NBER WORKING PAPER SERIES THE DYNAMICS OF THE AGE STRUCTURE, DEPENDENCY, AND CONSUMPTION. Heinrich Hock David N. Weil

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1 NBER WORKING PAPER SERIES THE DYNAMICS OF THE AGE STRUCTURE, DEPENDENCY, AND CONSUMPTION Heinrich Hck David N. Weil Wrking Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA March 2006 This paper has benefitted frm the feedback prvided by participants at the Suthern Ecnmic Assciatin, the Brwn University Department f Ecnmics Macr Lunch and the Brwn University Ppulatin Studies and Training Center Cllquium. Hck: Department f Ecnmics, 288 Bellamy Bldg., Flrida State University, Tallahassee, FL Tel: hhck@fsu.edu. Weil: Department f Ecnmics, Bx B, Brwn University, Prvidence, RI Tel: david_weil@brwn.edu The views expressed herein are thse f the authr(s) and d nt necessarily reflect the views f the Natinal Bureau f Ecnmic Research by Heinrich Hck and David N. Weil. All rights reserved. Shrt sectins f text, nt t exceed tw paragraphs, may be quted withut explicit permissin prvided that full credit, including ntice, is given t the surce.

2 The Dynamics f the Age Structure, Dependency, and Cnsumptin Heinrich Hck and David N. Weil NBER Wrking Paper N March 2006 JEL N. E10, E21, H55, J11, J13 ABSTRACT We examine the dynamic interactin f the ppulatin age structure, ecnmic dependency, and fertility, paying particular attentin t the rle f intergeneratinal transfers. In the shrt run, a reductin in fertility prduces a demgraphic dividend that allws fr higher cnsumptin. In the lng run, hwever, higher ld-age dependency can mre than ffset this effect. T analyze these dynamics we develp a highly tractable cntinuus-time verlapping generatins mdel in which ppulatin is divided int three grups (yung, wrking age, and ld) and transitins between grups take place in a prbabilistic fashin. We shw that mst highly develped cuntries have fertility belw the rate that maximizes steady state cnsumptin. Further, the dependency-minimizing respnse t increased lngevity is t raise fertility. In the face f the high taxes required t supprt transfers t a grwing aged ppulatin, we demnstrate that the actual respnse f fertility will likely be exactly the ppsite, leading t increased ppulatin aging. Heinrich Hck Department f Ecnmics Flrida State University 228 Bellamy Building Tallahassee, Flrida hhck@fsu.edu David N. Weil Department f Ecnmics Bx B Brwn University Prvidence, RI and NBER david_weil@brwn.edu

3 1 Intrductin This paper examines the dynamic interactin f the ppulatin age structure, ecnmic dependency and fertility, paying particular attentin t the rle f intergeneratinal financial transfers. Our gal is t anaylze hw changes in fertility affect a cuntry s demgraphic structure and hw, via its effect n the dependency burden faced by wrking age adults, the age structure in turn affects fertility. We develp a highly tractable cntinuus-time verlapping generatins mdel t analyze these dynamics. As will be seen belw, nne f the pieces that cnstitute ur mdel is new. Hwever, the feedback lp n which we fcus is f great ptential imprtance, and has s far nt been studied by ecnmists. The effect f ppulatin age structure n ecnmic utcmes is a widely studied tpic. Fr example, Cutler et al. (1990) and Elmendrf and Sheiner (2000) discuss the effect f ppulatin aging in the United States n feasible and ptimal paths f cnsumptin. Because much f the transfer f resurces t the elderly is channeled thrugh gvernments, ppulatin aging will have a particularly dramatic effect n gvernment finances (see Lee and Edwards, 2001). Blm, Canning and Sevilla (2001) similarly examine hw a demgraphic dividend resulting frm reductins in fertility - that is, a perid f several decades in which the rati f wrking age adults t dependent children and elderly is unusually high - affects verall ecnmic grwth in develping cuntries. In the abve literature, the imprtant chain f causality is frm the demgraphic t the ecnmic. That is, the underlying demgraphic inputs - mst ntably changes ver time in fertility - are either taken as exgenus r related t phenmena (such as declining child mrtality) that are utside the ecnmic mdel being examined. In this paper we clse the circle, and cncentrate ur attentin ntheinterdependence f fertility and ppulatin age structure, thrugh the channel f ecnmic dependency. Specifically, we 1

4 lk at hw changes in fertility affect ppulatin age structure and ecnmic utcmes, and hw these feed back t affect fertility. A mment s cnsideratin suggests that the prblem that we are interested in is inherently dynamic. Changes in fertility have effects n the ppulatin age structure that take generatins t play ut. Mst significantly, while the immediate effect f a decline in fertility is a reductin in a sciety s dependency burden, and s an expansin in the feasible level f cnsumptin, the lng-run effect f such a decline may be t actually raise dependency and lwer cnsumptin. Fertility and age structure are thus linked in a dynamical system. Such a system will have bth steady states (in which fertility is cnstant and the ppulatin age structure is stable) and a lng lasting dynamic respnse t external shcks. Our interest in the mutual dependency f fertility and a sciety s age structure is primarily mtivated by thinking abut the future prspects f sme f the mst develped cuntries in the wrld. Cuntries such as Italy and Japan are nw cming t the end f a decades-lng perid during which lw fertility prduced a transitrily lw level f ppulatin dependency. During the perid , rapid ppulatin aging will drastically impact cnsumptin pssibilities (and even mre drastically impact gvernment budgets). The effect f this cnsumptin crunch n fertility - and thus n the age structure f the ppulatin even further dwn the rad - is an issue that has nt yet been addressed by ecnmists and demgraphers. 1 The link between ppulatin age structure and fertility that we examine is nt the nly such channel f causatin. Mst famusly, Easterlin (1987) hypthesized that a 1 Micevska and Zak (2002) examine fertility during incme cntractins that were even larger than thse prjected t result frm ppulatin aging: The Great Depressin (in the United States and Germany) and the transitin frm cmmunism during the 1990s. In bth cases, incme declines prduced significant reductins in ttal fertility. Micevska and Zak argue that such a phenmenn can be explained by a Malthusian mdel in which the perceived level f subsistence cnsumptin is a functin f an individual s wn past cnsumptin. 2

