CIRJE-F-53 A One-Secor Neoclassical Growh Model wih Endogenous Reiremen Kiminori Masuyama Norhwesern Universiy December 2007 CIRJE Discussion Papers can be downloaded wihou charge from: hp://www.e.u-okyo.ac.jp/cirje/research/03research02dp.hml Discussion Papers are a series of manuscrips in heir draf form. They are no inended for circulaion or disribuion excep as indicaed by he auhor. For ha reason Discussion Papers may no be reproduced or disribued wihou he wrien consen of he auhor.
A One-Secor Neoclassical Growh Model wih Endogenous Reiremen By Kiminori Masuyama Final Manuscrip Absrac This paper exends Diamond s OG model by allowing he agens o make he reiremen decision. Earning a higher wage income when young no only enables he agens o save more. I also induces more agens o reire early and gives an addiional incenive o save more for reiremen. This leads o a higher capiallabor raio in he following period, and hence he nex generaion of agens earns a higher wage income when young. Due o his posiive feedback mechanism, endogenous reiremen magnifies he persisence of growh dynamics and even generaes muliple seady saes for empirically plausible parameer values. JEL Classificaion Numbers: D9, O Keywords: Diamond s Overlapping Generaions Model, Labor Force Paricipaion of he Elderly, Magnificaion Effecs, Muliple Seady Saes, Persisence, Povery Traps Correspondences: Kiminori Masuyama Deparmen of Economics Norhwesern Universiy 200 Sheridan Road Evanson, IL 60208 k-masuyama@norhwesern.edu
. Inroducion. The labor force paricipaion rae of he elderly declines wih economic growh boh in ime series and cross secions of counries. Alhough many facors including social securiy sysems undoubedly have major effecs on he reiremen behavior, he observed paerns are remarkably similar across counries, suggesing ha he rend for early reiremen may be a consequence of he rising income. 2 Early reiremen in urn affecs he growh process of he economy hrough is effecs on he aggregae saving and labor supply. To sudy such ineracive processes beween induced reiremen and economic growh, his paper develops a varian of Diamond s (965) one-secor neoclassical growh model of an overlapping generaions (OG) economy, in which each generaion lives for wo periods. The original Diamond model assumes ha he agens work in he firs period (when hey are young) and reire in he second (when hey are old). The presen paper endogenizes he labor force paricipaion decision in he second period, while mainaining he original assumpion ha he agens always work in he firs period. In order o generae he paerns consisen wih hose observed in boh he ime series and cross secions of counries, we impose resricions on he parameers of he uiliy funcion so ha he income effec of he higher wage when young, which encourages reiremen, dominaes he price effec of he higher wage when old, which discourages reiremen. 3 I urns ou ha endogenizing he labor force paricipaion of he elderly has significan effecs on he mechanics of economic growh. Earning a higher wage income in he firs period no only enables he agens o save more, bu i also induces more agens o reire in he second period, which provides an addiional incenive o save more for reiremen. This resuls in a higher capial-labor raio in he following period, which implies ha he nex generaion of agens earns a higher wage income in heir firs period. Due o his posiive feedback mechanism, See, for example, Fuchs (983) and Gruber & Wise (999). 2 See Fuchs (983) for an earlier survey of he empirical evidence for he income effec on reiremen. More recenly, Cosa (998) found, using he hisorical U.S. daa prior o he inroducion of he Social Securiy sysem, ha he rising income is a major facor responsible for he decline in he elderly s labor force paricipaion rae. 3 As Fuchs (983) poined ou, his assumpion is no inconsisen wih he well-known correlaion ha, wihin each cohor, he individuals wih lower income end o reire early, which has more o do wih he heerogeneiy of he workers. The key mechanism here is ha earning a higher wage income when young induces more workers o reire early, no ha more producive workers reire early. In Secion 6, we sugges a way of exending he presen model o generae his correlaion wihou changing he main resuls.
