On the Dynamics of the Age Structure, Dependency, and Consumption

Transcription

1 On the Dnamics f the Age Structure, Dependenc, and Cnsumptin Heinrich Hck Mathematica Plic Research, 600 Marland Avenue SW, Suite 550, Washingtn, DC , USA hhck@mathematica-mpr.cm David N. Weil Department f Ecnmics, Brwn Universit, Bx B, Prvidence, RI 091, USA and NBER, Cambridge, MA, USA Abstract We examine the effects f ppulatin aging due t declining fertilit and rising elderl life expectanc n cnsumptin pssibilities in the presence f intergeneratinal transfers. Our analsis is based n a highl tractable cntinuus-time verlapping generatins mdel in which the ppulatin is divided int three grups (uth dependents, wrkers, and elderl dependents) and life-curse transitins take place in a prbabilistic fashin. We shw that the cnsumptinmaximizing respnse t greater lngevit in highl develped cuntries is an increase in fertilit. Hwever, with larger transfer paments, the actual fertilit respnse will likel be the ppsite, leading t further ppulatin aging. JEL Classificatin Numbers: E1, H55, J13 Kewrds: Aging, cnsumptin, intergeneratinal transfers Direct crrespndence t Heinrich Hck, wh acknwledges financial supprt fr earl wrk n this paper frm an NICHD training grant (T3-HD07338) t the Ppulatin Studies and Training Center at Brwn Universit. This paper has benefited frm cmments b participants at the Brwn Universit Department f Ecnmics Macr Lunch and the 006 Cnference n Ppulatin Aging, Intergeneratinal Transfers, and the Macrecnm. We are particularl grateful t Alexia Prskawetz and tw annmus referees fr valuable feedback. All errrs are ur wn.

2 1. Intrductin This paper cnsiders hw changes in fertilit affect cnsumptin thrugh changes in the structure f dependenc and hw, via its effect n the dependenc burden faced b wrking-age adults, the age structure in turn affects fertilit. Highlighting the rle f intergeneratinal transfers, we develp a highl tractable cntinuus-time verlapping generatins mdel that allws us t graphicall analze f the relatinships between ecnmic dependenc and the age structure. The mdel als clarifies a ptentiall imprtant feedback lp between the tw. Our interest in the mutual dependenc f fertilit and a sciet s age structure is primaril mtivated b cnsidering the future prspects f sme f the wrld s mst develped cuntries. Cuntries such as Ital and Japan are nw cming t the end f a decades-lng perid during which lw fertilit prduced a transitril lw level f ppulatin dependenc. Over the next 0 ears, rapid ppulatin aging will drasticall reduce cnsumptin pssibilities (and even mre drasticall affect gvernment budgets). The effect f this cnsumptin crunch n fertilit and thus n the age structure f the ppulatin even further dwn the rad is an issue that has received little attentin frm ecnmists. The effect f ppulatin aging n cnsumptin is a widel studied tpic. Fr example, Cutler et al. (1990) and Elmendrf and Sheiner (000) discuss the effect f ppulatin aging in the United States n feasible and ptimal paths f cnsumptin. Given that a large share f resurce transfers t the elderl is channeled thrugh the state, ppulatin aging will have a particularl dramatic effect n gvernment finances (see Lee and Edwards 001). On the ther hand, Blm, Canning, and Sevilla (001) examine hw a demgraphic dividend resulting frm reductins in fertilit that is, a perid f several decades in which the rati f wrking- 1

3 age adults t dependent children and elderl is unusuall high affects incme per capita in develping cuntries. In the abve literature, the imprtant chain f causalit is frm the demgraphic t the ecnmic. The underling demgraphic inputs, mst ntabl changes ver time in fertilit, are either taken as exgenus r related t phenmena such as declining child mrtalit that are utside the ecnmic mdel being examined. A separate strand f literature cnsiders the impact f ld-age dependenc n fertilit thrugh the pensin sstem (Zhang 1995; Cign and Rsati 1996; Wigger 1999). In this paper, we clse the lp b cncentrating n the interdependence f fertilit and the ppulatin age structure arising frm ecnmic dependenc in the presence f intergeneratinal transfers. We clarif the shrt- and lng-term cnsumptin effects f ppulatin aging that result frm reduced fertilit, which are ptentiall quite different and are cntingent n the initial age structure. Drawing n related numerical wrk (Hck and Weil 007), we argue that current fertilit rates in the develped wrld are belw thse that wuld minimize lng-run ecnmic dependenc. Our mdel makes clear that, in such a situatin, reductins in cnsumptin assciated with greater elderl life expectanc might be ffset b increases in the fertilit rate. Ging bend previus wrk (ur wn and that f thers), we describe a feedback lp whereb rising ld-age dependenc reduces the dispsable incme f the wrking ppulatin, resulting in lwer fertilit and further ppulatin aging. 1 Cnsequentl, the privatel ptimal respnse t 1 The mechanism that we examine here shares the feature f Easterlin s (1987) mdel that reductins in living standards trigger cmpensating reductins in fertilit. Hwever, we fcus n the fiscal effect f transfers t the elderl n after-tax earnings rather than n the effect f chrt size n the pre-tax wage f ung wrkers in principle the tw mechanisms culd easil c-exist. Cerda s (005) mdel f scial securit crises is similar in spirit t urs, but ur mdel permits a brader understanding f the interrelatedness f cnsumptin pssibilities and age structure.

4 ppulatin aging is at dds with the cnsumptin-maximizing (r tax-minimizing) respnse. In a further extensin, we cnsider a plitical interactin between members f different age grups cncerning the genersit f gvernment transfers t the elderl. We shw that a plitical bargaining prcess can amplif the feedback frm ppulatin aging t fertilit, resulting in even lwer fertilit and et mre aging. In additin t identifing the feedback frm ppulatin age structure t fertilit via the channel f dependenc, a secnd cntributin f this paper is t cnstruct an analticall tractable dnamic mdel f ppulatin age structure. Specificall, we build a cntinuus-time verlapping generatins mdel in which ppulatin is divided int three grups (ung, wrking age, and ld) and transitins between grups take place in a prbabilistic fashin as in Blanchard (1985). Within this mdel, fertilit ma be taken as exgenus r made an endgenus functin f the ppulatin age structure. Our mdel is simple enugh t be analzed graphicall and et captures the dnamic adjustment f age structure, fertilit, and cnsumptin t a variet f external shcks. Our fcus is n is ppulatin aging due t increased life expectanc, but the basic framewrk can be extended t applicatins well bend thse pursued here. The rest f the paper is structured as fllws. In Sectin, we describe ur mdel f the ppulatin age structure and develp the basic dnamic equatins used in the subsequent analsis. Sectin 3 presents results frm the mdel under the assumptin that the fertilit rate is exgenus. In Sectin 4, we allw fertilit t be an endgenus functin f incme and then Gertler (1999) extends Blanchard s (1985) mdel, allwing fr tw age grups (wrking age and retired). Grafenhfer et al. (007) present a mdel f prbabilistic aging invlving eight age grups, which the use t quantitativel assess the effects f ppulatin aging n savings, cnsumptin, and the tax rate. Bth mdels, hwever, take fertilit as exgenus. Bmmier and Lee (003) develp a mdel f an ecnm with a cntinuus age distributin. The derive sme interesting aggregate stead-state results, but make limited prgress in analzing aggregate dnamics. 3

