Working Paper No Net Intergenerational Transfers from an Increase in Social Security Benefits

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1 Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion and NBER November 2006 The Levy Economics Insiue Working Paper Collecion presens research in progress by Levy Insiue scholars and conference paricipans. The purpose of he series is o disseminae ideas o and elici commens from academics and professionals. The Levy Economics Insiue of Bard College, founded in 1986, is a nonprofi, nonparisan, independenly funded research organizaion devoed o public service. Through scholarship and economic research i generaes viable, effecive public policy responses o imporan economic problems ha profoundly affec he qualiy of life in he Unied Saes and abroad. The Levy Economics Insiue P.O. Box 5000 Annandale-on-Hudson, NY hp:// Copyrigh The Levy Economics Insiue 2006 All righs reserved.

2 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan, Guan Gong, and Michael Hurd 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. We find ha, a leas for singles, increases in Social Securiy benefis are unlikely o be offse by bequess. JEL Classificaions: E21, H55 Keywords: Bequess, Social Securiy 1

3 INTRODUCTION 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, even when here is no beques moive and all bequess are accidenal. 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 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, which depends on he probabiliy of survival o each fuure period, he reurn o saving, budge 1 A similar model for couples is much more complex and will be an objecive of fuure work. 2

4 consrains, 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 CONSUMPTION 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 N ρ ρ ( ) ( ) 0 0. [1] Ω= uc e ad+ V w e md 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 N = 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 = bequeahable wealh a ; m = probabiliy densiy of dying a. 3

5 The consrains on he maximizaion are: iniial bequeahable wealh, w 0, is given; he nonnegaiviy consrain, w 0 is given by: ; and he rae a which bequeahable wealh changes dw rw c A d = +, [2] in which r = real ineres rae (consan and known), and 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 and abou 40 percen of hose aged 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) and 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. 4

6 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 years 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. ESTIMATION OF THE 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: and ha he beques funcion is: uc () = c γ γ 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, h, in [3]. 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). A subjecive survival probabiliy as measured in AHEAD is a responden s esimae of he 2 See Soldo, Hurd, Rodgers, and Wallace (1997) for a descripion of AHEAD. 5

7 probabiliy of surviving o a arge age. For example, a 71 year-old would be asked abou his or her subjecive probabiliy of surviving o age 85, while a 77 year-old would be asked abou survival o age 90. Following Gan, Hurd, and McFadden (2005), we esimaed individualized survivor curves ha depend on boh he life able and on subjecive survival. Briefly, we used a saisical mehod o combine hem such ha someone who repored subjecive survival chances greaer han he life able survival chances a he arge age would be given an individualized survival curve ha is greaer han he life able curve a all ages. In our esimaion a each age, h depended on he individualized survival curve. 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. EXPECTED BEQUESTS 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 (Social Securiy wealh). A beques happens when someone dies holding wealh. 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 6

8 $10,000. The consumpion pah in his figure differs from he consumpion pah in Figure 1 because i perains o a woman aged 65 and because i is based on he parameers in Table 1 of Gan e al. (2004). Our mehod of finding he effec of changes of Social Securiy on bequess is o firs conduc a simulaion such as ha which produced he wealh and consumpion pahs in Figure 2. Then we resimulaed 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 1989). 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 resuls 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 zero a age 93. Under his scenario, he expeced presen value of bequess is $32,500. Even hough wealh is exhaused sooner under his scenario, bequess are greaer because more wealh is held a ages 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 ineres rae) of Social Securiy benefis (Social Securiy wealh) is $107,600. According o he esimaed opimal consumpion pah, he expeced presen 7

9 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 have similar resuls, bu now he man is assumed o have hree children. A comparison of columns 1 and 3, which holds Social Securiy benefis consan bu changes he beques moive, shows 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 would 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 columns show, he expeced presen value of bequess falls by $2,600 raher han by $2,700, as in columns 1 and 2. Thus, he beques moive causes an addiional $100 in bequess ou of an increase in he presen value of Social Securiy benefis of $107,700. Table 3 has similar resuls, bu hey are for a 65 year-old woman. The difference in inpus ha cause 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. 8

