002), who explore the payouts from varous beneft clamng optons under U.S. Socal Securty system rules. Gustman and Stenmeer (2005) analyze, usng a structural estmaton model, how retrement and clamng patterns respond to Socal Securty ncentves. Yet these studes from the publc fnance lterature focus only on the retrement phase of the lfe cycle, and hence they do not ntegrate the portfolo choce problem wthn a full household optmzaton framework. Hubener et al. (2013) develop a mult-person portfolo choce model 2 For example, non-tradable rsk labor ncome by Vcera (2001) and Cocco et al. (2005), nterest rate rsk by Campbell and Vcera (2001), health rsk by French (2005), and rsk on housng expendtures by Gomes and Mchaeldes (2005). 3 For example, housng wealth by Cocco (2005), lfe annutes by Horneff et al. (2008) and Inkmann et al. (2011). 4 See Bode et al. (1992), Farh and Panageas (2007), Gomes at al. (2008), and Cha et al. (2011). 5 Earler work by Scholz et al. (2006, 2007) explored the mpact of chldren on wealth accumulatons wthn a lfe cycle framework, but t assumed exogenous labor supply/retrement dates and excluded portfolo choce decsons. 3

8 of a retred couple, allowng for nvestments n rsky stocks, annutes, and lfe nsurance purchases. Once agan, however, that paper s slent on the work lfe ssue and t adopts a smple Socal Securty beneft rule. In what follows, we develop the structure of the lfe cycle portfolo choce model for households wth uncertan famly status, tme budget constrants that depend on the arrval of chldren, and realstc Socal Securty beneft optons. Secton 3 presents the calbraton of the parameters, most mportantly the mpact of the arrval of chldren on avalable tme for work and the dynamcs of uncertan wage rates. In secton 4, we dscuss the man fndngs from the model smulatons and compare our model predctons about clamng wth emprcal evdence from HRS data. Secton 5 explores possble polcy reforms lke changes n beneft structures under Socal Securty rules. A fnal secton concludes and summarzes results. 2. The Lfe Cycle Optmzaton Model In our model, agents face the rsk of exogenous famly transtons throughout ther workng lves and nto retrement. In the followng, x (y) denotes a woman (man). Tme t = (0,, T) s measured n years. At tme t = 0 (assumng age 20 for women and 24 for men) the ndvdual starts workng lfe and can be ether sngle or marred. We assume that the four year age dfference between spouses s fxed over the lfe cycle. Each ndvdual has an uncertan lfe span and may lve for a maxmum o f T = 80 years. 2.1 Famly dynamcs The state varable famly s t s modeled at each pont n tme as a Markov chan wth 35 dscrete states. Before retrement, the possble famly states are never marred, marred couple, dvorced, and wdowed. We further dfferentate each of these states for the woman or the man. In addton, a household can have between zero and three chldren. We do not dstngush between never marred, dvorced, and wdowed sngle retrees. Possble retrement states for couples nclude only the wfe beng retred, only the husband beng retred, and both spouses beng retred. For modelng spousal benefts, t s also necessary to dfferentate these states wth respect to the age when the husband clamed hs own retrement benefts (see secton 2.5). A complete lst of all possble famly states s gven n Appendx A. The tme-dependent transton matrx Π j,t = Prob( s t = s t+1 = j ) for ths Markov chan s nfluenced by fve factors: mortalty, marrage, dvorce, fertlty, and chldren leavng the household. We abstract from multple brths and dvorces durng retrement. We only allow marred couples to receve chldren, and we treat three or more chldren as the 4

9 same famly state. 6 In the case of a dvorce, chldren are assumed to stay wth ther mothers. 7 At the end of our projecton horzon T, whch corresponds to an age of 100 for women and 104 for men, we set the survval probablty to zero. In the followng, we descrbe the model for couples and refer to the sngle case only when t s not a straghtforward smplfcaton of gnorng the absent partner. 2.2 Fnancal products Indvduals may select from three dfferent fnancal products to manage ther lqud wealth: rskless bonds, rsky stocks, and term lfe nsurance. Bonds are characterzed by a constant annual real gross rate of return R 0. The dstrbuton of the stock return R t s assumed to be lognormal and serally ndependent. In each perod t, the ndvdual ε {x, y} may purchase a one-year term lfe nsurance contract. If the nsured person des wthn the perod [t, t + 1], any survvng spouse or chldren receve the face value L t at tme t + 1. If the nsured person survves, no payments are dstrbuted, snce no cash value s bult up by the nsurance contract. Accordng to the actuaral prncple of equvalence, the premum LP t charged by the nsurance company equals the present value of the expected payout plus some expense loadngs δ t, whch s gven by LP t = (1 + δ t ) ൫1 p t ൯ R L t. (1) Here p t denotes the probablty from a mortalty table that ndvdual condtonal on beng alve at tme t survves to tme t + 1. The (age-dependent) loadng factor δ t reflects expenses covered by the nsurance company for admnstraton and to control for adverse selecton Tme budget Each ndvdual has an avalable tme budget Θ. Dependng on famly status and age, a certan amount of tme must be spent on chld care θ s,t. Before retrement, the worker can decde how much of the avalable tme he wll spend n the pad labor market τ t to generate labor ncome. Workng for pay nflcts (unpad) commutng tme τ t, trav. Tme remanng s utlty-ncreasng lesure l t. Accordngly, the tme budget equaton s gven as follows: 6 Ths lmts computatonal effort. Moreover, the margnal effects of an addtonal chld regardng consumpton scalng or chld care tme decrease wth the number of chldren. 7 The dfferent number of chldren for a dvorced husband matters only for chld support payments and affects the possble famly states to whch he may swtch. 8 Modelng lfe nsurance as mult-year contracts would requre at least one more state varable for each addtonal spouse, whch would make the model ntractable. See Hubener et al. (2013) for a dscusson of how sngle perod lfe nsurance contracts can substtute for longer-runnng contracts. 5