5 chrt s size was linked t its fertility thrugh its effect n the earnings f yung adults relative t thse f their parents: members f a large chrt wuld find themselves with incme that was lw relative t the standard f living they had grwn up with, and wuld adjust fertility dwnward t partially restre their standard f living. The mechanism that we examine here shares the feature f Easterlin s mdel that reductins in living standards trigger cmpensating reductins in fertility. Hwever, ur fcus is n the fiscal effect f transfers t the elderly n after-tax earnings, rather than n the effect f chrt size n the pre-tax wage f yung wrkers. Fr this reasn, the key aspect f ppulatin agestructurenwhichwefcusistheratifelderlydependentstwrkingageadults, rather than the rati f lder t yunger wrkers n which Easterlin fcuses. In principle the tw mechanisms culd easily cexist. In additin t identifying the feedback frm ppulatin age structure t fertility via the channel f dependency, a secnd cntributin f this paper is t cnstruct an analytically tractable dynamic mdel f ppulatin age structure. Specifically, we build a cntinuus time verlapping generatins mdel in which ppulatin is divided int three grups (yung, wrking age, and ld), and transitins between grups take place in a prbabilistic fashin àlablanchard (1985). 2 Within this mdel, fertility can be taken as exgenus r made an endgenus functin f the ppulatin age structure. The mdel is simple enugh t be analyzed graphically, and yet captures the dynamic adjustment f age structure, fertility, and cnsumptin t external shcks such as changes in ld-age mrtality, retirement age, r preferences regarding children. The mdel has applicatins well beynd thse pursued here, and als represents a new and cnvenient way f cnceptualizing demgraphic-ecnmic interactins. 2 Gertler (1999) extends Blanchard s mdel in a fashin similar t urs, but nly allws fr tw age grups (wrking age and retired). Grafenhfer et al. (2005) present a mdel f prbabilistic aging in which there are eight age grups, which they use t quantitatively assess the effects f ppulatin aging n savings, cnsumptin, and the tax rate. Bth mdels take fertility as exgenus. 3

6 The rest f this paper is structured as fllws. In Sectin 2 we lay ut ur mdel f the ppulatin age structure and develp the basic dynamic equatins that will be used in the subsequent analysis. Sectin 3 analyzes ur mdel under the assumptin that the rate f fertility is exgenus. We als shw hw the mdel can be used t calculate the rate f fertility that maximizes cnsumptin in steady state and discuss the cnsumptinmaximizing respnse f fertility t increases in elderly dependency. In sectin 4 we allw fertility t be an endgenus functin f incme, analyze the dynamics f the cmplete system, and cmpare the actual and cnsumptin-maximizing respnses f the ecnmy t an exgenus shck t ld-age mrtality. Sectin 5 cncludes. 2 A Dynamic Mdel f the Age Structure Ecnmists have lng recgnized the need t incrprate age hetergeneity int macrecnmic analysis. Samuelsn (1958) and Diamnd (1965) develped simple ecnmicdemgraphic mdels in which agents with a finite life span prgressed thrugh a small and discrete prgressin f ages. Overlapping generatins (OLG) mdels f this type have been used extensively in the ecnmic grwth literature. OLG grwth mdels typically assume a tw r three perid lifecurse, which allws fr a relatively clean analysis f fertility and ld-age dependency. Hwever, perids within such mdel represent lng spans f time in the real wrld (e.g years). As a result, the dynamics generated are lumpy, with jumps in rates and stcks ccurring nly between mdel perids. This is satisfactry fr an understanding f lng-term transitins and equilibria. Hwever, such mdels d nt allw changes in fertility and the ppulatin structure t be analyzed ver smaller time increments. Adding mre perids t a discrete time OLG mdel might allw fr smther dynamics, but adding mre time perids implies adding mre age grups. This increase in 4

7 the state space implies difficulties in aggregatin, making it mre difficult t cleanly analyze macrstructural relatinships. Under certain circumstance switching t cntinuus time may simplify matters. A classic example is Blanchard s (1985) mdel f perpetual yuth. 3 We develp here a smewhat stylized cntinuus-time mdel that brrws frm bth the traditinal OLG framewrk and that f Blanchard. Given ur interest in ecnmic dependency, rather than fcus n the lifecycle as a series f ages, we instead mdel it as a prgressin thrugh a series f stages f ecnmic life. Mst individuals fllw a pattern whereby they are first dependent n their parents, wrk fr sme amunt f time, and then retire. Accrdingly, we divide the ppulatin int three grups: A Y is the stck f yung peple wh have never wrked; A M is the stck f peple in the ecnmy wh are in their wrking years; finally, A O is the stck f peple wh nce wrked, but are nw retired. The triple (A Y,A M,A O ) characterizes the age structure f the ppulatin. Each individual i underges a mntne prgressin thrugh the age structure. We apply the Blanchard idea t this dem-ecnmic prgressin by assuming cnstant exit prbabilities frm each grup A j where j {Y,M,O}. Fr the yung and wrking, λ Y and λ M give the hazard f transitin t wrk and retirement, respectively. Amng the elderly, λ O is the prbability f dying. All f the flws are determined by the structural parameters, {λ j },excepttheflw f births int A Y,whichisgivenbyN (t). The fllwing system f equatins summarizes this mdel f the evlutin f the age structure: Ȧ Y (t) = N (t) λ Y A Y (t) (1) A M (t) = λ Y A Y (t) λ M A M (t) (2) Ȧ O (t) = λ M A M (t) λ O A O (t). (3) 3 In a mre recent paper, Bmmier and Lee (2000) develp a mdel f an ecnmy with a cntinuus age distributin, and are able t derive a number f interesting aggregate steady state results. Hwever, they make nly limited prgress in analyzing aggregate dynamics, and even these are made under the assumptin that the fertility rate is fixed at its steady-state value. 5