endogenizing he labor force paricipaion of he elderly magnifies he persisence of growh dynamics, hereby slowing down a convergence o he seady sae, and even leading o muliple seady saes for empirically plausible parameer values. The resul ha endogenous reiremen magnifies he persisence of growh dynamics suggess ha he mechanism discussed in his paper may offer an explanaion for he slow convergence puzzle. The presen mechanism, however, should no be viewed as an alernaive o he oher persisence mechanisms previously proposed in he lieraure. 4 I should be viewed insead as complemenary, because endogenizing reiremen also amplifies persisence caused by hese mechanisms. The possibiliy of muliple seady saes suggess ha here are wo-way causaliies beween early reiremen and economic developmen. Some readers may be surprised by he resul ha endogenous reiremen can generae muliple seady saes in a one-secor neoclassical growh model. Afer all, i is sraighforward o show ha inroducing he consumpion-leisure rade-off in he sandard one-secor neoclassical growh model wih he infiniely-lived represenaive agen (he RA model for shor) does no affec he seady sae wage rae. 5 The reason why muliple seady saes are possible is ha he persisence mechanism in his paper relies on he feaures of he OG model ha are missing in he RA model. Firs, he labor supply is composed by he inelasic labor supply by he young and he elasic labor supply by he old. I is essenial ha a high wage reduces he labor paricipaion rae among he old more han among he young. The inergeneraional ransmission mechanism would be absen if everyone s labor supply were equally elasic, as in he RA model. Second, in he Diamond OG model, no agen s asse holding is consan over ime even in he seady sae; each agen mus firs earn he wage income in order o save and own he capial sock, and he individual asse holding increases when young and declines when old. The seady sae in he Diamond OG model means ha he ime profile of he individual asse holding does no change from one generaion o anoher. The seady sae in he RA model, on he oher hand, means ha he agen s asse holding is consan over ime. To pu i anoher way, he propensiy o save ou 4 See Masuyama (2005) and he work cied herein for oher persisence mechanisms ha generae muliple seady saes. 5 To see his, i suffices o noe ha he wage rae is a funcion of he capial-labor raio in a one-secor neoclassical growh model, and ha he seady sae capial-labor raio in he RA model is uniquely pinned down by one of he seady sae condiions, βf'(k) =, where β is he discoun rae. See Blanchard and Fischer (989, Chaper 2) for a sandard reference for he RA model. 2
of he wage income depends on he agen s age in he Diamond model. A higher wage income by he young increases he aggregae saving, while a higher wage income by he old reduces he aggregae saving. This is no a feaure of he RA model. The above discussion should also explain why reiremen, no delayed labor force paricipaion, is imporan for he analysis. If he labor paricipaion rae of he young, insead of he old, were endogenized, he capial sock would move in he opposie direcion, causing reversal, raher han persisence. 6 There are a few relaed sudies. Feldsein (974) discussed he possibiliy ha social securiy may end up increasing he aggregae saving hrough induced reiremen, insead of reducing i as he sandard analysis migh sugges. To explore Feldsein s insigh formally, Hu (979) endogenized he labor supply by he elderly in he Diamond OG model, and examined he general equilibrium effecs of social securiy. However, he did so under he assumpions ha ensure he uniqueness of he seady sae. 7 Some readers may hink ha growh implicaions of endogenous reiremen canno be imporan because he elderly accouns for a small share of he labor supply, even when heir labor force paricipaion rae is high. Feldsein (974, p.924), for example, expressed such a view. To respond o such a skepicism wihin he wo-period overlapping generaions 6 Indeed, some previous sudies have endogenized he labor supply of he young in he Diamond OG model and demonsraed he possibiliy of endogenous cycles. See, for example, Reichlin (986). 