5 analze the dnamics f the cmplete sstem, cmparing the actual and cnsumptinmaximizing respnses f the ecnm t an exgenus shck t ld-age mrtalit. We als extend the mdel t allw fr a plitical interactin between the elderl and wrking-age adults that determines the magnitude f gvernment transfers t the elderl. Finall, Sectin 5 cncludes.. A Dnamic Mdel f the Age Structure Ecnmists have lng recgnized the need t incrprate age hetergeneit int macrecnmic analsis. Samuelsn (1958) and Diamnd (1965) develped ecnmic-demgraphic mdels in which agents prgressed thrugh a discrete set f ages. Overlapping generatins (OLG) mdels f this tpe have been used extensivel in the ecnmic grwth literature. OLG grwth mdels tpicall assume a tw- r three-perid lifecurse, which allws fr a relativel clean analsis f fertilit and ld-age dependenc. Hwever, perids within such mdels represent lng spans f time in the real wrld (e.g., 0 t 30 ears). Mre imprtant, mst mdels d nt allw fr variabilit in the time spent in each phase f life. We develp here a smewhat stlized cntinuus-time mdel that brrws bth frm the traditinal OLG framewrk and frm Blanchard s (1985) mdel f perpetual uth. Given ur interest in ecnmic dependenc, we mdel the lifeccle as a prgressin thrugh a series f stages f ecnmic life rather than cnceptualizing it as a series f ages. Mst individuals fllw a pattern whereb the are first dependent n their parents, then wrk fr sme amunt f time, and then retire. Accrdingl, we divide the ppulatin int three grups: AY () t is the stck f ung peple wh have never wrked at time t; A () t is the stck f peple in their wrking ears; and AO () t is the stck f peple wh nce wrked but are retired b time t. 4 M

6 T fcus n the dnamics f the age structure, we assume that utput is prduced slel b labr, which is supplied inelasticall b peple in their wrking ears. The ttal pl f resurces available fr cnsumptin is: Ω () t = W () t A () t, (1) M where W(t) is the prevailing wage at time t. A sstem f transfers frm the wrking supprts the ung and the elderl. As a result, we fcus n the uth and ld-age dependenc ratis: AY () t AO () t t ( ) = and t ( ) =. A () t A () t M M In the remainder f this sectin, we establish varius prperties f the equatins gverning the evlutin f the dependenc ratis, which prvide the basis fr ur mdel s dnamical sstem..1 Lifeccle Transitins Individuals prgress mntnicall thrugh the age structure (A Y, A M, and A O ) and face a cnstant exit prbabilit frm each grup. Fr the ung and wrking, λ Y and λ M give the hazard f transitin t wrk and t retirement, respectivel. Amng the elderl, λ O is the prbabilit f ding. All f the flws are determined b the transitin parameters, { λ j }, except the flw f births int A Y, which is given b N(t). The fllwing sstem f equatins summarizes this mdel f the evlutin f the age structure: A () t = N() t λ A () t, () Y Y Y A () t = λ A ( t) λ A () t, (3) M Y Y M M A () t = λ A ( t) λ A () t. (4) O M M O O 5

7 The parameters λ j give the inverse f the average time spent in each age grup, T j. 3. Dependenc Dnamics The equatin f mtin fr the ld-age dependenc rati ma be derived frm (3) and (4) as: ( ) t () = λ λ λ t () λ tt () (). (5) M O M Y Thus, the dnamics f ld-age dependenc are determined entirel b the age structure and the transitin parameters. Mvement f the uth dependenc rati additinall depends n the flw f births, which is a functin f the stck f fecund persns and the rate f fertilit amng the fecund. Frm () and (3), the equatin f mtin fr the uth dependenc rati ma be written as: Nt t () = () ( ( λ ) () [ ()] Y λm t λy t. (6) A () t M Fr simplicit, we assume that the fecund ppulatin is a fixed prprtin, φ, f the wrkfrce. Since all members f the wrkfrce are hmgeneus in ur mdel, an alternative interpretatin f φ is t decmpse it as φ = TF / TM, where T F < T M is the expected duratin f fecundit fr 3 In ur mdel, we ignre childhd and earl-adult mrtalit, assuming that death ccurs nl amng the retired. The mdel culd easil be adapted t incrprate these ther frms f mrtalit b changing the basic equatins t: A Y() t = N() t ( λy + μy) AY(), t A M () t = λyay() t ( λm + μm ) AM (), t A () t = λ A () t μ A (). t O M M O O With this set-up, λ j is the transitin prbabilit t the next grup, and μ j is the death rate in the grup. Analsis f this mdel wuld parallel that in the main text, with qualitativel similar results. 6