8 Table 1: Base Data fr the OECD12 Cuntry R T M T O Australia Belgium Canada France Germany Italy Japan Netherlands Spain Sweden United Kingdm United States Average Data Surces: OECD (2004, Table SS8) and United Natins (2005, Tables 7 and 22). We will use this system, under varius assumptins regarding hw fertility is determined, t characterize the evlutin f the age structure. 4 The parameters λ j give the inverse f the average time spent in each age grup, T j. In referring t the number f members in A j, we will use the term size, and when discussing the average amunt f time spent in grup j,i.e. T j we will call this the width f A j. The widths f the age grups can be used t fit the mdel t demgraphic data. 4 In ur mdel we ignre childhd and early adult mrtality, assuming that death nly ccurs amng the retired. The mdel culd easily be adapted t incrprate these ther frms f mrtality by changing the basic equatins t: A Y (t) = n (t) A M (t) (λ Y + µ Y ) A Y (t) ( 1) Ȧ M (t) = λ Y A Y (t) (λ M + µ M ) A M (t) ( 2) A O (t) = λ M A M (t) µ O A O (t). ( 3) With this set-up, λ j is the transitin prbability t the next grup, and the µ j is the death rate in the grup. Analysis f this mdel wuld parallel that in the main text, with qualitatively similar results. 6

9 Since ur paper is geared tward describing issues affecting the highly-develped wrld, we fcus ur attentin t 12 members f the Organizatin fr Ecnmic Cperatin and Develpment (the OECD12 ). 5 Sincedatantheagefentryintthelabrfrce (T Y )aredifficult t cme by, we assume that age 20 is a reasnable number. We set T M = R 20, where R is the average age f retirement and fr OECD12 member natins is available frm OECD (2004, Table SS8.1). 6 The United Natins (2005, Table 22) prvides data n life expectancy in five-year increments. Since the decline in life expectancy is rughly linear between ages 55 and 75, we cmpute life expectancy at retirement (T O ) thrugh linear interplatin f life expectancies at the surrunding five-year increments. 7 Table 1 presents the basic data (R, T M,T O ) fr the OECD12. As can be seen in the table, life expectancy after retirement is substantial. The prprtin f life spent as an elderly dependent ranges frm 0.22 in the United States t almst 0.3 in Italy. 2.1 Prductin and Dependency In rder t fcus ur attentin n the dynamics f the age structure, we assume that utput is prduced slely by labr, which is supplied inelastically by peple in their wrking years. The ttal pl f resurces available fr cnsumptin is Ω (t) =W (t) A M (t), 5 These cuntries are selected s that they are amng the tp twenty in bth GDP per capita and ppulatin amng the OECD natins. The members f ur OECD12 are: Australia, Belgium, Canada, France, Germany, Italy, Japan, the Netherlands, Spain, Sweden, the United Kingdm, and the United States. 6 The measure f R is the average f the fficial and effective retirement ages, the latter f which is based n bserved changes in participatin rates ver a 5-year perid fr successive chrts f wrkers (by 5-year age grups) aged 40 and ver (OECD 2004, ntes accmpanying Table SS8.1). 7 Since ur mdel des nt accunt fr sex differences, we cmpute R and T O as the gender-weighted average f the retirement age and life expectancy values fr each sex. Ppulatin weights are frm United Natins (2005, Table 7). 7

10 where W (t) is the prevailing wage at time t. The yung and the elderly are supprted thrugh a system f transfers frm the wrking. As a result, we fcus n the yuth and ld-age dependency ratis: y (t) = A Y (t) A M (t) and (t) = A O (t) A M (t). (4) In the remainder f this sectin we establish varius prperties f the equatins gverning the evlutin f the ld-age and yuth dependency ratis. These will serve as the basis fr ur subsequent analysis f dynamic equilibrium The ld-age dependency rati The equatin f mtin fr the ld-age dependency rati can be derived frm (2) and (3) as ȯ (t) =λ M (λ O λ M ) (t) λ Y y (t) (t). (5) The dynamics f (t) in (y, )-space are relatively straightfrward, and we present sme basic results here that are applicable thrughut the paper. Based n (5), the zer mtin lcus fr (t) is given by (t) ȯ=0 = λ M (λ O λ M )+λ Y y (t) Z (y (t)). (6) Hence, fr a given value f the yuth dependency rati, there is exactly ne equilibrium value f. Cnsidering the graphical prperties f the Z (y) in (y, )-space, the vertical intercept is given by λ M / (λ O λ M ) > 0. That the intercept is psitive fllws frm the fact that the expected time in retirement is less than the expected time wrking. Thus, since T O <T M,wehavethatλ O >λ M. Because Z 0 (y) < 0 and Z 00 (y) > 0, the zer mtin lcus fr is dwnward slping and cnvex with respect t the rigin. 8

11 Finally, nting that d [ȯ (t)] /d [ (t)] < 0, all equilibrium values f are stable. That is, fr a given value f the yuth dependency rati, the ld-age dependency rati steadily cnverges t the crrespnding pint n the zer mtin lcus Fertility and the yuth dependency rati The dynamics f the yuth dependency rati requires depend n the flw f births, which is a functin f the stck f fertile persns and the rate f fertility amng the fertile. We give here a fairly general dynamic equatin fr the yuth dependency rati, laying ut a basic assumptin regarding the stck f fertile f persns. In later sectins we specify therateffertilityamngthefertileingreaterdetail. Frm (1) and (2) the equatin f mtin fr the yuth dependency rati can be written ẏ (t) = N (t) A M (t) (λ Y λ M ) y (t) λ Y [y (t)] 2. (7) Fr simplicity, we assume that the fertile ppulatin is a fixed prprtin, φ, fthewrkfrce. Since all members f the wrkfrce are hmgenus in ur mdel, an alternative interpretatin f φ is t decmpse it as φ = T F /T M,whereT F <T M is the expected length f time spent fertile amng each member f M. 8 Letting n (t) represent the average fertility rate at time t amng the fertile, under this assumptin (7) reduces t ẏ (t) =φn (t) (λ Y λ M ) y (t) λ Y [y (t)] 2. (8) We will refer t φn (t) as the flw f births per wrker. The equatin f mtin fr the yuth dependency rati (8) des nt depend explicitly 8 An additinal set f equatins, cnsistent with thse gverning the ecnmic age structure, culd be specified t allw φ t vary ver time. Hwever, ding s makes analyzing the mdel substantially mre cmplicated, while adding very little qualitatively t ur analysis. 9