7 There is an exensive lieraure in labor economics and public finance, which analyzes he elderly s labor force paricipaion decision. This lieraure does no address general equilibrium implicaions of endogenous reiremen. Their main objecive is an empirical assessmen of he impac of social securiy sysems and pension plans on reiremen behavior, insead of undersanding growh implicaions of reiremen behavior. In an aemp o explain how social securiy provisions affec he iming of reiremen, his lieraure uilizes increasingly sophisicaed models of reiremen behavior, including hose of sochasic dynamic programming. On he oher hand, he presen model keeps reiremen behavior as simple as possible in order o mainain he racabiliy of general equilibrium analysis. For example, i is assumed ha he agens live only for wo periods, ha he labor supply when young is exogenous, ha here is no uncerainy, and ha reiremen is a zero-one decision, ec. One may hence be surprised o find ha he model of reiremen behavior developed below is no a special case of hose developed in his lieraure. The reason is ha he presen model is designed o evaluae he effec of reiremen on saving hrough a change in he ime profile of labor income, no hrough a change in he ineremporal preferences over consumpion. This requires ha he preferences for reiremen be weakly separable from he ineremporal preferences for consumpion. (Ineresingly, Feldsein (974, Fig. ) made his assumpion implicily in his graphic analysis.) I urns ou ha his condiion, when he reiremen decision is endogenized, implies ha he preferences canno be ineremporally separable. The recen lieraure on reiremen behavior, on he oher hand, imposes he ineremporal separabiliy of preferences so as o make he sandard ool of dynamic opimizaion readily applicable o he problem. I should also be noed ha he exising empirical sudies impose funcional forms ha rule ou he possibiliy of nonhomoheic preferences, which are oo resricive for he presen analysis. This, oo, is a reflecion of he difference in objecive. To evaluae he incenive effecs of social securiy provisions on he iming of reiremen, he income effec may no be imporan. For he presen sudy, which is concerned wih growh implicaions, i is essenial. 3
framework, he model assumes ha he old agen s effecive labor supply be a fracion of he young agen s, given by a parameer,. I urns ou ha he magnificaion effec of endogenous reiremen is independen of. Of course, his independence is due o he paricular funcional forms assumed in he paper. However, i suggess ha he effec of an increase in he elderly s share in labor supply on he magnificaion effec is ambiguous in general, which should be sufficien o refue he argumen ha he elder s share in labor supply mus be large enough o have significan effecs. The reason why he main resul does no have o depend on he elderly s share in he labor supply is ha we are dealing wih he effec of endogenous reiremen on growh dynamics. The relevan quesion is no only how much early reiremen causes he wage rae o increase, bu also how much increases in he wage rae induce early reiremen. Before proceeding, i is worh clarifying he measure of developmen adoped in his paper. The sandard measure of developmen, per capia income, is highly misleading when comparing counries ha differ significanly in heir labor force paricipaion raes. For example, per capia income in Japan may be higher han hose in some European counries, in par because he labor force paricipaion rae in Japan is much higher. I is possible ha, because of higher oupu per worker, many people in hese European counries can afford o reire early, which make heir per capia incomes lower han Japan s. In his case, he oupu per worker is a beer measure of developmen. The ulimae measure of developmen, of course, should be he sandard of living. In he model developed below, he lifeime uiliy of he agen is higher if and only if he wage rae, which moves ogeher wih he capial/labor raio and he oupu per worker, is higher. On he oher hand, higher per capia income does no necessarily imply he higher lifeime uiliy. I is for his reason ha he wage rae (or he capial/labor raio or he oupu per worker), insead of per capia income, is used as he measure of developmen. The res of he paper is organized as follows. Secion 2 ses up he framework. Secion 3 considers he special case, in which he agens do no care abou consumpion when young, so ha he only rade-off is beween consumpion and leisure when old. Under his assumpion, he young save all he wage income, independen of he reiremen decision. This helps us o explain he key mechanism wihou worrying abou he complicaions ha arise from he join saving/reiremen decision. In his case, however, endogenous reiremen does no creae large 4