8 each member f M. 4 Let n(t) represent the average fertilit rate amng the fecund. Under the abve assumptin, equatin (6) reduces t: ( ) [ ] t () = φnt () λ λ t () λ t (). (7) Y M Y We will refer t φn(t) as the flw f births per wrker. The equatin f mtin fr the uth dependenc rati (7) des nt depend explicitl n (t). Thus, if fertilit is determined in a manner unrelated t the ld-age dependenc rati, equilibrium analsis f the relative ecnmic age structure is straightfrward. This is the case in Sectin 3, where we take the fertilit rate as given. In Sectin 4, we allw fertilit t be endgenusl determined. Given that fertilit will be a functin f after-tax incme, transfers t the elderl thrugh the pensin sstem cause the ld-age dependenc rati t pla a rle in the dnamics f the uth dependenc rati. 3. Ecnmic Dependenc In this sectin, we fcus n hw changes in the age structure attributable t fertilit and mrtalit affect dependenc and cnsumptin in the presence f intergeneratinal transfers. 5 The traditinal demgraphic measure f dependenc is the supprt rati, which is simpl the rati f wrkers t ttal ppulatin. Hwever, cnsumptin tends t var substantiall ver the lifeccle (Cutler et al. 1990; Lee, Lee, and Masn 008), impling that prprtinate changes in uth and elderl dependenc will have different effects n resurce availabilit. Cnsequentl, we 4 An additinal set f equatins, cnsistent with thse gverning the ecnmic age structure, culd be specified t allw φ t var ver time. Hwever, such a specificatin results in a mdel that is substantiall mre cmplicated t analze while adding ver little qualitativel t ur analsis. 5 A cmpanin paper (Hck and Weil 007) cnsiders the additinal effects f changes in the retirement age. 7

9 summarize ecnmic dependenc using a cnsumptin-weighted measure. Nrmalizing the cnsumptin f wrkers t ne, let ρ Y and ρ O dente the relative cnsumptin f the uth and elderl. The extent f ecnmic dependenc is given b: et () = ρ t () + ρ t (). (8) Y O Given ur assumptins, the aggregate resurce cnstraint in equatin (1) implies that the cnsumptin f the wrking is c () t = η()w() t t, where M 1 η () t = (9) 1 + et ( ) is an aggregate cnsumptin index affecting all age grups prprtinatel. Increases in bth uth and elderl dependenc reduce per-capita resurces in the ecnm, but their effects n η are scaled b the relative cnsumptin requirements f the ung and ld. Belw, we cnsider the impact f changes in the fertilit rate n ecnmic dependenc and cnsumptin. We then cnsider the cnsequences f ppulatin aging due t increased life expectanc amng the elderl. First, hwever, we characterize the basic equilibrium f the mdel. 3.1 Equilibrium with Exgenus Fertilit With a fixed fertilit rate, the mdel prduces a dnamical sstem that cnverges t a stable equilibrium age structure determined b the underling demgraphic parameters. It is cnvenient 8

10 t express the flw f births as φ n= G T, where G is the grss reprductive rate (GRR). 6 / M Hlding G cnstant, slving (7) ields ne psitive rt fr the equilibrium uth dependenc rati. We express the psitive rt in terms f the expected time spent in each age grup (T j ) rather than as the exit parameters (λ j ): 1 T 1 T T = T T T Y Y Y M M M G, (10) which crrespnds t a vertical line in (,)-space. [Figure 1 abut here] Based n equatin (5), the equilibrium ld-age dependenc rati assciated with is: ( ) = Z = 1. (11) T T T T ( / 1 ) + ( / ) M O M Y This functin is dwnward-slping, cnvex with respect t the rigin, and has a vertical intercept equal t T /( T T ). The dnamics f the sstem impl that (, ) is a glball stable O M O equilibrium as depicted in Figure 1. Given the ne-t-ne relatinship in equatin (11), the functin Z () is a lcus f stable ppulatins when the demgraphic parameters are held 6 The GRR is the number f children individuals can expect t have, assuming that the are subject t the currentl prevailing rate f fertilit fr the entiret f their childbearing ears. Given ur assumptins regarding the stck f fecund persns, the GRR wrks ut t G = T n = φt n. F M 9

11 cnstant. 7 We refer t the slpe f this lcus as the marginal rate f demgraphic transfrmatin between the ung and elderl 3. Fertilit and Cnsumptin Much f the 0th centur was characterized b falling birth rates in the develped wrld. Our basic mdel can be used t analze the transitinal dnamics assciated with such declines in fertilit, alng with the changes in stead-state cnsumptin attributable t the assciated shifts in age structure. After a fall in fertilit, equatin (11) implies that the equilibrium uth dependenc rati falls. As shwn in Figure, the decline in maps t a higher equilibrium value f ld-age dependenc alng the lcus f stable ppulatins. [Figure abut here] Alng the transitin path, the declines in uth dependenc are initiall larger than the increases in ld-age dependenc, which reduces ecnmic dependenc. The result is an increase in the cnsumptin index η, reflecting the demgraphic dividend discussed b Blm, Canning, and Sevilla (001). The standard argument is that the dividend is transient and eventuall ffset b ppulatin aging. Hwever, a pint that has received little attentin in the literature is that the net effect n stead-state cnsumptin is a priri ambiguus. This can be seen quite clearl in ur mdel, which als demnstrates the existence f a threshld level f fertilit belw which further reductins ultimatel reduce stead-state cnsumptin. The inset f 7 A stable ppulatin is ne in which the relative size f the age grups is cnstant. When fertilit is at the r r replacement rate (G = 1), the equilibrium dependenc ratis are = T / T and = T / T, paralleling basic results frm stable ppulatin ther. 10 Y M O M

12 Figure shws a pssible set f paths fr,, and η after a decline in the fertilit rate in the case where the new equilibrium has lwer aggregate cnsumptin than the riginal equilibrium. Fllwing Weil (1999), is-dependenc lines assciated with a given level f ecnmic dependenc, e, are btained frm equatin (8) as: = ( e ρ )/ ρ I ( ). (1) Y O e Overlaing is-dependenc lines alng the lcus f stable ppulatins assciates each equilibrium ppulatin age structure with a level f ecnmic dependenc. Is-dependenc lines clser t the rigin in (,)-space are assciated with a lwer rate f ecnmic dependenc and a higher cnsumptin index, η. Because Z () is cnvex with respect t the rigin, maximizing m m cnsumptin ields a unique equilibrium pair (, ) as depicted in Figure 3. At this pint, the marginal rate f demgraphic transfrmatin is equal t the marginal rate f substitutin between cnsumptin f the ld and cnsumptin f the ung. [Figure 3 abut here] Frm equatins (10), (11), and (1), the cnsumptin-maximizing pint n the lcus f stable ppulatins is assciated with an ld-age dependenc rati f: G T T O ρ T,, = T T ρ T Y Y Y M M O M. (13) Because is a mntnicall negative functin f G, a unique fertilit rate, G m, that generates m m the equilibrium (, ), given values f the ther parameters. At initial equilibria crrespnding t higher fertilit rates G h > G m, i.e., thse in Regin A f the Z () lcus in 11