12 n (t). Thus, if fertility is determined in a manner unrelated t the ld-age dependency rati, equilibrium analysis f the relative ecnmic age structure is straightfrward. This is the case in Sectins 3 where we take the fertility rate as given. In Sectin 4 we allw fertility t be endgenusly determined. Since fertility will be a functin f after-tax incme, transfers t the elderly thrugh the pensin system cause the ld-age dependency rati t play a rle in the dynamics f the yuth dependency rati. 3 Fertility and Ecnmic Dependency In this sectin we analyze ur mdel assuming that the fertility rate is fixed and exgenus. (That is, n (t) =n fr all t.) We fcus n the relatinship between fertility, ld-age mrtality and a measure f ttal dependency in the ecnmy. The traditinal demgraphic measure f ttal dependency is simply the sum f yuth dependency, y, and ld-age dependency,. Hwever, as pinted ut by Cutler et al. (1990), the cnsumptin requirements f the yung and the elderly are nt necessarily the same. This implies that an increase in yuth dependency will nt generally have the same implicatins fr resurce availability as an increase in ld-age dependency. Cnsequently, we fcus n a measure f needs-weighted ecnmic dependency, e (t). In particular we assume that the children and elderly have cnsumptin needs equivalent t ρ Y and ρ O times that f the wrking, which implies that ecnmic dependency at a given mment is e (t) =ρ Y y (t)+ρ O (t). (9) Given ur labr-based prductin structure, we assume further, as in Weil (1999), that the cnsumptin f the all three grups is indexed t the wage. That is c Y (t) =ρ Y c M (t), 10

13 c O = ρ O c M (t), andc M (t) =η (t) W (t). Given the aggregate resurce cnstraint, c Y (t) A Y (t)+c M (t) A M (t)+c O (t) A O (t) =Ω (t), the cnsumptin index must satisfy η (t) = 1 1+e (t). (10) The index η affects the cnsumptin f all grups prprtinately. In the parlance f Cutler et al., η is the supprt rati, that is, the rati f the prductin capacity f the ecnmy t the cnsumptin needs f the ppulatin. Increases in yuth and elderly dependency reduce the per-capita resurces in the ecnmy, but their effects n η are prprtinal t the relative cnsumptin needs f the yung and ld. Belw we cnsider the impact f changes in the fertility rate n ecnmic dependency and cnsumptin. We then cnsider the cnsequences ppulatin aging due t increased life-expectancy amng the elderly. Befre prceeding we characterize the basic equilibrium f in ur mdel, given cnstant rate f fertility. 3.1 Equilibrium with Exgenus Fertility With a fixed fertility rate, the equatin f mtin fr the yuth dependency rati is ẏ (t) =φn (λ Y λ M ) y (t) λ Y [y (t)] 2. (11) Since the dynamic equatin des nt depend n (t), (11) can simply be examined fr equilibrium values f the yuth dependency rati. Setting ẏ (t) equal t zer yields ne psitive rt, dented by ȳ. Writing this in terms f the widths f the age grups (T j ), 11

14 Figure 1: Glbal Dynamics with Exgenus Fertility y (y) Z y rather than the exit parameters (λ j ), ȳ = 1 2 µ 1 T Y + T M v u 1 tã 2 µ! 1 T 2 Y + T Y φn. (12) T M The equatin f mtin fr y implies that this rt is stable. The crrespnding equilibrium ld-age dependency rati is given by ō = Z (ȳ) = 1 (T M /T O 1) + (T M /T Y )ȳ. (13) Graphically, Figure 1 prtrays the glbal dynamics f the ppulatin dependency ratis. The equilibrium pint is glbally stable and the ppulatin dependency ratis cnverge mntnically t the pint (ȳ, ō). 12

15 It is cnvenient t cnsider the flw f births per wrker in terms f the grss reprductive rate (GRR), a fertility cncept cmmnly used by demgraphers. Since individuals in ur mdel are the reprductive unit f analysis, the GRR is the number f children an individual can expect t have, assuming that they are subject t the currently-prevailing rate f fertility fr the entirety f their childbearing years. Given ur assumptins regarding the stck f fertile persns, this implies that the grss reprductive rate is simply G = T F n, and that the flwfbirthsperwrkeris φn = G T M, (14) Nte that when G equals ne, the ppulatin f wrkers is just replacing itself. Substituting in (14) fr the flw f births, the equilibrium yuth dependency rati can be rewritten in terms f the GRR: ȳ = 1 2 µ 1 T s 1 µ Y + 1 T 2 Y + T Y G. (15) T M 2 T M T M When the GRR is at the replacement rate the equilibrium dependency ratis simplify t ȳ r = T Y /T M,andō r = T O /T M. These results are fairly intuitive, and parallel thse in basic demgraphic thery. In a stable ppulatin with replacement fertility, the relative sizes f the age grups in equilibrium shuld simply be the ratis f their widths. 3.2 Declining Fertility and the Cnsequences fr Cnsumptin Much f the twentieth century was characterized by declines in perid fertility rates, with the exceptin f the pst-wrld War II baby bm. Cnsidering the implicatins fr the ppulatin dependency ratis, given a fall in G, equatin (15) implies that the equilibrium yuth dependency rati falls. As shwn in Figure 2, the decline in ȳ maps t a higher equilibrium value f the ld-age dependency rati alng the Z (y) lcus. 13

16 Cnsidering the transitin t an equilibrium with lwer fertility, initially the declines in yuth dependency are larger than the increases in ld-age dependency, s that ecnmic dependency (e) declines. This leads t an increase in the cnsumptin index η, reflecting the demgraphic dividend discussed by Blm, Canning and Sevilla (2001). Later in the transitin, there cmes a turning pint where the declines in e caused by y are mre than ffset by the increases in e, in which case η begins t increase as the ecnmy appraches the new equilibrium. 9 Hence, the cnsumptin benefits frm the demgraphic dividend are time-limited. The net effect f a fertility decline n equilibrium ecnmic dependency is apririambiguus. The inset f Figure 2 shws a pssible set f paths fr y,, andη after a decline in the fertility rate in the case where the new equilibrium has a higher level f ecnmic dependency (and a lwer level f cnsumptin) than the riginal equilibrium. As we will see, this likely represents the effects f further fertility declines in mst f the highly develped wrld. Fllwing Weil (1999), given a certain value f ecnmic dependency, ẽ, we can use (9) t graph is-dependency lines in (y, ) space as = ẽ ρ O ρ Y ρ O y Iẽ (y) (16) Since changes in fertility trace ut equilibrium ppulatin dependency ratis alng Z (y), we may cnsider the relatinship between the is-dependency lines and this lcus. Isdependency lines that are clser t the rigin are assciated with a lwer rate f ecnmic dependency and a higher cnsumptin index, η. Because Z (y) is cnvex with respect t the rigin, there is a unique equilibrium pair (ȳ m, ō m ) that is tangent t the Z (y) lcus. (See Figure 3.) The pair (ȳ m, ō m ) minimizes ecnmic dependency, which implies 9 The initial decline in e shuld be evident frm an analysis f the phase diagram. While is still clse t the Z (y) lcus, y is very far frm its steady state. Our assertin f a turning pint makes intuitive sense, and can be established based n numerical simulatins f the mdel using parameter values describing the OECD12 natins. 14