10 Θ = θ s,t + τ t + τ t, trav + l t (2) 2.4 Labor ncome Dependng on the tme devoted to pad work τ t, each agent earns uncertan labor ncome specfed as follows: Y t = τ t w s,t P t ε t, (3) Here w s,t denotes the wage rate whch depends on sex, age, and famly status. The varable ε t s an ndependent dentcally lognormal dstrbuted transtory ncome shock wth mean of one, and P t s the permanent component of the wage rate wth lognormal shock η t evolvng accordng to: P t+1 = P t η t (4) Note that, n the case of a couple, the transtory shock as well as the permanent ncome component s assumed to affect both spouses dentcally or, equvalently, both transtory and permanent shocks are perfectly correlated across partners. 9 The permanent ncome component P t (and ts shock η t ) have a mean of one, such that w s,t s the average wage for the gven combnaton of sex, age, and famly state. 2.5 Retrement ncome From age 62 onward, each spouse has the possblty of clamng Socal Securty retrement benefts, up to age 70 when clamng becomes mandatory. The retrement ncome payable to the ndvdual s equal to hs Prmary Insurance Amount (PIA), whch s based on lfetme earnngs wth adjustment for early or delayed beneft clamng. The Socal Securty retrement beneft s gven by:,ret ret Y t = PIA t λ ε t (5) wth λ beng the adjustment factor for early clamng reducton or delayed retrement credt ret (relatve to the Full Retrement Age), and ε t s a lognormal transtory shock wth a mean of one. In accordance wth U.S. practce, the PIA s based on the earnngs hstory. Usng a concave pece-wse lnear functon, the PIA s computed from the Average Indexed Monthly Earnngs (AIME), whch s the worker s average monthly labor earnngs over hs (wage 9 The modelng of dfferent ncome shocks requres one addtonal state varable whch ncreases the computatonal burden of solvng the model. 6

11 rules. 11 After both partners have clamed ther retrement benefts, the partner wth the lower apprecaton adjusted) best 35 years. To keep the model tractable, we use the PIAs for each spouse as state varables. To be precse, the state varable after clamng s the beneft amount, whch s the product PIA t λ of the PIA and the adjustment factor for early clamng reducton or delayed retrement credt. Hence we need not treat the clamng age as a dfferent state varable. 10 Further detals on how the PIAs are used as state varables can be found n Appendx C. After clamng retrement benefts, ndvduals stll have the opportunty to contnue workng untl age 70. If they do, they are taxed at a rate of 50 cents per dollar earned above the exempt amount of the retrement earnngs test, consstent wth the U.S. Socal Securty retrement ncome may elect to receve spousal benefts nstead of hs own benefts. These amount to 50% of the partner s benefts, unless the spousal benefts are clamed before reachng Full Retrement Age. In ths case, a permanent reducton of up to 30% apples. In contrast to own retrement benefts, clamng spousal benefts after the Full Retrement Age s not ncentvzed wth an ncrease of lfelong payments. Snce trackng the clamng age for spousal benefts requres an addtonal state varable, our model framework only allows for clamng spousal benefts at the Full Retrement Age. 12 After ths age, a partner receves spousal benefts, f these exceed the own already-clamed old age retrement benefts. Another rule s that f one partner clams after hs Full Retrement Age, the delayed retrement credt only ncreases hs own benefts, but not hs partner s spousal benefts. In order to exclude the delayed retrement credt for spousal benefts, we use separate retrement states for dfferent clamng ages of the husband. 13 When a spouse passes away, the survvng spouse may swtch to wdow(er) benefts. These are equal to 100% of the deceased spouse s benefts. In our model, ths s not an actve decson; nstead, these benefts are automatcally pad f the wdow(er) benefts are hgher. If retrement benefts have not yet been clamed, the PIA of the survvng spouse s substtuted n place of the PIA of the deceased spouse. Accordngly, we need not keep track of whether the wdow(er) s PIA results from own work hstory or that of a deceased spouse. 10 For a couple, there are 81 possble combnatons. 11 Survey evdence shows that most people do understand Socal Securty benefts are reduced by the earnngs test, but most are unaware that ther benefts foregone are pad back after the Full Retrement Age; see Brown et al. (2013). Nevertheless, ths has been true only snce the year If the spousal benefts exceed the wfe s own benefts at the Full Retrement Age, but she would lke to receve benefts from age 62 onwards, she can clam her own benefts at ths age, and swtch to her spousal benefts four years later. In ths way, she can avod a permanent beneft reducton. 13 Our results suggest that ths dfferentaton s only necessary for husbands, snce ther retrement benefts are never less than half ther wves benefts. 7

