Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis Li Gan Guan Gong Michael Hurd April, 2006 ABSTRACT When he age of deah is uncerain, individuals will leave bequess even if hey have no desired bequess simply because hey will hold wealh agains he possibiliy of living longer. Bequess are accidenal. Saring from a baseline level of Social Securiy benefis, an increase in benefis will cause consumpion o increase. However, consumpion may no increase by as much as he increase in Social Securiy, which would cause wealh o be greaer han under he baseline scenario. The higher wealh levels would ranslae ino greaer bequess. Therefore, an increase in Social Securiy benefis may no be a complee ransfer from he younger generaion o he older generaion: some of he increase in benefis may be bequeahed back o he younger generaion. Wheher his happens depends on he form of he uiliy funcion, he amoun of bequeahable wealh, and wheher here is a beques moive. The objecive of his paper is o quanify for single persons how much of an increase in Social Securiy benefis would be bequeahed back o he younger generaion.
1. Inroducion When he age of deah is uncerain, individuals will leave bequess even if hey have no desired bequess, simply because hey will hold wealh agains he possibiliy of living longer. Bequess are accidenal. Saring from a baseline level of Social Securiy benefis, an increase in benefis will cause consumpion o increase. However, consumpion may no increase by as much as he increase in Social Securiy, which would cause wealh o be greaer han under he baseline scenario. The higher wealh levels would ranslae ino greaer bequess. Therefore, an increase in Social Securiy benefis may no be a complee ransfer from he younger generaion o he older generaion: some of he increase in benefis may be bequeahed back o he younger generaion. Wheher his happens depends on he form of he uiliy funcion, he amoun of bequeahable wealh, and wheher here is a beques moive. The objecive of his paper is o quanify how much of an increase in Social Securiy benefis would be bequeahed back o he younger generaion. We will use an esimaed life-cycle model for consumpion by singles. 1 2. Life-cycle model A broad characerizaion of he siuaion a reiremen is he following. People reach reiremen wih an array of economic resources: a claim on Social Securiy; a claim on Medicare; pension righs; and bequeahable wealh. An appropriae heoreical framework o analyze his siuaion is he life cycle model of consumpion ha goes back o Modigliani and Brumberg (1954), wih exensions o accoun for a beques moive (Hurd 1989). In life cycle models of consumpion under uncerainy, individuals make choices in he curren period based on curren informaion and beliefs so as o maximize he expeced discouned presen value of uiliy. The expeced discouned presen value of uiliy is he sum of uiliy in he curren period based on curren choices and he curren sae of he world, and he expeced discouned presen value of fuure uiliy, 1 A similar model for couples is much more complex and will be an objecive of fuure work.
which depends on he probabiliy of survival o each fuure period, he reurn o saving, budge consrains, and opimal consumpion choices a each period in he fuure, and he value of financial bequess a he deah. We base he analysis on a somewha resriced version of he life-cycle model. Life-ime uiliy is based on ime-separable uiliy from consumpion and from bequess (Yaari 1965); he only uncerainy is dae of deah; resources are iniial bequeahable wealh, righs o pensions, and a sream of annuiies such as Social Securiy; bequeahable wealh canno become negaive, and, herefore, borrowing agains fuure annuiies is no allowed. Because i does no have a provision for he choice o work, i is applicable only o respondens afer hey ener reiremen or disabiliy. Model of consumpion by singles These assumpions lead o he following behavioral model for a single person: maximize expeced lifeime uiliy Ω over he consumpion pah { c } N ρ ρ Ω= u( c ) e a d + V ( w ) e m d. [1] 0 0 The firs erm is expeced discouned uiliy from consumpion, where u( ) = he uiliy flow from consumpion; ρ = he subjecive ime rae of discoun; a = he probabiliy of being alive a ; and N = he maximum remaining years of life ( a = 0) The second erm is he expeced discouned uiliy of bequess, where V ( ) = uiliy from bequess which may depend on he personal characerisics of poenial inheriors such as he economic saus of any children in an alruisic or sraegic beques model; w = m = bequeahable wealh a ; probabiliy densiy of dying a. N N