13 Figure 3, small declines in the birth rate are assciated with lwer stead-state ecnmic dependenc. In this case, cnsumptin will be higher at all future dates than in the initial stead state, as evident in Figure 4. [Figure 4 abut here] Fr cuntries alread in Regin B f the lcus f stable ppulatins, further fertilit reductins will result in a transient demgraphic dividend that cmpletel dissipates due t ppulatin aging higher elderl dependenc eventuall generates lwer stead-state cnsumptin. The numerical results f Hck and Weil (007) suggest that the fertilit rates bserved in man highl develped cuntries are alread belw the cnsumptin-maximizing rate. 8 Hck and Weil (007) als indicate that increases in fertilit rates in develped natins might be assciated with stead-state cnsumptin gains as large as 10 percent. Hwever, reversing demgraphic mmentum is als cstl in the shrt run. As shwn in Figure 4, in a cuntr with the same initial ecnmic dependenc as implied b G h, an increase in the fertilit rate frm sme lwer value G l < G m will initiall lead t higher ecnmic dependenc because f the larger rati f ung t wrking. The gains in cnsumptin cme nl later in the transitin as the expansin f the wrkfrce reduces the burden f elderl dependenc. 3.3 Lngevit and Cnsumptin Recent ppulatin aging in the develped wrld is largel the prduct f the demgraphic mmentum assciated with falling fertilit ver the last half centur. Hwever, this prcess has 8 The calculatins in Hck and Weil (007) use lifeccle cnsumptin data frm Japan and the United States as a benchmark and are derived frm a simulated numerical analgue f the mdel presented here. Based n a range f mrtalit and retirement prfiles that span what is bserved in OECD member natins, cnsumptin-maximizing fertilit rates are generall well abve replacement while actual fertilit rates are substantiall lwer. 1

14 been amplified b increases in life expectanc, which have becme mre and mre cncentrated amng the elderl. 9 Hlding the ther parameters cnstant, ur mdel generates relativel straightfrward dnamics when elderl life expectanc rises. Specificall, the lcus f stable ppulatins the Z () curve in Figure 1 shifts upward. The ld-age dependenc rati increases mntnicall tward the new equilibrium, driving dwn the aggregate cnsumptin index alng the wa. With n change in the fertilit rate, the uth dependenc rati remains unchanged. Belw, we discuss hw a multiplier effect might arise if the fertilit decisins f the wrking are affected b the decline in per-capita resurce availabilit assciated with rising elderl dependenc. Here, we fcus n the implicatins fr the cnsumptin-maximizing rate f fertilit. Our mdel predicts that, ceteris paribus, higher life expectanc after retirement is assciated with a higher cnsumptin-maximizing fertilit rate, G m. Greater lngevit increases the ttal cst f supprting the elderl, which raises the marginal rate f demgraphic transfrmatin. While higher fertilit increases equilibrium uth dependenc, it translates t marginall larger reductins in ld-age dependenc n the lcus f stable ppulatins. Thus, whereas an increase in fertilit frm the initial cnsumptin-maximizing rate wuld lwer cnsumptin (thrugh a higher value f ), this effect is mre than ffset b the eventual decrease in elderl dependenc. Fr cuntries with a fertilit rate initiall higher than G m, such as G h in Figure 5, the cnsumptin-maximizing rate draws clser t the existing fertilit rate. As Figure 5 9 In the United States, fr example, life expectanc at age 65 has risen b apprximatel 3.5 ears since 1950 (Bard f Trustees, Federal Old-Age and Survivrs Insurance and Disabilit Insurance Trust Funds 005) and is prjected t rise b an additinal fur ears b 050 (Lee and Carter 199). Other highl develped cuntries have seen reductins in ld-age mrtalit at least as large and can be expected t fllw a similar trend in the future (Munnell, Hatch, and Lee 004). 13

15 demnstrates, nt nl des the best attainable level f cnsumptin fall, s des the ptential equilibrium gain frm a reduced fertilit rate. 10 Fr cuntries with a fertilit rate lwer than G m (e.g., G l in Figure 5), the cnsumptin-maximizing fertilit rate mves further awa as ld-age mrtalit declines. Althugh the best attainable level f cnsumptin decreases, the ptential equilibrium gains frm a higher fertilit rate are greater. Hence, the cntinued decline in childbearing rates exhibited in man highl develped cuntries is at dds with the fertilit strateg that wuld maximize future equilibrium cnsumptin. As we shw belw, the burden f ld-age dependenc itself ptentiall amplifies the discrepanc between actual and cnsumptinmaximizing fertilit. [Figure 5 abut here] 4. The Feedback frm Aging t Fertilit Thus far, we have taken the dramatic declines in fertilit in the develped wrld as given. A variet f mechanisms have been prpsed t explain this phenmenn, including a shrinking gender gap in wages (Galr and Weil 1996), labr market frictins (Adsera 005), rigid gender nrms (de Laat and Sevilla-Sanz frthcming), and changing fertilit preferences (Gldstein, Lutz, and Testa 003). 11 While shrinking famil size nrms ma be a prduct f histrical declines in fertilit (Lutz, Skirbekk, and Testa 007), we cnsider an alternative mechanism whereb the age structure ma interact with fertilit. In pa-as-u-g pensin sstems, rising elderl dependenc reduces the per-capita resurces available t the wrking. Based n standard 10 This can be established analticall since the marginal rate f demgraphic substitutin rises fr an value f as T O increases. 11 Khler, Billari, and Ortega (00) prvide a mre cmprehensive review. 14

16 ecnmic mdels f childbearing, this ma translate int additinal reductins in fertilit, further ppulatin aging, and even lwer dispsable incme. We develp a basic extensin t ur mdel that allws fr a dnamic interactin amng ld-age dependenc, fertilit, and the age structure. We retain the prductin structure described abve, where wrkers prvide labr inelasticall in return fr a wage f W(t). Wages are subject t a prprtinal tax, τ, that funds public transfers t the ld and the ung. We assume that the elderl are supprted entirel thrugh public transfers in the frm f a pensin sstem while the ung receive supprt in the frm f bth public and private expenditures. 1 We als assume that the gvernment budget is balanced perid b perid. In realit, f curse, gvernments are able shift t expenditure burdens intertemprall, which, in develped cuntries, usuall invlves passing n debt t future generatins. Cnstructing a ther f hw gvernments chse the path f brrwing ver time wuld take us far bend the scpe f the current paper. But, in the cntext f ur mdel, slwing (r negative) ppulatin grwth attributable t lw fertilit can significantl wrsen the psitin f debtr gvernments b shifting the burden f pament t smaller future chrts. 13 Thus, the negative effects f aging n fertilit, which we examine in the cntext f balanced budgets, are likel t be exacerbated b pre-existing gvernment debt. 1 Althugh smewhat stlized, this distinctin mirrrs the substantial difference in the pattern f transfers t the ld and ung bserved in highl develped cuntries. Fr example, private transfers accunt fr ver 61 percent f the transfer-based cnsumptin f persns under age 0 in the United States, but nl 11 percent f ttal transfers t persns ver age 65 (Masn et al. 009). One cnsequence f this assumptin is that we abstract frm a selfinterested ld-age supprt mtive fr childbearing f the tpe highlighted b Cign (199). 13 Tda, Japan is the best example f a cuntr where falling ppulatin size interacts in a txic fashin with a high level f per-capita gvernment debt. 15