12 Ths qute realstc formulaton of Socal Securty beneft optons dffers from and extends substantally the typcal approach taken n pror portfolo choce lfe cycle studes. That s, the usual approach untl now has been to assume that the worker s retrement beneft s gven by a fxed proporton of hs last labor ncome as of a prespecfed date. 14 Moreover, pror studes have not modeled spousal or survvor payments, gnorng the possblty that one spouse could clam frst on her own account, and later swtch to alternatve beneft payment optons. 2.6 Wealth dynamcs Besdes determnng how much tme to spend workng, each perod the household must also decde how much of ts lqud wealth (W t ) to spend on consumpton (C t ), lfe x nsurance premums for (LP t, LP y t ) for the wfe (husband) x (y), and how to allocate savngs to bonds B t and stocks S t. The household s lqudty constraned, so that t cannot borrow to fnance consumpton and lfe nsurance purchases: W t = C t + LP t x + LP t y + B t + S t LP t x 0 LP t y 0 S t 0 B t 0 (6) (7) Next perod s lqud wealth s gven by any remanng wealth ncludng captal market returns, labor ncome (Y t ), and Socal Securty benefts (Y,ret t ), less ncome taxes accordng to proportonal rates θ labor and θ ret and housng expenses h s,t : W t+1 = S t R t+1 + B t R 0 + (1 I t+1 ) L x t + ൫1 I t+1 ൯ L t x y y = x y x,ret + ቀ൫Y t + Y t ൯ (1 θ labor ) + ൫Y + Y y,ret t t ൯ (1 θ ret )ቁ ൫1 h s,t ൯ (8) x y The ndcator varables I t and I t are equal to one f the correspondng spouse s alve at tme t and zero otherwse. Other cash flows mght occur due to famly state transtons. If one of the spouses des, the remanng spouse j receves the payment from the lfe nsurance contract L t. If one chld leaves the household, we assume the parents must pay college fees (here desgned as a lump sum). Furthermore, a dvorced woman wth chldren receves chld support payments, whle a dvorced husband wth chldren must devote a certan fracton of hs ncome for chld support. 14 See for nstance Cocco et al. (2005) and Love (2010). Cha et al. (2011) do ncorporate a flexble retrement age and a delayed retrement credt, but ther study does not track lfetme earnngs. Also t takes the perspectve of a sngle representatve worker nstead of a mult-person household wth uncertan famly status, as here. 8

13 2.7 Preferences and optmzaton We mpose that the household has a tme-addtve utlty functon wth constant relatve rsk averson γ, and t derves utlty from a composte good consstng of consumpton C t and effectve lesure l t. Dependng on the number of adults and chldren present n the household, total consumpton s normalzed by a scalng factor φ s (see Love, 2010 and Hubener et al., 2013). For a sngle adult, effectve lesure s dentcal to tme devoted to lesure, whereas for a couple, effectve lesure s gven by the geometrc mean of both spouses lesure tmes: 15 l t = ටl x t l y t (9) The relatve mportance between consumpton and lesure s gven by a modfed Cobb-Douglas functon, whereby the preferences for lesure are governed by the parameter α. The household s expected lfetme utlty can be expressed by the recursve Bellman equaton J t ൫W t, P t, PIA x y t, PIA t, s t ൯ 1 γ 1 C t = max ቊ l t x y x y C t, τ t,τ t,s t,b t,lp t,lp t 1 γ ൬ α ൰ φ s (10) + β E t J t+1 ൫W t+1, P t+1, PIA t+1, PIA t+1, s t+1 ൯ ൧ቋ, x y where β represents the tme preference rate. The value functon s governed by the state x varables fnancal wealth W t, the permanent ncome component P t, PIA t and PIA y t, and the famly state s t. The controls are consumpton C t, workng tme τ t, nvestments n stocks S t or x y bonds B t, and premums for lfe nsurance purchases LP t and LP t. The expectaton of the household s future value functon s the sum over all possble famly states weghted usng the transton probablty Π st,s t+1. E t [ J t+1 (W t+1, A t+1, s t+1 ) ] = s Π st,s E t [ J t+1 (W t+1, A t+1, s t+1 = s) ] (11) An excepton s the case of dvorce, the only nstance n whch a household s splt nto two separate unts, each wth a dfferent utlty functon. In ths case, the ndvdual value functons are equally weghted: 15 Just as total consumpton of the couple s normalzed to the ndvdual level usng a scalng factor, Formula (9) scales both spouses total lesure tme to an ndvdual level. Instead of takng an arthmetc mean, by usng the geometrc mean we ensure a fnte elastcty of substtuton between the lesure tmes of both partners. Ths avods corner solutons (.e., only partner works wthout havng own lesure tme) and ensures that partners seek to balance ther ndvdual tme devoted to lesure. 9