17 Figure 2: The Effects f a Decline in the Fertility Rate y 1 y 2 t 2 1 t 2 η 1 η 2 1 G falls t y 2 y 1 Z (y) y that cnsumptin f all members f the ecnmy is maximized. At this pint, the slpe f the Z (y) lcus must equal the slpe f the is-dependency line, which based n (13), (15), and (16) implies that µ ō G, T Y, T s µρy O TY =. (17) T M T M ρ O T M Given that ō is a mntnically negative functin f G, there is a unique fertility rate, G m, that generates this equilibrium, given values f the ther parameters. At equilibria crrespnding t higher fertility rates, i.e. thse in Regin A f the Z (y) lcus in Figure 3, small declines in G are assciated with lwer equilibrium ecnmic dependency and higher cnsumptin. The ppsite ccurs in Regin B, which is assciated with lwer fertility rates. 15

18 Figure 3: Ecnmic Dependency and the Z (y) Lcus I e" (y) I e' (y) Ie m(y) Regin B m Regin A y m Z (y) y Fr a cuntry with a fertility rate G h >G m (i.e. in Regin A), as the fertility rate declines tward the cnsumptin-maximizing rate, cnsumptin increases at all future times, assuming that the ther parameters remain stable. On the ther hand, fr a cuntry with a fertility rate belw the cnsumptin-maximizing rate the transitin t G m is cstly. As shwn in Figure 4, in a cuntry with the same initial ecnmic dependency as implied by G h and a fertility rate f G <G m, ecnmic dependency must initially increase, implying a reductin in η. It is nly later in the transitin, as the elderly dependency rati declines, that the cnsumptin benefits frm the increase in fertility are realized. Given a set f widths f the age grups and relative cnsumptin needs, ascertaining whether an ecnmy is in Regin A amunts t checking whether the bserved fertility rate (G a ) is at r abve the cnsumptin-maximizing fertility rate. We undertake this 16

19 Figure 4: Transitin t Cnsumptin-Maximizing Equilibrium frm Abve and Belw I e' (y) Ie m(y) η ( e) < η( e' ) y( G l ) ( G m ) y y( G h ) Z (y) y calculatin fr the OECD12 natins. Based n measures f educatin, health and private cnsumptin expenditures, Elmendrf and Sheiner (2000) estimate that ρ Y =0.62 and ρ O =1.37 in the United States. 10 It seems reasnable that ρ Y and ρ O shuld fllw a similar pattern in the ther highly-develped natins. Taking the values f ρ Y and ρ O frm Elmendrf and Sheiner, and using the base data n age widths frm Table 1, we cmpute G m fr the OECD12 natins based n equatin (17). In the first clumn f Table 2, we reprt the actual grss reprductive rate and in the secnd we give ur calculatin f G m. In eleven ut f the twelve natins, the current GRR is less than the cnsumptin-maximizing fertility rate, and fr mst the difference is quite substantial. It is nly in the United States that the currently-bserved fertility rate exceeds G m. 11 Hence, 10 These update the estimates f the relative cnsumptin needs given by Cutler et al. (1990). 11 That the cnsumptin-maximizing rate is s lw in the U.S. is largely a functin f the relatively shrt life expectancy after retirement. Similarly, G m is the highest in Italy where life expectancy after retirement is the lngest. We discuss the relatinship between G m and life expectancy mre explicitly belw. 17

20 Table 2: Actual and Cnsumptin-Maximizing Grss Reprductive Rates and Ptential Cnsumptin Gains fr the OECD12 Cuntry G a G m Ptential gain in η (%) Australia Belgium Canada France Germany Italy Japan Netherlands Spain Sweden United Kingdm United States Average Ntes: G a is frm United Natins (frthcming); G m and the percentage gain in the cnsumptin index, η, are calculated as described in the text. based n present demgraphic data, the equilibria fr mst highly-develped ecnmies fall in Regin B f the Z (y) lcus. This suggests that recent declines in fertility will ultimately be assciated with higher equilibrium rates f ecnmic dependency and lwer levels f cnsumptin. Cnversely, equilibrium cnsumptin culd be increased in mst cuntries thrugh increases in the fertility rate. In rder t determine the equilibrium benefits frm mving t the cnsumptinmaximizing rate f fertility, we use equatins (13) and (15) t calculate the steady-state ppulatin dependency ratis fr the OECD12 cnsistent with the age widths in Table 1 and the values f G a and G m reprted in Table 2. Weighting these by the relative cnsumptin needs gives us the equilibrium level f ecnmic dependency assciated with the actual fertility rate and with the cnsumptin-maximizing fertility rate. We 18

21 dente these as ē a and ē m, respectively. Finally, equatin (10) allws us t cmpute the per-capita cnsumptin index η assciated with these tw values. The third clumn f Table 2 reprts the percentage increase in η (ē m ) relative t η (ē a ). The average ptential steady-state cnsumptin gain in the OECD12 is apprximately ne percent. Cuntries with fertility rates furthest away frm their respective values f G m bviusly have the mst t gain. Fr example, if the fertility rate in Italy were t rise t its cnsumptinmaximizing rate, the steady-state per-capita cnsumptin index wuld rise by almst 6.6 percent. Hwever, since mst f the cuntries under cnsideratin have fertility rates belw their cnsumptin-maximizing rate, the transitin t G m wuld entail a transitry perid f higher ecnmic dependency and lwer cnsumptin. A number f explanatins have been prpsed fr the currently lw levels f fertility in the mst-develped natins, which we have s far taken as given. 12 Nnetheless, the future path f fertility in these lwest-lw fertility cuntries is a matter f sme debate amng demgraphers, and there is limited thery t guide predictins. At the same time, it is almst a certainty that ld-age mrtality will decline in the future. In Sectin 3.3, we explre the cnsequences f falling mrtality amng the elderly in terms f ecnmic dependency and cnsumptin, and in Sectin 4 we cnsider a mechanism whereby increased life expectancy itself may affect the path f fertility. 3.3 The Effects f Falling Old-Age Mrtality Old-age mrtality has fallen substantially ver the past fifty years, and is expected t decline further in the future. In the United States, fr example, life expectancy at age 65 has risen by 3.25 years since 1955 and is expected t rise by apprximately the same amunt ver the next 50 years. 13 Other highly-develped cuntries have seen cmparable 12 See, e.g. Khler, Billari and Ortega (2002). 13 Here we use the simple, rather than gender-weighted, average f sex-specific life-expectancies. The data n perid life expectancies and cme frm the Bard f Trustees, Federal Old-Age and Survivrs 19