10 As wih he 65 year-old man, he woman s expeced bequess decline when Social Securiy benefis are increased from $10,000 o $20,000 per year. As shown in he righhand 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, column Hurd (1989). The ime rae of discoun,, 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. Deleed: ρ 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 no 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. Resuls for couples are unlikely o be subsanially differen simply because a he deah of a spouse, abou 75% of he wealh goes o he surviving spouse (Hurd and Smih 2001). A ha poin, he surviving spouse follows he consumpion pah generaed by he 9

11 singles model. As we have seen, he singles model does no produce any imporan beques offse o an increase in Social Securiy benefis. An unanswered quesion, however, is he role of inervivos ransfers. They are fairly large, and perhaps hey would be increased in response o an increase in Social Securiy benefis. To answer ha quesion would require he specificaion and esimaion of a considerably more complex model han he one used here. 10

12 REFERENCES Gan, Li, Guan Gong, Michael Hurd, and Daniel McFadden Subjecive Moraliy Risk and Bequess. NBER Working Paper Gan, Li, Michael Hurd, and Daniel McFadden Individual Subjecive Survival Curves, in David Wise (ed.) Analyses in he Economics of Aging. Chicago: Universiy of Chicago Press. Hurd, Michael D Moraliy Risks and Bequess. Economerica 57(4): Hurd, Michael D. and Kahleen McGarry "Evaluaion of he Subjecive Probabiliies of Survival in he HRS. Journal of Human Resources 30(0): S268 S The Predicive Validiy of Subjecive Probabiliies of Survival. The Economic Journal 112(482): Hurd, Michael D. and James P. Smih Anicipaed and Acual Bequess. in David Wise (ed.) Themes in he Economics of Aging. Chicago: Universiy of Chicago Press. Modigliani, Franco and R. Brumberg Uiliy Analysis and he Consumpion Funcion: An Inerpreaion of Cross-secional Daa. In K. Kurihara (ed.) Pos Keynesian Economics. New Brunswick, NJ: Rugers Universiy Press. Soldo, Beh, Michael D. Hurd, Willard Rodgers, and Rober Wallace Asse and Healh Dynamics Among he Oldes-Old: An Overview of he Survey. Journal of Geronology 52B(May): Yaari, M Uncerain Lifeime, Life Insurance, and he Theory of he Consumer. Review of Economic Sudies 32(2):

13 FIGURE 1 Consumpion pah consumpion Social Securiy 12

14 FIGURE 2 Consumpion and wealh pahs Wealh Consumpion Social Securiy 13

15 FIGURE 3 Response o increase in Social Securiy iniial $30.2k higher $32.5k 0 iniial higher wealh consumpion annuiy wealh consumpion annuiy 14

16 Table 1 Ineres rae and uiliy funcion parameers Gan e al (2004) Hurd (1989) r γ ρ α e e-7 * a e e-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 (in housands) 65 year-old male, parameers from Table 1 (Gan e al) No children Three children Iniial Social Securiy Increased Social Securiy Iniial Social Securiy Increased Social Securiy Iniial bequeahable wealh Iniial Social Securiy benefis Social Securiy wealh Expeced PV consumpion Expeced PV bequess

17 Table 3 Bequeahable wealh, Social Securiy benefis, expeced presen value of consumpion and bequess (in housands) 65 year-old female, parameers from Table 1 (Gan e al) No children Three children Iniial Social Securiy Increased Social Securiy Iniial Social Securiy Increased Social Securiy Iniial bequeahable wealh Iniial Social Securiy benefis Social Securiy wealh Expeced PV consumpion Expeced PV bequess Table 4 Bequeahable wealh, Social Securiy benefis, expeced presen value of consumpion and bequess (in housands) 65 year-old male, parameers from Table 1 (Hurd) No children Three children Iniial Social Securiy Increased Social Securiy Iniial Social Securiy Increased Social Securiy Iniial bequeahable wealh Iniial Social Securiy benefis Social Securiy wealh Expeced PV consumpion Expeced PV bequess Table 5 Bequeahable wealh, Social Securiy benefis, expeced presen value of consumpion and bequess (in housands) 65 year-old female, parameers from Table 1 (Hurd) No children Three children Iniial Social Securiy Increased Social Securiy Iniial Social Securiy Increased Social Securiy Iniial bequeahable wealh Iniial Social Securiy benefis Social Securiy wealh Expeced PV consumpion Expeced PV bequess

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