14 1 E t [J t+1 ] = 2 E t [J t+1 (s t+1 = dvorced woman )] + 1 E t [J t+1 (s t+1 = dvorced man)] (12) 2 If one spouse des, the desre to provde for the survvng partner s reflected n the correspondng value functon of the survvng spouse. If the last or both spouses de, they may wsh to provde for ther chldren or leave a bequest. The strength of ths bequest motve s gven by the parameter B s,t. The correspondng utlty s gven by remanng wealth normalzed by the bequest parameter and multpled by the avalable tme budget: 16 1 γ 1 W t J t = Θ α ቇ for b s,t > 0 1 γ ቆ b s,t f both spouses have ded (13) J t = 0 for b s,t = 0 Between ages 62 and 69, each spouse has the opportunty to clam hs Socal Securty benefts. At age 70, no further delayed retrement credt can be earned and clamng s mandatory. Table A1 n Appendx A lsts the possble retrement states to whch transtons by clamng benefts are possble. If the utlty of a retrement state exceeds the utlty of the current state calculated from equaton (10), the utlty of the current state s replaced by the hgher value and the couple swtches to the retrement state Model Calbraton and Parameterzaton 3.1 Famly process calbraton The drvers of famly state transtons are marrage hazards, dvorce hazards, fertlty, chldren leavng the household, and mortalty. We calbrate our probabltes for marrages and dvorces usng the Urban Insttute s Model of Income n the Near Term, Verson 6 (Smth et al., 2010). In ths model, current age and sex s related to marrage and dvorce hazard rates, the number of prevous marrages, and the duraton of the current marrage tme snce last marrage. To parameterze the transton probablty matrx, we smulate a populaton of 1,000,000 people wth an ntal marrage rate of 20%, 18 for whch we track the number and duraton of marrages. These then evolve accordng to the MINT hazard rates. We derve the transton probablty Π j,t by dvdng the number of transtons n our smulated populaton 16 The multplcaton wth some lesure s necessary for the bequest utlty beng measured n the same unts as the utlty from consumpton and lesure. To use the tme budget Θ s equvalent to normalzng Θ = 1 and 1 ൬ W 1 γ t 1 γ B s,t usng J t = ൰ as utlty from bequest. 17 If there are several retrement states to whch the couple could swtch, the state wth the hghest utlty s chosen. 18 A marrage rate of 20% for 20 year old women and 24 year old men s n lne wth the MINT study and a bt hgher than the Natonal Health Statstcs Report (Copen et al., 2012), whch reports a marrage rate of 17.3% for women and 11.3% for men age But f we add the cohabtaton rates (most comparable to the marred couple famly state) of 18.7% and 15.0%, our assumpton s on the low sde. 10