The consrains on he maximizaion are: iniial bequeahable wealh w 0 is given; he nonnegaiviy consrain, is given by: w 0 ; and he rae a which bequeahable wealh changes in which r = dw rw c A real ineres rae (consan and known), and d = +, [2] A = flow of annuiies a ime. The nonnegaiviy consrain on bequeahable wealh can be jusified by a legal ban on borrowing agains Social Securiy benefis. In addiion, in he daa very few are observed wih negaive wealh, and hose few end o have negaive wealh as he resul of negaive business wealh. This is likely o be he resul of unanicipaed losses raher han borrowing for consumpion purposes. The imporance of aking accoun of he corner soluion ( w = 0 ) is seen from he fracion of single elderly wih approximaely zero nonhousing wealh. In 1993, abou 19 percen of hose aged 70-79 and abou 40 percen of hose aged 90-100 had wealh of less han $1,000. The model places considerable emphasis on annuiy income, which is based on he empirical observaion of is imporance: in 1994, 94 percen of he elderly (65 or over) had some annuiy or pension income (including Social Securiy); 79 percen had more han half of heir income from annuiies or pensions. The soluion o he single s problem is (Hurd 1989): du = u( h + ρ r) hv for w > 0; d c = A for w = 0, [3] where w 0 is given and ( ) u = du c dc = marginal uiliy of consumpion a ime ;
h = m a = moraliy risk (moraliy hazard) a ime ; and ( ) V = dv w dw = marginal uiliy of bequess a ime. The model does no admi an analyical soluion because of he boundary condiion and because of he beques moive. The opimal consumpion pah mus be found numerically: condiional on he specificaion of he uiliy funcion, he equaion of moion of consumpion is given implicily by [3], and he level is found from he lifeime budge consrain. A ypical soluion as found in prior esimaion based on he Reiremen Hisory Survey (Hurd, 1989) is shown in Figure 1. This is he consumpion pah for a man aged 75 wih iniial bequeahable wealh of $100,000 and Social Securiy income of $10,000. By age 92 all bequeahable wealh has been consumed and he consumpion pah will follow he pah of Social Securiy. Once he opimal consumpion pah has been found, prediced wealh { ŵ } is calculaed from he equaion of moion of wealh [2]. Therefore, for each individual he model can be used o forecas consumpion and wealh. Income can be forecas from observed annuiy income and from capial income as rw. Esimaion of he Model In previous work we have esimaed he model of consumpion by singles (Gan, Gong, Hurd and McFadden, 2004). We specified ha he uiliy funcion is he consan relaive risk aversion uiliy funcion uc () = 1 1 1 c γ γ and ha he beques funcion is V( w) = ( α + α n) w 0 1
if n, he number of children, is posiive; oherwise V( w ) = 0. We esimaed his model over wo waves of daa from he Asse and Healh Dynamics sudy (AHEAD). 2 An imporan deerminan of he consumpion pah is moraliy risk in [3]. h While prior work simply used life ables o consruc h for each individual, we use individual repors on subjecive survival probabiliies (Hurd and McGarry, 1995, 2002). Following Gan, Hurd and McFadden (2005) we esimaed individualized survivor curves ha depend on boh he life able and on subjecive survival. A each age he individualized survival curve. h depended on Our preferred esimaion resuls produced he parameer values condiional on an assumed real ineres rae of 0.04 as shown in Table 1. These differ from hose in Hurd (1989), which are also shown in he Table. In our simulaions we will use boh ses of parameers. Expeced bequess Our model solves for he opimal consumpion pah condiional on iniial bequeahable wealh, Social Securiy benefis (Social Securiy wealh), age, sex and he number of children. Then, using he equaion of moion of wealh we find he opimal pah of bequeahable wealh. From hese pahs we can calculae he expeced presen value of consumpion and of Social Securiy benefis. The expeced beques a some fuure ime τ condiional on surviving o ime is jus wm τ τ, ha is wealh held a τ imes he probabiliy of dying a τ. The expeced presen value of bequess is jus he discouned sum of he wm τ τ. From hese calculaions we form a lifeime balance shee. On he receip side here is iniial bequeahable wealh plus Social Securiy wealh; on he expendiure side here is he expeced presen value of consumpion and he expeced presen value of bequess. Figure 2 shows he consumpion and wealh pahs for a single woman iniially aged 65. Her iniial bequeahable wealh is $100,000 and Social Securiy income is $10,000. The consumpion pah in his figure differs from he consumpion pah in 2 See Soldo, Hurd, Rodgers and Wallace (1997).