enough persisence o generae muliple seady saes, unless he share of capial is implausibly large or unless i ineracs wih oher persisence-enhancing mechanisms. Secion 4 allows he agens o also care abou consumpion when young. This makes he saving and reiremen join decisions and inroduces a reiremen moive for saving. In his case, endogenous reiremen creaes persisence large enough o generae muliple seady saes for empirically plausible parameer values. Secion 5 inroduces a form of heerogeneiy among agens. Secion 6 suggess some direcions for fuure research. 2. The Framework Time is discree and exends o infiniy. There is a single final good, he numeraire, which can eiher be consumed or invesed. I is produced compeiively by a sandard consan-reurno-scale echnology, Y = F(K, L ). Le k K /L denoe he capial-labor raio, and f(k ) F(k,) denoe he producion funcion in is inensive form, which is increasing and concave in k. The facor markes are compeiive, and boh capial and labor earn heir marginal values, as follows. () r = R(k ) f(k ). (2) w = W(k ) f(k ) k f(k ). The economy is populaed by overlapping generaions of he equal size, normalized o be one. Each generaion lives for wo periods. Aside from he fac ha hey may live in differen periods, he agens are homogenous (unil secion 5). The young in period supplies one uni of labor inelasically, and earns wage income, w. The agen may consume some of he wage income, c y, and save he res, s = w c y, in capial. When he agen becomes old in period, s/he earns capial income, r s. In addiion, he agen may supplemen he capial income by coninuing o work and earning wage income, equal o w, where is he effecive uni of labor supply by he old. Alernaively, he agen may reire. The old agen s labor force paricipaion is a zero-one decision; e = 0 if s/he reires and e = if s/he works. 8 The agen s old 8 The assumpion ha he reiremen is a zero-one decision is made no only because i is realisic for many individuals, bu also because i allows for an analyical soluion. Gong and Liu (2006) exended he presen model o allow for e o be a coninuous variable, and showed numerically ha he resuls in his paper essenially carry over. 5
consumpion is equal o c o = r s + w e. The agen s choice can hus be described as he soluion o he following maximizaion problem: (M) Given w, choose c y 0, c o 0, and e {0,} o maximize U(c y, c o, e ) subjec o c o = r s + w e = r ( w c y ) + w e. The parameer,, is inroduced o allow for he possibiliy ha he elderly accouns for only a small share of he labor supply, even if heir labor force paricipaion rae is high. I urns ou ha he main resuls obained below are independen of. In equilibrium, he agen may be indifferen beween working and reiring when old, so ha, in spie of he homogeneiy, he old generaion s labor force paricipaion rae, x, may ake a value beween zero and one, and he labor supply in period is L = (+x ). The supply of capial in period is equal o he oal saving made by he young in period. An agen s saving in period is generally deermined joinly wih wheher s/he reires in period. If we indicae his dependence by s (e ), hen he gross saving by he young generaion in period, and hence he capial sock available in period is given by K = s (0)(x )+s ()x. The capial-labor raio is herefore k = K /L = {s (0)(x )+s ()x }/(+x ). Noe ha his model would be idenical o Diamond s original model, if he agen is forced o reire, e = 0, and hence he labor force paricipaion rae by he old is exogenously equal o zero (x = 0). 3. Consumpion/Reiremen Trade-off when Old Le us consider a special case, where he uiliy funcion does no depend on c y. Tha is, he agen cares only consumpion and leisure when old. Then he agen saves all he wage income, s = w, independen of he reiremen decision, and hence he oal supply of capial in period is simply K = w. This simplificaion has wo advanages. I allows us o focus on he reiremen/consumpion rade-off faced by he old. I also helps us o see how he reiremen decision by he curren old generaion will affec all he fuure generaions wihou he complicaion ha arises from he join saving-reiremen decision. 6
The preferences of he old agen in period are now described by U(c o, e ), which is sricly increasing in c o, unbounded from above, i.e., U(c o, e ) +, as c o +; and saisfies U(c o,) < U(c o, 0) for all c o > 0. These assumpions imply ha one can uniquely define a posiive-valued funcion,, by (3) U(c o +(c o ), ) U(c o, 0). Conversely, for any posiive-valued funcion,, U(c o, e ) = c o + (c o )(e ) saisfies all he above condiions. Therefore, wihou furher loss of generaliy, we can se U(c o,e ) = c o + (c o )(e ). Or equivalenly, we can choose any posiive-value funcion, (c o ), as a primiive of he model, insead of he uiliy funcion, U(c o, e ). One can inerpre as he value of leisure (or he compensaing differenial for working) in he second period. An agen who has earned w and saved K = w when young receives r K = r w when old. If s/he reires, he uiliy level is U(r K,0) = U(r K + (r K ), ). On he oher hand, if s/he coninues o work, i is U(r K + w, ). Therefore, an old agen in period chooses o work if (r w ) < w ; s/he chooses o reire if (r w ) > w ; and s/he is indifferen (r w ) = w. Given w = K, he equilibrium in period is given by eqs. () and (2), and (4) k = K /(+x ) = w /(+x ), = if rw w, (5) x [0,] if rw w, = 0 if rw w. These condiions joinly deermine he mapping from w o w, w = (w ), which can be applied ieraively o solve for he equilibrium rajecory of he economy, for any iniial condiion, w 0 = K. 3.A. Exogenous Reiremen Before proceeding, le us firs consider he case where he old generaion s labor force paricipaion rae is given exogenously x [0,], for all. Tha is, a fracion x of he old generaion in period reires, and he res says in he labor force. The Diamond overlapping generaions model is a special case where x = 0 for all. I is well known ha he dynamics in 7