15 from state to state j at age t by the number of paths n state. In the MINT model the number of chldren does not affect hazard rates, so these transtons are ndependent of the number of chldren. Fertlty-drven transtons probabltes are determned n a subsequent step. For the transtons between famly states wth dfferent numbers of chldren, we use 2009 values of the all race fertlty rates from the Natonal Vtal Statstcs Reports (Martn et al., 2011). Reported fertlty rates are adjusted for the fact that n our model, only marred couples have chldren. 19 We assume that chldren leave the household when they turn age 18. Snce our state varables track only the numbers of chldren but not ther age, we agan smulate a populaton wth the already-calbrated fertlty, marrage, and dvorce transtons, and we track the ages of the chldren and make them leave the household after 18 years. The transton probablty to states wth one fewer chld Π j,t s gven by the number of paths at age t wth a chld turnng 18 n state, dvded by the total number of paths n state. Mortalty transtons to wdow or wdower states are gven by sex and age-dependent one-year survval probabltes, for whch we use the U.S populaton lfe table n the Natonal Vtal Statstcs Report 2005 (Aras, 2010).We assume survval probabltes are ndependent of famly status. 3.2 Tme budget and chld care tme Each spouse s assumed to have a tme budget of Θ = 16 hours per day, and the possble work week conssts of fve days (relevant for dstngushng between full, part tme, and over-tme work). We further assume a year to have 52 weeks (relevant for transformaton to annual values) and a month beng 1/12 of a year (relevant for determnng the AIME and PIA). To calbrate state and age dependent chld care tme θ s,t we use data from waves of the Amercan Tme Use Survey. 20 The U.S. Census Bureau conducts the ATUS as an extenson to the Current Populaton Survey (CPS). Two to fve months after households complete the last CPS ntervew, they are elgble for the ATUS. One adult per household s randomly selected to do the ntervew; ths structure precludes us from analyzng emprcally the nteracton of couples tme allocatons. The 24-hour tme dares are collected usng 19 The Natonal Vtal Statstcs Reports gve the fertlty rate of the complete populaton f tot, the fertlty rate of unmarred women f u, and the fracton r of unmarred brths to all brths. The fertlty of marred women s then derved by: f m = 1 r 1 r ftot fu 20 A good descrpton of the ATUS can be found n Hamermesh et al. (2005) 11

16 telephone ntervews, when the respondents report each actvty of the prevous day and ts correspondng duraton. The ntervewer assgns each reported actvty a code categorzed nto 17 top-level categores wth several sub-categores. After the frst wave of 2003 wth 20,720 respondents, there were about 13,000 respondents n each subsequent wave. 21 In the pror lfe cycle lterature wth endogenous labor supply (Gomes et al., 2008; Cha et al., 2011), tme s only dvded nto (ncome generatng) workng tme and lesure. In ths sense, lesure cannot be seen exclusvely as recreatonal, snce t ncorporates both pure lesure and home producton (Gronau, 1977). Smlar to chldren s effects on consumpton, represented n our model by a consumpton scalng factor φ s, chld care tme θ s,t s ntended to capture the effect of chldren on the parents tme budgets. But consderng only tme drectly devoted to chldren would underestmate the truth, snce other actvtes may take longer wth chldren present n the household (for example, cleanng the house or cookng for more people). In ths sense θ s,t cannot be vewed as chld care only, but rather t s the margnal effect of chldren on all actvtes related to home producton. Accordngly, for the calbraton of θ s,t, we consder the followng ATUS actvtes as home producton tme: Carng For & Helpng Household Members 22, Household Actvtes, Consumer Purchases, Carng For & Helpng Nonhousehold Members, Professonal & Personal Care Servces, 23 Household Servces, Government Servces & Cvc Servce, and all travel related to those actvtes. 24 We dvde the ATUS respondent sample nto four subgroups: marred women, marred men, sngle women, and sngle men, and we drop observatons older than age 66. Next, we nclude only those observatons where the age dfference to the youngest chld s at least 18 years and at most 45 (55) years for women (men). 25 Fnally, we exclude the tme dares flled out on holdays or weekends. Naturally, we nclude observatons wth and wthout chldren to dentfy the effect of chldren. We regress the tme spent on the aforementoned actvtes on a set of dummes for the number of chldren (wth one dummy representng three or more 21 Slghtly more than half the dares are recorded on the weekend or a holday. 22 Ths ncludes all 19 actvtes related to chldren lke physcal care, supervsng chldren s actvtes, and playng wth them. Even though the latter can be seen as recreatonal lesure, we choose not to exclude t due to ts drect reference to the effect of chldren on avalable tme. 23 Note that these are the tme costs to make use of the servce, as for example watng on a baby stter. 24 We exclude the followng actvtes: Personal Care, Work & Work-Related Actvtes, Educaton, Eatng & Drnkng (wthout food preparaton), Socalzng, Relaxng, & Lesure, Sports, Exercse, & Recreaton, Relgous & Sprtual Actvtes, and Volunteer Actvtes. Our model assumes a day has 16 wakng hours and hence we exclude personal care, whch s manly sleepng tme besdes washng, dressng, and groomng. Educaton s excluded due to ts close relaton to work and all the other actvtes are recreatonal lesure. 25 There s no ndcator as to whether the chldren n the home are the bologcal chldren or not. These restrctons should mnmze the observatons of people lookng after ther underage sblngs and grandparents lookng after ther grandchldren. 12