17 The pensin sstem replaces after-tax wages at a rate, β, while transfers t the ung are a prprtin, α, f the net pa f the wrking. Given the aggregate resurce cnstraint (1), the balanced-budget tax rate is: α t () + βt () τ () t =. (14) 1 + αt ( ) + βt ( ) Higher rates f uth and elderl dependenc reduce after-tax incme fr wrkers and result in lwer per-capita public transfers. 4.1 Privatel Optimal Fertilit At each mment in time, fecund wrkers allcate their after-tax incme t their wn cnsumptin and t bearing children. The price f gds and services is numeraire, and we assume that childrearing requires a fixed quantit f market-based childcare services. Cnsequentl, the price f childbearing is p () t = ξw() t. 14 Assuming a lg utilit functin, fecund wrkers face the fllwing ptimizatin prblem: ct (), nt () n [ ct] θ [ nt] max ln ( ) + ln ( ) (15) subject t ct () + p() tnt () = wt (), (16) n 14 Given the stchastic nature f aging in ur mdel, parental paments are made upfrnt int a trust fund that pas fr χ units f childcare services per perid fr the life f the child. T ensure that children alwas receive the requisite care, an actuariall neutral trust fund will set ξ = χ T Y / (1 g T Y ), where g is the expected rate f wage grwth. Fr the path f trust fund paments t remain bunded, g must be less than 1/ T Y. Taking 5 ears as an upper limit n the average length f uth dependenc, g must be less than 4 percent ver the lng run, which is in line with bserved grwth rates. 16

18 where c(t) is the cnsumptin, n(t) is the annual flw f births per fecund wrker, and [ τ ] wt () = 1 () t W() t is the take-hme wage f the wrking. Slving equatins (15) and (16) fr the fertilit rate, we have: n * () t = ψ ψ[ 1 τ() t ] 1 + αt ( ) + βt ( ) =, (17) with ψ = θ /[ ξ(1 + θ )]. The flw f births per fecund wrker satisfies sme usual prperties: fertilit is a psitive functin f the relative preference fr children (θ) and a negative functin f childrearing csts (ξ). The fertilit rate is als determined b the tax rate, which is, in turn, a functin f the existing age structure f the ppulatin. 4. Sciall Optimal Fertilit The indirect utilit functin f all agents in the ecnm is mntnicall declining in the tax rate (hlding α and β fixed) because cnsumptin f the ung and elderl is prprtinal t wrkers after-tax incme. As a result, stead-state scial welfare is maximized when the tax rate is minimized. Similar t the analsis in Sectin 3, the fertilit rate that ields the tax rate minimizing age structure, m n τ, is the implicit slutin t: m TY T O α T nτ,, = T T β T M M M Y. (18) Hwever, the privatel chsen fertilit rate will generall differ frm the tax-minimizing rate, depending n preferences and private cst f children. Given that transfers t the elderl represent such a large fractin f public expenditures, the calculatins in Hck and Weil (007) 17

19 suggest that current fertilit rates in mst develped natins are well belw thse that wuld minimize taxes in equilibrium. The privatel chsen fertilit rate als des nt take int accunt the effects f adult fertilit chices n the tax rate via the age structure, resulting in a cllective actin prblem f srts. We can cmpare privatel ptimal fertilit decisins t the utcme chsen b a scial planner wh seeks t maximize the stead-state utilit f fecund wrkers, incrprating the effects f fertilit decisins n the dependenc ratis int the ptimizatin decisin. Substituting the budget cnstraint (16) int the bjective functin (15) and taking int accunt the relatinship between the tax rate and the age structure in (14), the scial planner s prblem is: 1 max ln ξn +θ ln n 1+ α( n) + β( n) [ n]. (19) Rearranging the first-rder cnditin t (18), we can write: ( ˆ) n* n n where ˆn is the scial planner s fertilit rate and n* ( ˆ) chse, cnditinal n the stead-state ( ( n ˆ), ( ˆ) ( ) + β ( ) ( n) ( n) 1 α nˆ nˆ nˆ = ˆ+, (0) ξ( 1+ θ) 1+ α ˆ + β ˆ n ). n is the fertilit rate that wrkers wuld Using the curvature f the lcus f stable ppulatins in (11), the term in braces n the right-hand side f (0) has the same sign as: ( ˆ) ( ˆ) ( ˆ) ( ˆ) TM α n + β n = n α β n TY. 18

20 Given that () is psitive, the privatel chsen fertilit rate (cnditinal n the scial planner s preferred age structure) is lwer than ˆn when: ( ˆ) T M α β n < 0 (1) T Y and is therwise higher. Cnsider first the extreme case with n public expenditures n the ung. When α = 0, the left-hand side f (1) is unambiguusl negative. Wrkers chse t lw a fertilit rate relative t the scial planner s rate because the fail t accunt fr the future increased burden f aging induced b their fertilit decisins. In the ppsite case, with n transfers t the elderl ( β = 0 ), wrkers chse t high a fertilit rate because the d nt accunt fr the externalit impsed n ther wrkers via publicl funded child care. With public transfers t bth the elderl and the ung, the privatel chsen rate f fertilit cincides with the scial planner s rate when (1) hlds with equalit. Based n equatin (18), this cnvergence ccurs if and nl if individuals chse the tax-minimizing fertilit rate. Mre generall, when privatel chsen fertilit differs frm the tax-minimizing rate ( m n τ ), the scial planner, in essence, balances the indirect utilit gain frm mving tward the taxminimizing rate against the direct lss f utilit frm mving awa frm the private equilibrium. In particular, it can be shwn that ˆn lies in the interval between is the stead-state rate f fertilit chsen b private actrs. 15 m n τ and n *, where n * * ( * = n n ) Given the calculatins described abve, the mre likel scenari is that * m n < nˆ < n τ. In this case, the privatel chsen fertilit rate wuld nt nl be lwer than the tax-minimizing rate, it 15 A prf is available n request. 19