22 Figure 5: The Effects f a Decline in Old-age Mrtality with Fertility Held Cnstant 2 1 t η( e 1 ) I e 2(y) 2 I e 1(y) η( e 2 ) T O increases t 1 y 1,2 Z 2 ( y) Z 1 ( y) y r greater reductins in ld-age mrtality ver the past half-century and can be expected t fllw a similar trend in the future. 14 Assuming a cnstant rate f fertility and cnstant values f the ther parameters, Figure 5 demnstrates the effects f an increase in life expectancy amng the elderly in the cntext f ur mdel. As life expectancy rises frm TO 1 t TO,theZ 2 (y) lcus shifts upward, leaving yuth dependency unchanged. The ld-age dependency rati increases mntnically tward the new equilibrium, while the cnsumptin index η declines due t the higher burden f elderly dependency. The cnsumptin-maximizing fertility rate is a functin f the life expectancy f the elderly, as described in equatin (17). Ceteris paribus, as life expectancy rises s des G m because the ttal cst f supprt per elderly dependent rises. While an increase in fertility Insurance and Disability Insurance Trust Funds (2005, Table V.A3). Fr the prjected gain in life expectancy, we use the intermediate variant. Other independent mrtality prjectins (e.g. Lee and Carter 1992) predict even higher values f life expectancy. 14 See, e.g., Munnell, Hatch and Lee (2004). 20

23 Figure 6: The Effects f a Decline in Old-age Mrtality n the Cnsumptin-Maximizing Equilibrium Larger ptential equilibrium gains frm increasing fertility I m(y) e 2 I m(y) e 1 Smaller ptential equilibrium gains frm reducing fertility Z 2 ( y) y( G l ) y 1m y 2m y( G h ) Z 1 (y) y wuld cause y t rise in equilibrium, the reductin in η due t higher yuth dependency is mre than ffset by the eventual decrease in the mre cnsumptin-intensive elderly dependency. Fr cuntries that initially have a fertility rate higher than G m (such as G h in Figure 6), the cnsumptin-maximizing rate draws clser t their existing fertility rate. Asseeninthefigure, nt nly des the best-attainable level f cnsumptin fall, s des the ptential equilibrium gain frm reducing the fertility rate. 15 Fr cuntries with a fertility rate lwer than G m (e.g. G in Figure 6), the cnsumptin-maximizing fertility rate mves further away as ld-age mrtality falls. Althugh the best-attainable level f cnsumptin falls, the ptential equilibrium gains frm a higher fertility rate are greater. 15 This can be established analytically since the Z (y) lcus gets steeper fr any value f y as T O increases. 21

24 Repeating the numerical calculatin frm Sectin 3 fr all OECD12 under the assumptins that T O increases by 3.25 years and that actual fertility rates remain unchanged results in an average ptential increase in η f just ver 3 percent. Further, the United States jins the eleven ther natins in having a fertility rate belw the cnsumptinmaximizing rate. As we will see belw, the burden f ld-age dependency itself may cause fertility rates t decline (rather than rise), leading t lng-run utcmes that are even less efficient frm a cnsumptin standpint. 4 Old-Age Dependency and Fertility In ur analysis f ecnmic dependency and cnsumptin thus far, we have taken the fertility rate as exgenus. This ignres the vast ecnmics literature that regards fertility, fr the mst part, as the result f a decisin making by ptential parents wh weigh the csts and benefits f having children. Hence, in additin t differences in cnsumptin needs, there is a mre prfund asymmetry between yuth dependency and elderly dependency. The ecnmic burden f yuth dependency is partially the result f the fertility chices f the wrking. On the ther hand, adults have n chice ver the number f elderly persns in the ecnmy, but are typically required t supprt them thrugh public pensin systems. In a pay-as-yu-g pensin system, rising elderly dependency reduces the per-capita resurces available t the wrking. Given standard ecnmic mdels f child-bearing decisins, this ptentially prvides a partial explanatin fr declines in the fertility rate ver the last half-century. Drucker (1990, p.7) takes a strnger stand n this issue, arguing that the primary cause f lw and declining fertility in the develped wrld is that its yunger peple are n lnger able t bear the increasing burden f supprting a grwing ppulatin f lder, nnwrking peple. Bldrin et al. (2005) prvide sme empirical evidence in supprt f this idea, em- 22

25 plying a panel sample frm 1960 t the present fr eight Western Eurpean cuntries. Cntrlling fr measures f the genersity f the pensin system and infant mrality, they find a strng and negative relatinship between fertility and the share f the ppulatin 65 and ver (p 65 ). In the eight cuntries in their sample the average increase in p 65 between 1960 and 2000 was apprximately 4.7 percentage pints. 16 Applying their regressin cefficient t this difference implies a predicted decline f 0.15 in the GRR, which was ver a quarter f the average actual reductin in the GRR in the sample. Thus, increases in elderly dependency ptentially explain a gd deal f the recent declines in fertility. We develp here an extensin t ur basic mdel that allws this srt f dynamic interactin between ld-age dependency, fertility and the age structure. We retain the prductin structure described abve, where wrkers prvide labr inelastically in return fr a wage f W (t). Wages are subject t a prprtinal tax τ, which is used t fund public transfers t the ld and t the yung. We assume that the elderly are supprted entirely thrugh public transfers in the frm f a pensin system, while the yung receive bth public supprt and direct transfers frm their parents. Althugh smewhat stylized, this distinctin mirrrs the substantial difference in the pattern f transfers t the ld and yung bserved in reality. Accrding t Masn et al. (2005), private transfers accunt fr ver 61% f the transfer-based cnsumptin f persns under the age f 20 in the United States, but nly 11% f the ttal transfers t persns ver the age f 65. The pensin system replaces after-tax wages at a rate β, and we assume that percapita public transfers t the yung are equal t a fixed prprtin, π, f the net transfers 16 Our data n the share f the ppulatin age 65+ and fertility rates drawn frm United Natins (2000 and frthcming). 23