17 chldren), and a second/thrd 26 order polynomal n the number of years untl the youngest chld turns age 18. The results of the OLS regresson appear n Table 1, wth a graphcal representaton n Fgure 1. In general, women allocate more tme than men to home producton; for them, chldren also cause a hgher tme ncrease n non-market actvtes, compared to men. Sngle women do spend less tme on these actvtes than marred women, but the effect of (at least the frst two) chldren s about the same for both female groups. Table 1 and Fgure 1 here. For someone wth no chldren, the set of chld dummes and the age of the youngest chld are set to zero, so the regresson constant term reflects tme spent on home producton when no chldren are present. As mentoned above, θ s,t captures only the margnal effect of chldren on home producton tme, so rather than settng θ s,t to the estmated home producton tme for each famly state s, we set θ s,t to the dfference n home producton tme wth reference to someone havng a smlar martal status but wth no chldren (e.g., marred couple wth two chldren versus a marred couple wth no chldren). As already dscussed, our state varables do not drectly track the ages of chldren at home. Instead, for the calbraton of transton probabltes, we smulate a populaton keepng track of the chldren s ages. For each path, chld care tme s calculated accordng to the regresson results, 27 and the value θ s,t s derved by computng the mean over all paths for correspondng famly state s at (parent s) age t. We also use the ATUS for calbratng the tme needed to commute to work. The sample mean for those who worked at least an hour for pay and travelled to work less than x four hours on the dary day s τ t, trav = 0.64 hours for women and τ t, trav = 0.79 hours for men. y 3.3 Wage rate calbraton We estmate the determnstc component of the wage rate process w s,t and the varances of the permanent and transtory wage shocks η t and ε t usng the waves of the Panel Study of Income Dynamcs. Besdes age, sex, and educaton, we are especally nterested n the effect of the famly status and work hours on the hourly wage. In the data set, some respondents drectly report a wage n terms of dollars per hour; for the remander of the observatons, we nfer the hourly wage by dvdng annual ncome by annual work hours. 26 For all other subgroups except marred women, the coeffcent of thrd order n chld age s not sgnfcantly dfferent from zero, so we reduced the order of the polynomal for them. 27 Snce the number of sngle men wth chldren s small, we use the regresson results of sngle women for wdowed men wth chldren. 13

18 Annual work hours are gven by the hours worked per week 28 multpled by 52. All dollar values are n $2009. For the explanatory varables n the wage rate equaton, we use a polynomal up to second order n the respondent s age, a vector of dummy varables for the number of chldren under 18 n the household and whether a spouse s present n the household, and a set of dummes representng work: full tme for pay (between 20 and 40 hours per week), part tme (more than zero but less than 20 hours per week), or overtme (over 40 hours per week). For couple households, we treat each spouse as a dfferent observaton. By usng wage or nferred wage as the dependent varable, we automatcally lmt the sample to the workng populaton for whch we can nfer ths quantty. We also elmnate all observatons wth hourly wage rates of below $5 (whch contradcts mnmum wage laws) and extreme observatons above the 99 th percentle. Furthermore, we dvde the sample accordng to sex and educaton nto four subgroups: men wth a hgh school educaton, women wth a hgh school educaton, men wth a college educaton, and women wth a college educaton. Table 2 shows the results of an OLS regresson of the natural logarthm of hourly wages. 29 In Fgure 2, the age dependence s presented for the four subgroups who work full tme and have a spouse but no chldren. For both educaton groups, men have hgher wages than women 30 and the gap wdens wth age. For all subgroups, lvng together wth a spouse ncreases the wage sgnfcantly, by over 10%. Havng chldren slghtly ncreases men s wages (sgnfcant only for those wth a college educaton), whle t sgnfcantly decreases women s wages by about 5% for two and 10% and more for three or more chldren. There are large wage reductons for part-tme work (up to 20%), and the reductons are twce as large for men as for women. Workng overtme yelds hgher wage rates for all four groups, wth the largest bonus for the hgh school educated. Table 2 and Fgure 2 here 2 2 For estmatng the varances σ η and σ ε of shocks η t to permanent ncome and ε t, to transtory ncome, we follow the well-establshed procedure of Carroll and Samwck (1997). The dea s that the resdual of the observed log wage n the PSID and the predcted log wage from our regresson results can be attrbuted to permanent ncome and transtory shocks ln P t + ln ε t. Let r,d = (ln P t+d + ln ε t+d ) (ln P t + ln ε t ) be the dfference of these resduals of waves beng d years apart for ndvdual. Under the assumpton of serally 28 For waves 95-07, hours worked n the ndvdual s Man Job were reported; n wave 09, only hours worked n All Jobs were reported so Man Job could not be nferred. Yet there s no sgnfcant dfference n the sample mean and standard devaton of hours worked, or the wage regressons, f the 09 wave s omtted. 29 Dummes for each wave are also ncluded as explanatory varables (results avalable on request). 30 There s an excepton for college educaton, where women earn more between ages