Figure 1 because i perains o a woman aged 65 and because i is based on he parameers in Table 1 of Gong e al. Our mehod of finding he effec of changes of Social Securiy on bequess is o firs conduc a simulaions such as hose ha produced he wealh and consumpion pahs in Figure 2. Then re-simulae he model bu increasing Social Securiy benefis by some given amoun. A comparison of he change in he expeced presen value of bequess, he expeced presen value of consumpion and Social Securiy wealh will show how much of he increase has been used for bequess and for consumpion. Figure 3 shows an example of hese simulaions. The baseline or iniial simulaion is for a woman aged 65 wih hree children. The parameers are from Table 1 (Hurd). The baseline wealh, consumpion and Social Securiy benefis pahs are shown in he hicker lines. Baseline Social Securiy is $10,000 per year; baseline iniial wealh is $100,000. Baseline iniial consumpion is abou $12,900 per year and increases o $17,100 a age 81 and hen declines unil age 94 when wealh is exhaused. Afer his age she would consume $10,000. Wealh declines coninuously unil age 94 when i reaches zero. Under his scenario expeced bequess are $30,200. The simulaion wih Social Securiy benefis of $20,000 are shown in he hin lines. Iniial consumpion is $21,000 under his scenario. I increases unil age 82 afer which i declines unil age 93 when wealh is exhaused. Should his women survive unil 93 she would consume Social Securiy benefis of $20,000 unil he end of her life. Wealh increases unil age 69 afer which i declines coninuously reaching a age 93. Under his scenario he expeced presen value of bequess is $32,500 housand. Even hough wealh is exhaused sooner under his scenario, bequess are greaer because more wealh is held a ages 75-85 when he probabiliy of deah is large. Table 2 shows a summary of hese kinds of simulaions. The able perains o a 65 year-old man. In he wo lef-side panels are resuls under he assumpion ha he has no beques moive (no children). In he lef-mos panel his iniial bequeahable wealh is $100,000 and annual Social Securiy benefis are $10,000. The expeced presen value (discouned a 4% real) of Social Securiy benefis (Social Securiy wealh) is $107,600. According o he esimaed opimal consumpion pah he expeced presen value of consumpion is $193,200 and he expeced presen value of bequess is jus $12,800.
Thus he model predics ha 12.8% of iniial wealh will be bequeahed, and because here is no beques moive hese bequess are accidenal. The nex panel shows similar figures bu when Social Securiy benefis are $20,000 each year. Social Securiy wealh is wice as large, bu he expeced presen value of consumpion increases by $109,000 which is more han he increase in Social Securiy wealh. Because he lifeime budge consrain mus be saisfied, he expeced presen value of bequess mus decline, and, indeed, a direc calculaion shows ha i does. Therefore, for his example, an increase in Social Securiy benefis is enirely consumed by he receiving cohors, and hey even consume a lile more ou of heir own bequeahable wealh. The ne effec is a decrease in bequess. The righ wo columns has similar resuls bu now he man is assumed o have hree children. A comparison of columns 1 and 3 show ha he beques moive is weak: he expeced presen value of bequess increases by jus $200 or 2.6%. We should no expec, herefore, ha increases in Social Securiy benefis will cause a change in he expeced presen value of bequess ha is much differen from he comparisons of columns 1 and 2. And ha is he case: As he las wo column show, he expeced presen value of bequess falls by $2,600 raher han by $2,700 as in columns 1 and 2. Table 3 has similar resuls bu hey are for a 65 year-old woman. The difference in inpus ha causes he difference in resuls is ha women face subsanially lower moraliy risk and have greaer life expecancy. Thus Social Securiy wealh is abou $15,000 higher. This lower risk causes he opimal consumpion level iniially o be lower bu consumpion is achieved over a longer lifespan so ha oal consumpion is higher han for a 65 year-old man. Because consumpion is iniially lower wealh is held for a longer ime. Even hough more wealh is held, i is held earlier in life when moraliy risk is fairly low. Thus, compared wih a man bequess are lower. As wih he 65 year-old man, expeced bequess decline when Social Securiy benefis are increased from $10,000 o $20,000 per year. As shown in he righ-hand columns, bequess decrease even when here is a beques moive. Tables 4 and 5 have resuls similar o hose in Tables 2 and 3 excep ha differen parameers are used for he model, hose shown in Table 1, Hurd. The ime rae of discoun, ρ, is much lower so ha he sar of he consumpion pah is lower, causing
more wealh o be held. A consequence is ha bequess are subsanially higher even wihou a beques moive (lef columns): hus 28%-29% of bequeahable wealh is accidenally bequeahed. Even so, increasing Social Securiy causes bequess o decrease by $2,200 for he 65 year-old man and $400 for he 65 year-old woman. When here is a beques moive, an increase in Social Securiy benefis does cause an increase in bequess. In he case of he 65 year-old woman, he increase is by $2,100 which is an increase of 6.8%. However, Social Securiy wealh increased $132,900 so all bu a rivial fracion of he increase in Social Securiy benefis was used for consumpion. Conclusion Alhough in principle an increase in Social Securiy benefis could resul in subsanial increases in bequess wheher hey are accidenal or no, he empirical finding is ha hey do no or a leas subsanially. In fac, under a model of life-cycle consumpion by singles, which was esimaed over wo differen daa ses, beques acually decrease in he absence of a beques moive. Only in one of he esimaed models ha allowed for a beques moive did bequess increase, and even hen he increase was rivial. We explored many more cases such as variaion in he level of Social Securiy benefis, he number of children and he age of he single person (no shown). In no simulaion did we observe any significan increase in bequess in response o an increase in Social Securiy benefis. We conclude ha, a leas for singles, increases in Social Securiy benefis are unlikely o be offse by bequess.