26 t the elderly. The tax rate at time t is determined by the aggregate resurce equatin A Y αw (t)(1 τ (t)) + A M (t) W (t)(1 τ (t)) + A O βw (t)(1 τ (t)) = Ω (t), (18) where α = πβ. Slving (18) fr the balanced-budget tax rate gives τ (t) = αy (t)+β(t) 1+αy (t)+β(t). Higher rates f yuth and ld-age dependency bth increase the tax rate. As a result, increases in bth frms f dependency decrease the take-hme pay f the wrkers and the size f per-capita public transfers t the yuth and elderly. Given values f α and β, we can slve fr the tax-rate minimizing grss reprductive rate, G m τ. Similar t the analysis in Sectin 3, G m τ is the implicit slutin t µ ō G m τ, T Y, T s O α T Y =. (19) T M T M β T M Because f the large share f the cnsumptin f children that is privately funded, (α/β) will be substantially less than (ρ Y /ρ O ), which plays a crrespnding rle in equatin (17). Since ō is a negative functin f G, this will imply that the tax-rate minimizing rate f fertility is substantially higher than that which maximizes cnsumptin in the framewrk f Sectin 3. We d nt attempt t quantify G m τ, fcusing instead n the divergence between the privately chsen rate f fertility and the tax-minimizing rate. At each mment in time the fertile ppulatin f wrkers allcate their after-tax incme t their wn cnsumptin and t bearing children. The price f gds and services is numeraire. The nature f ur mdel f the age structure makes it difficult t link children t their parents, s we assume that childrearing expenses are paid up frnt 24

27 int a trust fund. Further, the cnsumptin needs f children at any given mment are indexed t the prevailing wage, equalling χw. This implies that, cnditinal n a given level f fertility and a tax rate (i.e., age structure), the cnsumptin f the parents will als be multiple f the pre-tax wage. 17 With a cnstant rate f wage grwth, g, t ensure that children always receive the requisite cnsumptin, an actuarially-neutral trust fund will set the price f children as p n = ξw (t), where ξ = χt Y / (1 gt Y ). 18 Fr simplicity we assume a lg utility functin, which implies fertile wrkers face the fllwing ptimizatin prblem max ln [c (t)] + θ ln [n (t)] (20) c(t),n(t) subject t c (t)+ξw (t) n (t) =w (t), (21) where c (t) is the cnsumptin, n (t) is the annual flw f births per fertile wrker, and w (t) =[1 τ (t)] W (t) is the take-hme wage f the wrking. Slving fr the flw f births ñ (t) =ψ w (t) W (t) = ψ 1+αy (t)+β(t), (22) where ψ = θ/(ξ(1 + θ)). The flw f births per fertile wrker satisfies sme usual prperties: fertility is a psitive functin f the relative preference fr children (θ) and a 17 The cnsumptin requirement f children culd be directly indexed t that f the parents, and we culd als include a time cst f children. Incrprating these wuld yield qualitatively similar results t the mdel we present, s we ignre them fr ease f expsitin. 18 Fr the path f trust fund payments t remain bunded, g must be less than 1/T Y.Taking25years as an upper limit n the average length f yuth dependency, this implies that g must be less than 4% ver the lng run, which is quite reasnable. 25

28 negative functin f childrearing csts (ξ). The privately-ptimal fertility rate is inefficient relative t the that which wuld be chsen by a scial planner whse gal is t maximize the equilibrium utility f fertile wrkers. T see this, cnsider the ptimizatin prblem f the scial planner wh takes int accunt the bth the utility that fertile wrkers get frm fertility and cnsumptin as well as the effects f fertility decisins n the dependency ratis. Substituting in the budget cnstraint (21) int the bjective functin (20), the scial planner s prblem can be written max ln 1 ˆn 1+αy (ˆn)+β(ˆn) ξˆn + θ ln [ˆn]. (23) The first rder cnditin fr (23) can be written 1 [ αy 0 (ˆn) β 0 (ˆn)] ˆn ˆn =ñ (ˆn)+ ξ (1 + θ) (1 + αy (ˆn)+β(ˆn)) 2, (24) where ñ (ˆn) is the fertility rate that wrkers wuld chse, cnditinal n the equilibrium ( (ˆn),y(ˆn)). Based n the equilibrium equatins (12) and (13), the term in brackets n theright-handsidef(24)isequalt [ αy 0 (ˆn) β 0 (ˆn)] = α + β [ (ˆn)] 2 (T M /T Y ) y 0 (ˆn). Since y 0 ( ) is negative, the scial planner s chsen fertility rate, ˆn, is higher than the wrker s private chice based n the age structure implied by ˆn when β [ (ˆn)] 2 (T M /T Y ) α<0, and is lwer therwise. Cnsequently, ˆn equals ñ (ˆn) nly when (19) hlds, which implies that ˆn = n m τ. That is, a scial planner ptimizing the fertile wrkers equilibrium utility 26

29 wuld set the fertility rate equal t that which minimizes the tax rate. As ld-age mrtality declines, there will be a greater divergence between privately-chsen fertility rates and the tax-minimizing rate. The dynamic analysis f Sectin 3 can easily be augmented t analyze the effects f increased life expectancy amng the elderly. Due t a standard incme effect, changes in affect fertility thrugh the tax rate. As a result, the dynamics f the yuth dependency rati will be affected by the ld-age dependency rati. Putting the flw f births per fertile wrker, ñ, int the equatin f mtin f the yuth dependency rati (equatin (8)), the graph f the zer-mtin lcus f y in (y, )-space is ẏ=0 = 1 φψt Y (1 + αy) Z β (1 T Y /T M ) y + y2 y (y). This lcus represents equilibrium values f y thatarestable(becausedẏ/dy < 0). Further, it is dwnward slping and cnvex with respect t the rigin, asympttes t infinity as y appraches zer, and crsses the hrizntal axis fr large enugh values f y. Finally, there is a single crssing in the psitive rthant between the Z y (y) and Z (y) lci, which implies that a glbally stable equilibrium in the ppulatin dependency ratis, depicted in Figure We depict the transitin assciated with a fall in mrtality in this mdel f endgenus fertility in Figure 8. As discussed abve, lwer rates f ld-age mrtality translate directly int increases in the rate f elderly dependency due t lnger life expectancy after retirement. If fertility rate were t remain unchanged, ld-age dependency wuld rise frm ō 1 t 0, and the yuth dependency rati wuld remain cnstant at ȳ 1. Hwever, increases in ld-age dependency drive up the tax rate in ur mdel, given the gvernment s balanced budget cnstraint. Since wrkers will have less dispsable incme, the 19 A prf f the single crssing between the lci is available upn request. 27