19 uncorrelated and ndependent shocks, ths dfference has a varance of σ η 2 + 2σ ε 2. Regressng the squared dfferences r 2,d on the tme span d between waves and a constant vector of 2 s yelds an estmate for these varances. The results of our calbraton appear at the bottom of Table 2. Snce we assume dentcal shocks for both spouses, we splt the sample only by educaton but not by sex. Compared to Love (2010) who based hs emprcal analyss on a broader defnton of household ncome (ncludng publc transfers and unemployment compensaton, as well as labor ncome), our estmate of the varance of permanent shocks σ η 2 s lower for the less educated and about the same for the college educated. Our varance of transtory shocks σ ε 2 s consderably lower for both educatonal groups. The broader defnton of household ncome used by Love (2010) versus ours (wage rates) mght be more prone to fluctuatons. The effect of unemployment n partcular s not ncorporated n our model or regressons, snce we cannot nfer wage rates when ndvduals are unemployed. For retrement ncome, whch s purely a publc transfer n our model, these conceptual dfferences no longer apply. Therefore, for the varance of transtory shocks to retrement ncome we set σ ε 2 ret = from Love (2010). 3.4 Other parameters Emulatng several other studes n the lfe cycle lterature, we use the consumpton scalng factor φ s = (A K) 0.7 proposed by Ctro and Mchael (1995), wth A beng the number of adults and K beng the number of chldren n the household. Our calbraton of bequest strength b s,t s motvated more by the provsonal motve (.e., spousal protecton) than a pure bequest motve to cover chldren (Hubener et al., 2013). We set b s,t to zero for any famly states wthout chldren present n the household. Otherwse, we assume that an annuty must be purchased that fnances the consumpton for each left-behnd chld untl hs 18 th brthday, plus four more years for college. 31 As the age of chldren s not explctly tracked n our model, we agan use the same smulaton technque as before for the famly transton probabltes and chld care tme estmaton to derve mean values of b s,t for famly state s at each age t. We choose a relatve rsk averson of γ = 5 and set the tme dscount factor to β = The lesure preference parameter s gven by α = 0.8, snce for ths value, the 31 Abstractng from dscountng wth the rskless rate, a 15- and a 17-year old chld yeld bequest factors of b = 5 (0.7 2) (0.7 1) 0.7 = 7.89, snce consumpton must be fnanced for fve years for both chldren and another two years for the youngest chld. 15

20 lfe. 33 When a couple dvorces, the partner wth lower retrement beneft clams s enttled to optmal lfe cycle profles for hours worked per week roughly match the average work hours n the PSID data used for the calbraton of the wage rate (also see Appendx B). The rsk-free rate s set to R 0 = 1.02, and we assume an equty premum for stock returns of E[R t ] R 0 = 4%. The standard devaton of stocks returns s 20%. Lfe nsurance contracts are prced accordng to the 2001 Commssoners Standard Ordnary (CSO) Mortalty Table, whch was developed by the Socety of Actuares and the Amercan Academy of Actuares (2002). As n Gomes et al. (2008), labor earnngs are taxed at a rate of θ labor = 30% and retrement benefts at θ ret = 15%. Several other parameters are calbrated followng Love (2010): for nstance, we use hs estmaton of housng costs h s,t from PSID data; for chld support, dvorced men are assumed to pay 17%/25%/30% of ther labor ncome for 1/2/3+ dependent chldren; dvorced women wth chldren receve the correspondng fracton of a sngle man s ncome as f he works for 40 hours per week; f a chld turns age 18, the household pays 40% of ts ncome for college costs 32 upon hs departure; n the case of dvorce, wealth s splt evenly between spouses after deductng 10% of assets for dvorce costs. When a sngle ndvdual marres, we must make some assumptons about the new partner. Frst, we post that the new partner has the same permanent wage rate component P t as the sngle ndvdual had before. Second, the PIA of the new husband s an age-dependent multple of the wfe s PIA rangng between 1.04 n ther early 20 s and 1.13 just before retrement. Thrd, the fnancal wealth brought by the husband nto the couple s wealth s also an age-dependent multple of the wfe s wealth rangng between 1.06 early and 1.19 late n spousal benefts, and after the former partner s death, to wdow(er) benefts. Our model does not track the PIA of former partners, so we ncrease the PIA of a dvorced woman (man) to 70.85% (58.23%) of the former partner f her (hs) own PIA s smaller. Ths s motvated by the followng consderaton: an annuty payng $50 per year to a woman as soon as her former husband reaches full retrement age, and $100 after hs death, as long as the woman lves, has the same actuarally far present value as an annuty payng the woman $70.85 per year 32 Based on a study by Turley and Desmond (2006), Love assumes college costs of 10% of the famly s ncome for four years. Snce the famly states n our model do not contan any nformaton on the number or even the ages of chldren already havng left the household, we have to model ths payment as a lump sum upon the chld s leavng. 33 We derve these multples by assumng that both partners have worked full tme up to ths age. The rato of the PIAs resultng from ths work hstory yelds the frst multple. Smlarly, the second multple s calculated from the rato of correspondng average lfetme ncome. 16