he Diamond model may be complicaed unless addiional resricions are imposed on he producion funcion. Since he goal here is o provide a benchmark for he case of endogenous reiremen, we impose such resricions so ha, wihou he endogeneiy of reiremen, he dynamics is well-behaved. More specifically, i is assumed ha he producion funcion is a Cobb-Douglas, f(k) = Ak, where (0,) is he capial share. Then, (2) becomes w = ()A(k ). From (4) and w = K, he dynamics is described as (6) w = ()A(k ) = ( ) ( x ) A( w ) Figure illusraes he dynamical sysem, given by (6), under he assumpion ha he labor force paricipaion rae is consan over ime, x = x. For any x, he mapping is globally concave and he dynamics has a unique seady sae. A higher x shifs he mapping down, reducing he wage rae and capial sock in he seady sae. The parameer ha governs he persisence of he dynamics is equal o he capial share,, independen of x. 3.B. Endogenous Reiremen We are now ready o examine he effec of endogenous reiremen. The dynamics are now described by (5) as well as (6), wih w = ()A(k ) and r = A(k ). In order o obain a closed-form soluion for he mapping, le us consider (c o ) = (c o ), or equivalenly, U(c o,e ) = c o +(c o ) (e ), where > 0 and (, ). The assumpion, >, ensures ha, as he economy develops, he income effec of a higher wage when young, which encourages reiremen, dominaes is price effec of a higher wage when old, which discourages reiremen. Some algebra yields ( ) ( ) ( ) (7) w = (w ) ) ( w ) where A( w ) if w (0,w ], ( A if w (w,w + ), ( ) A( w ) if w [w +, ), 8
(8), ( ) (9) ( ) ( ) and (0) w ( ) ( ) A < w +. A The choice of parameerizaion in (7)-(0) was made so ha he reader can see how he map, given by (7), depends on A and, by inspecion. The map is also illusraed in Figures 2a and 2b, which assume ha he map inersecs wih he 45 line in he inerval, (w, w + ). (I is easy o find a se of parameer values ha ensures he exisence of such an inersecion by adjusing, say,.) If w (0,w ], he old generaion, having earned and saved lile when young, does no reire: he labor force paricipaion rae is x =. If w [w +, ), he old generaion, having earned and saved enough when young, chooses o reire: x = 0. Thus, he dynamics of he economy in boh he lower and higher ranges is similar o he case of an exogenous labor force paricipaion rae. In paricular, he persisence parameer is equal o. In he middle range, (w, w + ), he labor force paricipaion rae changes wih w. Some algebra shows ha he capial-labor raio and he labor force paricipaion rae in his range change wih K = w, as follows. () k = A (2) x = A w = w w w w w = w w,. An increase in he wage rae leads o a decline in he paricipaion rae, and o a more-hanproporionae increase in he capial-labor raio. The homogeneiy of he agens makes i simple o derive he map in he middle range, where x (0,). I is given by he condiion ha he agens are indifferen beween working and 9
reiring when old. A large (a higher value of reiremen), by increasing, reduces he equilibrium labor force paricipaion rae and increases he capial-labor raio, as seen in () and (2). Therefore, i shifs he map,, upwards in he middle range, while i has no effec on he map in boh he low and high ranges. The middle range iself moves o he lef in response. An exogenous increase in A, he oal facor produciviy, shifs he map upward everywhere. However, he shif is bigger in he middle range, and he range iself moves o he lef, because he old generaion earns more capial income ou of saving, which reduces he labor force paricipaion rae and he capial-labor raio, as seen in () and (2). 3.C. Persisence Noe ha, in he middle range, he elasiciy of he map is equal o >, i.e., higher han he case wih a consan x. In oher words, he persisence of he dynamics is magnified by he facor o >. Eq. (8) shows ha he magnificaion facor is increasing in. When he value of reiremen rises sharply wih economic growh, a higher wage rae would be needed o keep he old generaion in he labor force. Noe also ha is independen of. This feaure of endogenous reiremen, he magnificaion of persisence, has imporan implicaions for he growh process. If <, he case depiced in Figure 2a, he economy sill converges o he unique seady sae. However, he speed of convergence is slower, as he economy raverses hrough he middle range. If =, he map generaes a uni-roo dynamics in he middle range. If >, he map becomes convex in he