30 Figure 7: Glbal Dynamics with Endgenus Fertility Z y ( y) y Z (y) y Figure 8: The Effects f a Decline in Old-age Mrtality with Endgenus Fertility Z y ( y) y 1 2 y t 2 1 t 2 ' 1 n 2 n T O increases t 1 Z 2 ( y) y 2 1 y Z 1 ( y) y 28

31 rate f childbirth amng the fertile will fall. We have already shwn in Sectin 3 that declines in fertility lead, ceteris paribus, t increases in ld-age dependency. As a result, there is a multiplier effect n declines in ld-age mrtality, whereby increases in ld-age dependency reduce fertility, which further increase ld-age dependency. Hwever, as the fertility rate falls, s des the yuth dependency rati. The cnsequent reductins in public expenditures n the yung acts as a brake n the feedback cycle between rising ld-age dependency and falling fertility. Nnetheless, the net result is a transitin t an equilibrium with an ld-age dependency rati ō 2 that is even higher than 0. Further the rati f yung t wrking falls frm ȳ 1 t ȳ 2 due t the declines in fertility induced by the rise in elderly dependency. As life expectancy amng the elderly rises, the tax-minimizing fertility rate (n m τ G m τ /φ) will increase due t the mechanism discussed in Sectin 3.3. Hwever, the respnse f the fertile wrkers t an increase in life expectancy is t reduce their fertility. As a result, in the highly develped cuntries where ñ is initially less than n m τ, the privatelyptimal fertility respnse f wrkers t a decrease in ld-age mrtality ends up reducing the well-being f future wrkers as well as decreasing the pl f resurces available t all f the ther individuals in the ecnmy. 5 Cnclusin In this paper we have analyzed the dynamic evlutin f a cuntry s ppulatin age structure and fertility rate. Our particular cncern was with the feedback frm ppulatin age structure t fertility via the channel f ld age dependency. In a cuntry with a high level f ld-age dependency, wrking age individuals will see a large fractin f their labr incme redistributed t the elderly. In respnse, wrking age individuals will lwer fertility. 29

32 The dynamic aspects f the prblem that we study are particularly imprtant because the shrt and lng-run effects f changes in fertility n dependency are s different. In the shrt run, a reductin in fertility unambiguusly lwers a sciety s dependency burden by reducing the number f children relative t wrking age adults. In the lng run, reductins in fertility raise a cuntry s f ld age dependency rati, ptentially unding the reductin in yuth dependency. Our calculatins indicate that develped natins with the lwest levels f fertility have already passed the pint where reductins in fertility raise rather than lwer the lng-run dependency burden. T cnduct ur analysis we cnstructed a new cntinuus-time verlapping generatins mdel that divides the ppulatin int three age grups: dependent yung, wrking age, and dependent elderly. Individuals in each age grup face cnstant hazards f transitining int the next grup (r int death, in the case f the elderly). The mdel allws us t examine the dynamic evlutin f age structure f the ppulatin and the cnsumptin dependency burden in respnse t changes in fertility (in the case where fertility is exgenus) r the jint evlutin f fertility, age structure and dependency t exgenus shcks such as ld age mrtality. Using ur mdel, we shw that fr cuntries that are already belw the level f fertility that maximizes cnsumptin in the steady state, the actual and ptimal respnses f fertility t a shck t ld-age survival have ppsite signs. In respnse t greater ld age survival, such cuntries will effectively dig themselves int a deeper demgraphic hle by cutting fertility in rder t maintain cnsumptin in the shrt run. The mdel that we present is smewhat stylized in the interests f analytic tractability and ease f expsitin. It culd be extended alng tw majr dimensins in rder t imprve its realism and frecast accuracy. First, it wuld be useful t expand the number f age grups beynd the three basic nes we use. Mving beynd three age grups wuld 30

33 mean that the dynamics f the mdel culd nt be examined analytically, hwever, and we wuld be frced t take a cmputatinal apprach alng the lines f Auerbach and Ktlikff (1987) and Grafenhfer et al. (2005). Secnd, the mdel culd als be extended t incrprate intertempral ptimizatin n the part f husehlds, which wuld entail shifting cnsumptin between perids in respnse t anticipated changes in demgraphics. Such intertempral shifting, either thrugh investment in physical capital r purchase f freign assets, culd allw agents t save fr their wn ld age, r the cuntry as a whle t save fr the perid when there will be a high level f ld-age dependency. Althugh this wuld have little effect n the steady state f the mdel, it wuld significantly influence the mdel s dynamics. As discussed in Cutler et al. (1990) and Elmendrf and Sheiner (2000), the ability f the ecnmy as a whle t insulate itself against demgraphic shcks by building extra capital is limited by bth depreciatin and capital s declining marginal prduct. Attempts t smth cnsumptin in the face f demgraphic shcks may als result in a run-up f price f capital assets when everyne is trying t save and a crrespnding asset meltdwn when everyne is trying t dissave. 20 We suspect that the extensins discussed abvewuldntalterthefundamental predictins f ur mdel, which are relatively grim. Demgraphers have been wrestling fr decades with explaining belw-replacement fertility in sme f the wrld s richest cuntries. Lw fertility is viewed as a trubling utcme in and f itself. The failure f a cuntry s citizens t replace themselves is cited as evidence f sme srt f scial illness. Irnically, this hand-wringing has taken place during a perid when the ecnmic cnsequences f lw fertility were psitive. Our results suggest that ver the next several decades, as the effect f lw fertility n cnsumptin pssibilities turns frm psitive t negative, there will be a further reductin in fertility, which will, in the lng run, prduce 20 See Abel (2003) and Lim and Weil (2003). 31