21 (because of the age dfference and the asymmetry n the mortalty rates, the correspondng value for men s only $58.23). For the pecewse lnear functon convertng the AIME nto the PIA, we use the offcal specfcaton for the Socal Securty bend ponts. For the frst $744 of the AIME, 90 cents per dollar are transferred nto the PIA, for values over ths and up to $4,483, 32 cents per dollar are transferred and for every addtonal dollar earned, on average, the PIA ncreases by 15 cents (n 2009 dollars). We set the exempt amount of annual ncome for the Retrement Earnngs test to $14, The deducton (bonus) for clamng early (late) old age retrement beneft s calculated accordng to Socal Securty clamng rules. As of the Full Retrement Age, defned here as age 66, retrement benefts as a fracton of the PIA are gven by Table 3. Table 3 here 4. Optmal Decsons on Savng, Work, Clamng, Lfe Insurance, and Investments In ths secton, we frst analyze the household s optmal behavor over the lfe cycle. In partcular, we are nterested n how famly status affects fnancal decsons (stocks, bonds, lfe nsurance demand), work effort, and the optmal tme to exercse the Socal Securty clamng opton. Next, we dscuss detals on the smulaton method of our lfe cycle model wth changng famly status. In Secton 4.2, we present patterns of average consumpton, wealth, holdngs n stocks, work hours, and Socal Securty clamng ages. We dscuss these patterns for women and men n a sngle or couple household. Further analyss on how educaton and the number of chldren nfluence optmal decsons s presented n Secton 4.3. Fnally, we nvestgate whether the predctons on clamng patterns from our model are consstent wth emprcal HRS data. 4.1 Smulatons We use the optmal controls of the baselne parameterzaton of our lfe cycle model to generate 100,000 smulated lfe cycles reflectng realzatons of stock returns, wage rates, and martal status. We assume that 59.3% (40.7%) of the smulated households have a wage rate profle correspondng to hgh school (college) educaton, whch represents the dstrbuton of educatonal status n the 2009 wave of the PSID. We dvde the sample of smulated lfe cycles equally nto female and male paths. At the start of the smulatons, 80% are sngles and 34 For addtonal nformaton on Socal Securty beneft rules, see Myers (1985) and 17

22 20% are already marred, whle later n lfe each ndvdual randomly moves between the 35 famly states. Each household s endowed wth a startng fnancal wealth as f each household member would have worked 40 hours per week n the prevous perod. We present the results n the usual way as n the lfe cycle lterature, so we generate smulated paths condtonal on survval. To do so, we modfy the transton matrx Π j,t for the smulaton by settng the mortalty of women n female paths and men n male paths to zero 35 and rescale the other probabltes such that they sum up to 1. Ths procedure keeps the same number of paths even at hgh ages. If a sngle agent marres, we make the same assumptons about the new spouse as n the optmzaton regardng age dfference, permanent ncome, wealth, and PIA. In the case of dvorce, we follow only the ex-wfe (ex-husband) n a female (male) path and gnore the other spouse. For the reportng of aggregate quanttes over all paths, as for example, average wealth, each path s weghted wth the survval probablty up to the consdered age. Ths gves female paths a slghtly hgher weght n comparson to male paths, especally at advanced ages. When sex dependent quanttes lke hours worked by women (men) are consdered, we only report the average over female (male) paths. We also report results for subsamples, e.g. sngle or couple households. In ths case, we use averages over all paths n that famly state at the reported age. Thus the samples are not constant at all dfferent ages. For example, an ndvdual beng a sngle woman at one age drops out of the sample of sngle households for later ages, when she marres. Ths agent may reenter the subsample at a hgher age, f a dvorce occurs. Table 4 provdes some basc nformaton about the average composton of the smulated populaton dynamc at dfferent ages. Table 4 here 4.2 Optmal lfe cycle profles Fgure 3 reports the average lfe cycle profles for the complete populaton of sngles and couples wth ether a hgh school or a college educaton. Panel A shows average consumpton, lfe nsurance demand, wealth level, and nvestments n equtes. Panel B reports average work hours for men and women, and Panel C the frequency of clamng Socal Securty benefts. Here we see that fnancal wealth bulds up gradually untl age 55 when t amounts to about $160,000 on average, and thereafter people start to draw down these 35 However, the optmal decsons of the agents take mortalty nto account. The mortalty of the spouses n couple states s not zero and states of wdowhood are thus possble n the smulaton. 18