middle, so ha growh acceleraes. Furhermore, here may exis muliple seady saes, wo sable and one unsable, as depiced in Figure 2b. In he lower sable seady sae, he wage is low and people do no reire. In he higher sable seady sae, he wage is high and people reire. Two oherwise idenical economies, if heir iniial posiions are separaed by he unsable seady sae, converge o differen seady saes. When an economy is rapped in he lower of he wo sable seady saes, a variey of he governmen policies can be used o lif he economy ou of he rap and o move oward he higher sable seady sae. This can be done by making he elderly reire eiher by force or by subsidy. (Indeed, he subsidy does no need o be condiioned on reiremen; even a simple, 0
uncondiional ransfer o he elderly can induce he elderly o reire because reiremen is a normal good.) I should be poined ou, however, ha he above menioned policies are no Pareo-improving. These policies mus reduce he welfare of eiher he curren old generaion (if hey are forced o reire) or he fuure generaions (if he axes are imposed on hem o finance he subsidies). The reason is simple. The model has neiher exernaliy (because all he ineremporal linkages operae hrough markes) nor dynamic inefficiencies (because he seady sae ineres rae is posiive, and hence is higher han he seady sae growh rae of he economy, which is equal o zero). Therefore, he equilibrium allocaion of he economy described above, even in he case where he economy is rapped in he lower sable seady sae, is Pareo-efficien and hus canno be Pareo improved by means of axes, subsidies, ransfers, or oher sandard correcive policy measures. How big is he magnificaion effec of endogenous reiremen? As i sands, he model does no generae quaniaively significan effecs. Even when is aken arbirarily large, canno be greaer han /(), which is equal o 0.5 for = /3 and o 2/3 for = 0.4. In order for he map o become convex in he middle o generae muliple seady saes, i is necessary o have > /2. This merely suggess ha his simple model canno explain a quaniaively large persisence. The nex secion will consider he case where he agens also care abou consumpion when young. This makes saving and reiremen join decisions, and inroduces a reiremen moive for saving. I will be shown ha, by affecing he saving decision, endogenous reiremen can increase persisence enough o generae muliple seady saes for an empirically plausible value of. Alernaively, endogenous reiremen can be combined wih oher mechanisms for persisence. I is worh noing ha endogenous reiremen no only supplemens oher mechanisms for persisence, bu also enhances heir power of generaing large persisence. As an illusraion, le us suppose ha he oal facor produciviy, A, is now endogenous, and evolves according o (3) A = A 0 (K ).
The idea is ha he level of aggregae capial sock can also be viewed as a proxy for knowledge capial. Through knowledge spillovers, he capial sock affecs he oal facor produciviy of he economy, and measures he exernaliy effec of knowledge spillovers. Since his effec is purely exernal, he agen does no ake i ino accoun when making decisions. Aggregae exernaliies of his kind have been suggesed by Romer (986) and ohers in he endogenous growh lieraure as a way of generaing persisence in dynamics. By insering (3) ino (6) and by recalling K = w, i can be shown ha he dynamics would follow (4) lnw = cons. + ln w, when he old generaion s labor force paricipaion rae is exogenous and consan. To generae acceleraing growh and he possibiliy of muliple seady saes, has o be much larger han for any plausible value of. For example, = /3 implies ha mus be more han wice as large as. Even wih = 0.4, mus be more han 50% larger han. Wih endogenous reiremen, on he oher hand, he dynamics follow, from (7) and (3), (5) ln w = cons. + w ln = cons. + ln ( ) in he middle range. Noe ha endogenous reiremen no only generaes persisence in addiion o he exernaliy effec, bu also enhances he exernaliy effec. Hence, muliple seady saes are possible for a plausible value of, and a much smaller. (For example, for = /3, > /3 w would suffice for a sufficienly large. For = 0.4, > 0.2 would suffice.), 4. Inroducing a Join Saving-Reiremen Decision Le us now inroduce he reiremen moive for saving. As saed before, he problem of he agen who becomes old in period can be described as: (M) Given w, choose c y 0, c o 0, and e {0,} o maximize U(c y, c o, e ) subjec o c o = r s + w e = r (w c y ) + w e. For he res of he analysis, wo addiional resricions will be imposed on he uiliy funcion. 2