21 wuld als be lwer than the fertilit rate assciated with the highest pssible level f equilibrium utilit amng fecund wrkers. This suggests a lng-term gain frm pr-natal plicies fr all agents in the ecnm. Child subsidies accmpanied b additinal lump-sum taxes cncentrated n the fecund prvide a thereticall attractive tl t induce private decisin makers t internalize the effects f their fertilit decisins n the aggregate tax rate. 16 While an interventin seeking t raise the fertilit rate wuld lead t a tempraril higher tax burden (see the discussin surrunding Figure 4), we shw belw that the dnamic respnse f agents t ppulatin aging increases the lng-run gap between actual and tax-minimizing fertilit. Thus, if the are t be implemented, pr-famil plicies shuld be intrduced sner rather than later. 4.3 Dnamics with Endgenus Fertilit The analsis f Sectin 3 can easil be augmented t analze the effects f ppulatin aging attributable t increases in elderl life expectanc. The basic mechanism is that changes in elderl dependenc,, affect the tax rate and therefre dispsable incme. Fecund wrkers respnd b reducing their childbearing, which results in an indirect link between the uth and elderl dependenc ratis. Putting the privatel ptimal flw f births per fecund wrker, n *, int the equatin f mtin f the uth dependenc rati (7), the lcus f (stable) equilibrium values f in (,)-space is given b: 1 = φψt + Z ( 1 α ) ( ) Y = 0. β ( 1 TY / TM ) + 16 In the case where all expenditures n children are privatel funded, as in van Grezen, Leers, and Meijdam (003), subsidies wuld be pegged t the child s expected lifetime cntributin t the pensin sstem. In ur mdel, child subsidies wuld be determined b the expected reductin in the tax burden assciated with childbearing. 0

22 The lcus is dwnward-slping and cnvex with respect t the rigin, asmpttes t infinit as appraches zer, and eventuall crsses the hrizntal axis. A single crssing in the psitive rthant between the Z () and Z () lci implies a glball stable equilibrium in the ppulatin dependenc ratis, as shwn in Figure 6. [Figure 6 abut here] The transitin assciated with an increase in lngevit is depicted in Figure 7. Lwer rates f ld-age mrtalit translate directl int increases in elderl dependenc due t lnger life expectanc after retirement. If the fertilit rate were t remain unchanged, ld-age dependenc wuld rise frm 1 t, and the uth dependenc rati wuld remain cnstant at 1. Hwever, increases in ld-age dependenc result in a higher tax rate, given the gvernment s balanced budget cnstraint. Because wrkers will have less dispsable incme, the rate f childbirth amng the fecund will fall. [Figure 7 abut here] We shwed in Sectin 3 that declines in fertilit lead, ceteris paribus, t increases in ldage dependenc. As a result, there is a multiplier effect assciated with declines in ld-age mrtalit: increased ld-age dependenc reduces fertilit, which further increases ld-age dependenc. Hwever, as the fertilit rate falls, s des the uth dependenc rati. The cnsequent reductins in public expenditures n the ung acts as a brake n the feedback ccle amng rising ld-age dependenc, rising taxes, and falling fertilit. 17 Nnetheless, the net result 17 Cerda (005) describes a similar mechanism. Hwever, in the absence f public transfers t the ung in his mdel, lw fertilit ma prduce ever-rising parll tax rates, eventuall resulting in a scial securit melt-dwn. 1

23 is a transitin t an equilibrium with an ld-age dependenc rati f that is even higher than. The rati f ung t wrking falls frm 1 t as a cnsequence f the fertilit respnse f wrkers. As life expectanc amng the elderl rises, the tax-minimizing fertilit rate will increase per the mechanism discussed in Sectin 3.3. In highl develped cuntries where n * is initiall less than m n τ, the multiplier effect n lngevit means a higher stead-state tax rate than wuld have resulted had there been n fertilit adjustment (τ in Figure 7). Thus, the privatel ptimal fertilit respnse t a decrease in ld-age mrtalit is at dds with the welfare-maximizing respnse. Wrkers reduce fertilit, resulting in an lder age structure, a higher tax burden, and lwer utilit amng future wrkers and dependents. 4.4 Dnamics with Endgenus Fertilit and Endgenus Pensins Thrughut the paper, we have assumed that a set f fixed supprt parameters gvern public transfers frm the wrking t the ther age grups. In the mdel with endgenus fertilit, fixed public supprt ratis (α and β) ma be cnsidered the result f a special case f a plitical bargaining prcess between the elderl and wrking. In this sectin, we cnsider hw the dnamics f the mdel change if we extend the basic mdel t allw fr a mre general class f endgenus bargaining utcmes regarding the pensin rate, β. 18 Incrprating richer bargaining dnamics suggests that an endgenus pensin rate will exacerbate the cnsumptin-reducing effects f an aging ppulatin ver the lng run. 18 Fr analtic simplicit, we cntinue t treat public supprt fr the ung, α, as fixed and exgenus, althugh the mdel culd accmmdate bargaining ver bth parameters.

24 We assume that the elderl and wrking cmpete in a plitical sstem that allcates resurces based n a Tullck-Hirshleifer cntest success functin (CSF). Taking α as given, the ttal pl f resurces t be split amng the tw grups is wam + β wao, where w cntinues t represent the after-tax wage and β is the replacement rate fr ld-age transfers. Fllwing Hirshleifer (1989), a general frm f the CSF t determine the share f resurces allcated t the elderl is based n the relative ppulatin sizes as fllws: μ ( π AO ) μ ( ) ( π ) β wao = wa + β wa A + A M O M O μ, () where π dentes the relative effectiveness f the elderl at bargaining and μ > 0 represents the decisiveness parameter. Rearranging equatin (), we have: β μ μ 1 = π ο, (3) frm which it is apparent that the cnstant value f β assumed in ur earlier analsis is equivalent t μ = 1. Fr 0< μ < 1, the replacement rate wuld depend negativel n the relative size f the ld-age ppulatin. Althugh there have been sharp dwnward adjustments in the replacement rate in respnse t a burgening elderl ppulatin, the relatinship des nt seem credible fr a cntinuus bargaining prcess ver the lng run. As μ tends t infinit, the prcess becmes a majrit-takes-all sstem f resurce allcatin, which als des nt seem realistic. Hence, we fcus n the case where 1 μ < ; if the first inequalit is strict, then there are increasing returns t minrit vting blcs. Extending the main dnamic mdel t allw fr an arbitrar decisiveness parameter changes the zer-mtin lcus fr s that: 3