References Gan, Li, Guan Gong, Michael Hurd and Daniel McFadden, 2004, Subjecive Moraliy Risk and Bequess, NBER Working Paper 10789. Gan, Li, Michael Hurd and Daniel McFadden, 2005, Individual Subjecive Survival Curves, in Analyses in he Economics of Aging, David Wise, edior, Chicago: Universiy of Chicago Press, pp. 377-402. Hurd, Michael D. 1989 Moraliy Risks and Bequess. Economerica 779-813. Hurd, Michael D. and Kahleen McGarry, 1995, "Evaluaion of he Subjecive Probabiliies of Survival in he HRS," Journal of Human Resources, 30, 1995, S268- S292. Hurd, Michael D. and Kahleen McGarry (2002). The Predicive Validiy of Subjecive Probabiliies of Survival. The Economic Journal, Vol 112, No 482 (Ocober 2002). Modigliani, Franco and R. Brumberg. 1954. Uiliy Analysis and he Consumpion Funcion: An Inerpreaion of Cross-secional Daa. In Pos Keynesian Economics, K. Kurihara (ed.). Rugers Universiy Press. Soldo, Beh, Michael D. Hurd, Willard Rodgers, and Rober Wallace, 1997, Asse and Healh Dynamics Among he Oldes-Old: An Overview of he Survey, Journal of Geronology, 1997, vol. 52B, (May) pp 1-20. Yaari, M. 1965. Uncerain Lifeime, Life Insurance, and he Theory of he Consumer. Review of Economic Sudies 32: 137-150.
Figure 1 Consumpion pah 25 20 15 10 5 0 75 80 85 90 95 100 consumpion Social Securiy
Figure 2 Consumpion and wealh pahs 100 80 60 40 20 0 65 70 75 80 85 90 Wealh Consumpion Social Securiy
Figure 3 Response o increase in Social Securiy 125 100 75 50 25 iniial $30.2k higher $32.5k 0 iniial higher 65 75 85 95 wealh consumpion annuiy wealh consumpion annuiy
Table 1 Ineres rae and uiliy funcion parameers Gan e al (2004) Hurd (1989) r 0.04 0.03 γ 0.986 1.12 ρ 0.058-0.011 α 0 3.8067e-7 3.8067e-7 * a 1.0431e-6 1.0431e-6 * 1 *Hurd esimaed a beques parameer o indicae any children, bu no an addiional parameer for he number. For ha reason we will use he beques parameers from Gan Table 2 Bequeahable wealh, Social Securiy benefis, expeced presen value of consumpion and bequess (housands) 65 year-old man, parameers from Table 1 (Gong e al) No children Three children Iniial Social Securiy Social Securiy Iniial Social Securiy Social Securiy Iniial bequeahable wealh 100.0 100.0 100.0 100.0 Iniial Social Securiy benefis 10.0 20.0 10.0 20.0 Social Securiy wealh 107.6 215.3 107.6 215.3 Expeced PV consumpion 193.2 302.2 193.0 302.1 Expeced PV bequess 12.8 10.1 13.0 10.4 Table 3 Bequeahable wealh, Social Securiy benefis, expeced presen value of consumpion and bequess (housands) 65 year-old woman, parameers from Table 1 (Gong e al) No children Three children Iniial Social Securiy Social Securiy Iniial Social Securiy Social Securiy Iniial bequeahable wealh 100.0 100.0 100.0 100.0 Iniial Social Securiy benefis 10.0 20.0 10.0 20.0 Social Securiy wealh 122.6 245.2 122.6 245.2 Expeced PV consumpion 212.1 335.8 211.9 335.2 Expeced PV bequess 9.8 7.5 10.1 7.7
Table 4 Bequeahable wealh, Social Securiy benefis, expeced presen value of consumpion and bequess (housands) 65 year-old man, parameers from Table 1 (Hurd) No children Three children Iniial Social Securiy Social Securiy Iniial Social Securiy Social Securiy Iniial bequeahable wealh 100.0 100.0 100.0 100.0 Iniial Social Securiy benefis 10.0 20.0 10.0 20.0 Social Securiy wealh 115.7 231.5 115.7 231.5 Expeced PV consumpion 189.5 307.7 188.2 304.1 Expeced PV bequess 27.6 25.4 29.0 29.2 Table 5 Bequeahable wealh, Social Securiy benefis, expeced presen value of consumpion and bequess (housands) 65 year-old woman, parameers from Table 1 (Hurd) No children Three children Iniial Social Securiy Social Securiy Iniial Social Securiy Social Securiy Iniial bequeahable wealh 100.0 100.0 100.0 100.0 Iniial Social Securiy benefis 10.0 20.0 10.0 20.0 Social Securiy wealh 132.9 265.8 132.9 265.8 Expeced PV consumpion 206.2 340.3 204.6 336.3 Expeced PV bequess 29.2 28.8 30.9 33.0