Firs, c y and c o are assumed o be weakly separable from e, so ha he ineremporal preferences for consumpion are independen of he reiremen decision. Wihou such a resricion, he effec of he reiremen decision on saving can be arbirary. The resul can change according o how he marginal rae of subsiuion beween consumpion in wo periods depends on e. Reiremen may reduce marginal uiliy of some consumpion iems, such as business suis, while raising ha of oher iems, such as books. Such inrospecion, however, may no be useful for he level of aggregaion ha we are dealing wih. Alhough reiremen may increase marginal uiliy of books, reading books may reduce he reired person s need for oher iems, hereby reducing marginal uiliy of consumpion in general. The assumpion of weak separabiliy, while resricive, seems o offer he mos naural benchmark. 9 Noe ha he weak separabiliy assumpion does no eliminae he reiremen moive for saving, because he reiremen decision affecs he ime-profile of labor income. Raher, i means ha reiremen canno affec saving by changing he ineremporal preferences over consumpion. Second, in order o focus on he magnificaion effec of endogenous reiremen, i is useful o impose he resricion on he preferences in such a way ha when he labor force paricipaion rae is consan, x = x, he persisence parameer is equal o, independen of x. This is saisfied if and only if ineremporal preferences are Cobb-Douglas. These wo resricions joinly imply ha he uiliy funcion can be wrien in he form, U(c y, c o, e ) U(z, e ), wih z Z(c y, c o ) = y o c c, where (0,] is a consan, independen of e. Noe ha he preferences here are no ineremporally separable. Given he preferences assumed, he uiliy maximizaion yields, (6) s = w c y w = w ( ) e. r Eq. (6) shows ha he agen s saving is coningen on he reiremen decision. Those who decide o reire save more han hose who decide no o. The difference, he reiremen moive 9 A leas, he resuls under his assumpion should be considered neural. Ineresingly enough, he assumpion of weak separabiliy beween he reiremen decision and wo period consumpions was made implicily in Feldsein s 3
for saving, would be larger if is smaller. Eq. (6) also shows he posiive correlaion beween he asse holding of he agens a he beginning of heir second period and heir reiremen decision. This should no be inerpreed as saying ha he wealhy reires early, because he wealh and he reiremen are joinly deermined in his model. The value of z also depends on he reiremen decision, as follows. w r (7) z = w e r. Hence, he opporuniy cos of reiremen is equal o w( r ). Therefore, he reiremen decision is given by = if ( r ) w w( r ) (8) e {0,} if ( r ) w w( r ) = 0 if ( r ) w w( r ) where, he value-of-reiremen funcion, is defined in a manner similar o (3). Tha is, i is a posiive-valued funcion, uniquely defined by U(z+(z), ) U(z, 0), for any U(z, e), which is sricly increasing and unbounded from above in z, and U(z, ) < U(z, 0) for all z > 0. Or equivalenly, we can se U(z, e) = z + (z)(e), wihou furher loss of generaliy. The aggregae saving by he young in period, which is equal o K, can be obained from (6) by aggregaing across all he agens, as follows. w (9) K = w ( ) x. r Since he labor supply is L = (+x ), his implies ha,,, (20) k = w ( ) ( w / r ) x x. Noe ha he reiremen decision by he old affecs he capial-labor raio hrough wo differen channels, even afer conrolling for is effecs on he facor prices. One channel is he labor supply effec, which appears in he denominaor of (20). The oher is he saving for reiremen (974) graphic analysis, when he drew he indifference curves defined over he space, (C, C 2 ) in his noaion, independen of he agen s reiremen behavior. 4
effec, capured in he second erm of he numeraor. Boh effecs work in he same direcion. A higher labor force paricipaion rae by he elderly hus implies a lower capial-labor raio, and hence a lower wage for he nex generaion. The labor force paricipaion rae saisfies in equilibrium, = if ( r ) w w( r ) (2) x [0,] if ( r ) w w( r ) = 0 if ( r ) w w( r ) Given w, he equilibrium in period is given by eqs. (), (2), (20) and (2).,,. 4.A. Exogenous Reiremen Before proceeding, le us ake a brief look a he case of exogenous reiremen. Under he assumpion of he Cobb-Douglas producion funcion, (20) implies ha he wage rae in period is given by, (22) w = ()A(k ) = ( ) Aw ( ) ( ) / ) x When x = x, he map is globally concave, and he economy converges o he unique seady sae, as depiced in Figure. Noe also ha he persisence of he dynamics is equal o, independen of x.. 4.B Endogenous Reiremen Le us now look a he case where he labor force paricipaion rae is endogenous. Again, le us assume ha f(k) = A(k), (0,), and (z) = (z), or equivalenly, U(z, e) = z + (z) (e), where > 0, (, ). Then, eqs. (2) and (22) imply ( )( *) A( w ) if w (0,w ], A ) if w (w,w + ), (23) w = (w ) ) ( ) A( w where ( ( ) A( w ) if w [w +, ), 5