25 1 φψt Y = ( 1 α ) Z ( ) 0 = +. (4) π ( 1 TY / TM ) + This lcus satisfies man f the same basic prperties in the general case as in the special case f μ = 1: it is mntnicall decreasing, asmpttes t infinit as appraches zer, and crsses the hrizntal axis fr a sufficientl high value f. 19 The decisiveness parameter affects the slpe f Z ( ) 1 μ, which has imprtant implicatins fr the dnamics f fertilit and the age structure as lngevit increases. As is evident frm Figure 7, a flatter Z ( ) lcus wuld amplif the ppulatin-aging multiplier effect arising frm higher elderl life expectanc. Based n equatin (4), it is straightfrward t shw that: 1 sgn Z ( ) = sgn e Z ( ) μ. (5) A larger decisiveness parameter will result in a flatter slpe f Z ( ) fr lw values f, but a steeper slpe fr sufficientl high values f. Dente the critical value f at which the relatinship changes as c. Cnsider the differential impact f increased lngevit n tw cuntries that are initiall identical in their age structure and fundamentals, differing nl accrding t the plitical decisiveness parameter. If bth cuntries still have a relativel ung age structure, s that > c, plitical bargaining ma dampen the multiplier n lngevit in the cuntr whse pensin rate depends psitivel n the size f the elderl vting blc ( μ > 1) relative t the similar cuntr with a fixed pensin rate ( μ = 1). Nnetheless, the ppulatin will cntinue t 19 In general, the Z ( ) lcus either stas belw the hrizntal axis r becmes undefined after the initial crssing; we ignre the trivial set f values f μ fr which the lcus wuld start t rise after meeting the hrizntal axis. 4

26 age as life expectanc increases, and all cuntries will eventuall reach a pint after which c <. Further increases in lngevit will lead t a flatter Z ( ) lcus in the cuntr with μ > 1, relative t the cuntr with μ = 1, impling a relativel strnger multiplier effect n ldage mrtalit. Thus, in the lng run, a cuntr whse pensin rate depends psitivel n the relative size f the elderl vting blc will experience accelerated ppulatin aging, as cmpared t an therwise similar cuntr with a fixed pensin rate. 0 Fr a sufficientl high decisiveness parameter, lng-run transitins ma nt be cmpletel smth because multiple equilibria ma appear in aging ppulatins. 1 Figures 8 and 9 shw hw this prcess might unfld in respnse t increases in life expectanc that lead t upward shifts in the Z () lcus frm an initiall lw level. In Figure 8, an ecnm starting ut in the 1 1 neighbrhd f an initial equilibrium (, ) will mve tward (, ). Anther equilibrium becmes supprtable as the ppulatin ages because a high value f μ implies larger ptential swings in the allcatin f resurces when relative grup sizes change a secnd, plitical multiplier effect. A large and exgenus increase in the relative number f elderl culd result in a melt-dwn transitin t (, ) invlving drastic increases in pensins and taxes and b b similarl drastic reductins in fertilit. Barring such a shck, (, ) represents a natural cntinuatin f the demgraphic and plitical frces that generated the initial equilibrium. a a a a [Figure 8 abut here] 0 If the decisiveness parameter is less than ne, then the multiplier effect n lngevit will be amplified in the shrt run but dampened in the lng run. 1 It was nt pssible t analticall characterize the parameter cnfiguratins resulting in multiple equilibria, but numeric simulatin ielded the cnfiguratins in Figures 8 and 9 when using reasnable parameter values and a high value f μ. 5

27 As elderl life expectanc increases past a certain pint, hwever, the unavidable cnsequence f natural demgraphic and plitical frces is a melt-dwn transitin. As shwn in Figure 9, increased lngevit will eventuall shift the Z () lcus upward s that the bttm-right equilibrium disappears. This ccurs because the elderl vting blc becmes s large that it triggers the plitical multiplier. The surge in pensins paid t the elderl results in a rapid increase in taxes and a sharp slwdwn in fertilit, which, in turn, leads t further ppulatin 3 3 aging. The sstem stabilizes at the upper-left equilibrium (, ), which is characterized b far fewer uth and a substantiall larger elderl ppulatin. [Figure 9 abut here] We argued abve that fertilit rates in the develped wrld are alread belw the sciall ptimal rate. Develped cuntries face an even mre difficult rad ahead if a melt-dwn transitin is still t ccur. The wrking ma seek t avid a melt-dwn b reducing the wage replacement rate f the pensin sstem, which ma, fr example, be achieved in the cntext f ur mdel b wrking t increase their plitical effectiveness relative t the elderl (decreasing π). Hwever, attempts t refrm the pensin sstem are better undertaken befre the plitical multiplier triggers a transitin t an ecnm with a substantiall larger and mre frmidable elderl vting blc. 5. Cnclusin In this paper, we have analzed the dnamic evlutin f a cuntr s ppulatin age structure and fertilit rate. Our particular fcus was n the feedback frm the ppulatin age structure t fertilit via the channel f ld-age dependenc. In a cuntr with a high rate f ld-age 6

28 dependenc, wrking-age individuals will see a large fractin f their labr incme redistributed t the elderl. In respnse, wrking-age individuals will reduce their fertilit. The dnamic nature f ur analsis is particularl imprtant because the shrt- and lngrun effects f changes in fertilit n dependenc differ markedl. In the shrt run, a reductin in fertilit unambiguusl lwers a sciet s dependenc burden b reducing the number f children relative t wrking-age adults. In the lng run, reductins in fertilit raise a cuntr s ld-age dependenc rati, ptentiall unding the reductin in uth dependenc. Our calculatins in Hck and Weil (007) indicate that develped natins with the lwest levels f fertilit have alread reached the pint where further reductins in fertilit raise rather than lwer the lng-run dependenc burden. We based ur analsis n a cntinuus-time verlapping generatins mdel that allws us t examine the dnamic evlutin f the age structure and cnsumptin dependenc in respnse t changes in fertilit (in the case where fertilit is exgenus) r the jint evlutin f fertilit, age structure, and dependenc in respnse t exgenus shcks such as ld-age mrtalit. We have shwn that, fr cuntries alread belw the level f fertilit that maximizes cnsumptin in the stead state, the actual and ptimal respnses f fertilit t changes in ld-age mrtalit have ppsite signs. In respnse t greater elderl lngevit, such cuntries will effectivel dig themselves int a deeper demgraphic hle b reducing fertilit in rder t maintain shrt-run cnsumptin. We then extended the mdel t allw fr a plitical interactin between members f the elderl and wrking-age generatins in rder t determine the magnitude f gvernment transfers t the elderl. As the fractin f the ppulatin that is elderl increases, we expect the genersit f such transfers t expand, amplifing the feedback frm ppulatin aging t lwer fertilit. 7

29 Our mdel is smewhat stlized in the interests f analtic tractabilit and ease f expsitin. It culd be extended alng tw majr dimensins t imprve its realism and frecast accurac. First, it culd be useful t expand the number f age grups bend the three basic nes cnsidered here. Mving bend three age grups, hwever, wuld prevent us frm examining the dnamics f the mdel analticall, frcing us t take a cmputatinal apprach alng the lines f Auerbach and Ktlikff (1987), Grafenhfer et al. (007), and Hck and Weil (007). Secnd, the mdel culd be extended t incrprate intertempral ptimizatin n the part f husehlds, which wuld entail shifting cnsumptin between perids in respnse t prjected changes in demgraphics. Such intertempral shifting, whether thrugh investment in phsical capital r purchase f freign assets, culd allw agents t save fr their wn ld age r the cuntr as a whle t save fr perids f high ld-age dependenc. Althugh intertempral reallcatin wuld have little effect n the mdel s stead state, it wuld significantl influence the mdel s dnamics. As discussed in Cutler et al. (1990) and Elmendrf and Sheiner (000), the abilit f the ecnm as a whle t insulate itself against demgraphic shcks b amassing extra capital is limited b bth depreciatin and capital s declining marginal prduct. Attempts t smth cnsumptin in the face f demgraphic shcks ma als spur a run-up f the price f capital assets when everne is tring t save and a crrespnding asset meltdwn when everne is tring t dissave. We suspect that the extensins discussed abve wuld nt alter the fundamental predictins f ur mdel, which are relativel grim. Demgraphers have been cncerned fr decades abut persistentl belw-replacement fertilit in the develped wrld. Mrever, gvernment interventins seeking t blster the fertilit rate have met with mixed success (Grant et al. 004). See Abel (003) and Lim and Weil (003). 8

30 Over the next several decades, as the effect f lw fertilit n cnsumptin pssibilities shifts frm psitive t negative, it will becme increasingl difficult t muster supprt fr pr-famil plicies in view f the assciated increases in shrt-term uth dependenc. In the absence f such effrts, ur results suggest that the fiscal burden f elderl dependenc will result in further declines in fertilit, which will prduce additinal decreases in cnsumptin ver the lng run. 9

31 References Abel A (003) The Effects f a Bab Bm n Stck Prices and Capital Accumulatin in the Presence f Scial Securit. Ecnmetrica 71(): Adsera A (005) Vanishing Children: Frm High Unemplment t Lw Fertilit in Develped Cuntries. American Ecnmic Review 95(): Auerbach AJ, Ktlikff LJ (1987) Dnamic Fiscal Plic. Cambridge Universit Press, Cambridge, UK Blanchard OA (1985) Debt, Deficits, and Infinite Hrizns. The Jurnal f Plitical Ecnm 93():3-47 Blm DE, Canning D, Sevilla J (001) Ecnmic Grwth and the Demgraphic Transitin. Natinal Bureau f Ecnmic Research Wrking Paper 8685, Cambridge, MA Bard f Trustees, Federal Old-Age and Survivrs Insurance and Disabilit Trust Funds (005) Annual Reprt f the Bard f Trustees f the Federal Old-Age and Survivrs Insurance and Disabilit Insurance Trust Funds. Gvernment Printing Office, Washingtn, DC Bmmier A, Lee RD (003) Overlapping Generatins Mdels with Realistic Demgraph. Jurnal f Ppulatin Ecnmics 16(1): Cerda R (005) On Scial Securit Financial Crises. Jurnal f Ppulatin Ecnmics 18(3): Cign A (199) Children and Pensins. Jurnal f Ppulatin Ecnmics 5(3): Cign A, Rsati FC (1996) Jintl Determined Saving and Fertilit Behaviur: Ther and Estimates fr German, Ital, UK and USA. Eurpean Ecnmic Review 40(8):

32 Cutler DM, Pterba JM, Sheiner LM, Summers LH, Akerlf GA (1990) An Aging Sciet: Opprtunit r Challenge? Brkings Papers n Ecnmic Activit 1(1):1-73 de Laat J, Sevilla-Sanz A (frthcming) Wrking Wmen, Men s Hme Time and Lwest Lw Fertilit. Feminist Ecnmics 17(3) Diamnd PA (1965) Natinal Debt in a Neclassical Grwth Mdel. American Ecnmic Review 55(5): Easterlin RA (1987) Birth and Frtune: The Impact f Numbers n Persnal Welfare. Secnd Editin, Universit f Chicag Press, Chicag Elmendrf DW, Sheiner LM (000) Shuld America Save fr its Old Age? Fiscal Plic, Ppulatin Aging, and Natinal Saving. Jurnal f Ecnmic Perspectives 14(3):57-74 Galr O, Weil DN (1996) The Gender Gap, Fertilit, and Grwth. American Ecnmic Review 86(3): Gertler M (1999) Gvernment Debt and Scial Securit in a Life-Ccle Ecnm. Carnegie- Rchester Cnference Series n Public Plic 50(1): Gldstein JR, Lutz W, Testa MR (003) The Emergence f Sub-replacement Famil Size Ideals in Eurpe. Ppulatin Research and Plic Review (5-6): Grafenhfer D, Jaag C, Keuschnigg C, Keuschnigg M (007) Ecnmic Ageing and Demgraphic Change. In: Lutz W (ed) Vienna Yearbk f Ppulatin Research 007. Austrian Academ f Sciences Press, Vienna Grant J, Hrens S, Sivadasan S, van het L M, DaVanz J, Hale L, Gibsn S, Butz W (004) Lw Fertilit and Ppulatin Ageing Causes, Cnsequences, and Plic Optins. RAND Eurpe, Leiden Cambridge Berlin 31