(24), ( )( ) (25) *, ( ) ( )/ (26) ( ) ( ) ( ), ( *) (27) w A ( ) < w + A ( ). I is easy o verify ha (23) is an exension of (7), by leing =. This exension keeps he qualiaive feaures of (7). In paricular, here is a middle range in which he labor force paricipaion rae changes and he persisence parameer is greaer han. Here he magnificaion facor is given by (24). In his range, he capial-labor raio and he labor force paricipaion rae change as follows. w (28) k = A ( ) (29) x = A = w w, ( ) w = ( )( ) w w w w. I can also easily be seen ha he effecs of A and are similar o before. The effec of a change in, on he oher hand, is difficul o evaluae; as seen in (23)-(27), appears everywhere. Neverheless, wha ineress us mos is how he magnificaion effec of endogenous reiremen changes wih he inroducion of he reiremen moive for saving. Eq. (24) shows ha a smaller makes larger. For example, le = /2, so ha he agen pus equal weigh on each period. Then, for a sufficienly large, he map becomes convex in he middle, for any > /3. If = /3, which implies he agen s discoun rae is abou 2.2% per year, if he period lengh is 30 years, hen for = /3, >.4 is enough for he convexiy. Noe, again, ha is independen of. 5. Heerogeneiy of Agens 6
In he model presened above, he homogeneiy of he agens and he zero-one naure of he reiremen decision ensured ha he price elasiciy of he labor supply by he old generaion (as a group) is infinie. This implies ha, if he old generaion s labor force paricipaion rae is beween zero and one, he agens mus be indifferen regarding reiremen. This grealy simplifies he analysis. This supply-side condiion alone deermines he map, w =(w ), in he middle range. The demand facors for labor need o be evoked only o pin down he value of x. Generally speaking, he presence of heerogeneiy or allowing for a parial reiremen would make he analysis more difficul because boh he supply and demand sides would hen need o be aken ino accoun o derive he map. This secion discusses one relaively simple way of making he price elasiciiy of he elderly s labor supply finie wihou losing he racabiliy of he model, by inroducing a form of heerogeneiy ino he analysis. Suppose now ha he agens differ in heir preferences. More specifically, (z) has he following simple form: (z) = m if z <, and (z) = M if z, where m is a sufficienly small posiive number and M is a sufficienly large, bu finie number. The agens differ in heir values of, and le G() be he disribuion funcion. Then, from (8), e = if and only if r, and hence he labor force paricipaion rae is ( ) G A ( ) ( w) w (30) x G( r) w. w < The equilibrium dynamics are now deermined joinly by (22) and (30). Clearly, he case of an exogenous reiremen is a special case, where G is consan. In his case, he persisence parameer is. When G is degenerae and has a mass on a single poin, eq. (30) shows ha x ( ) (0,) requires ha w w mus be consan. This implies ha he persisence parameer is equal o /{()}. (Noe ha he magnificaion facor in his case is /{()}, which coincides wih (24) wih = +.) Excep hese wo exreme cases, one canno solve for he map in a closed form (unless G is allowed o depend on parameers, such as,, and ). Neverheless, some qualiaive effecs of endogenous reiremen can be seen from (22) and (30), which joinly imply he dynamics in he following form: 7
( (3) w = ) w w w H, where H is a funcion, ha is increasing if and only if G is increasing. A oal differeniaion of (3) yields (32) dlog( w ) d log( w ) ( ), where is he elasiciy of H. Since () <, he magnificaion facor, given in he bracke, is greaer han one whenever > 0. Hence, in any range where G is increasing, he map is seeper and he persisence parameer is bigger han he case of an exogenous reiremen, as illusraed in Figure 3. A large amoun of heerogeneiy (i.e., a smaller ) makes he magnificaion effec smaller, bu expands he range in which he magnificaion effec operaes. I should also obvious ha, depending on he shape of G, here can be any number of seady saes. 6. Concluding Remarks This paper endogenized he reiremen decision in Diamond s overlapping generaions model and sudied he inerdependence beween he labor force paricipaion by he elderly and economic growh. Earning a higher wage income in he firs period no only enables he agens o save more, bu i also induces more agens o reire in he second period, which provides an addiional incenive o save more for reiremen. This resuls in a higher capial-labor raio in he following period, which implies ha he nex generaion of he agens earns a higher wage income in heir firs period. Due o such posiive feedback mechanism, he endogeneiy of reiremen magnifies he persisence of growh dynamics, hereby slowing down a convergence o he seady sae, and even generaing muliple seady saes for empirically plausible parameer values. Obviously, here are many ways in which he model can be exended. Only a few will be suggesed below. Firs, while secion 5 discusses he case where he agens differ in heir reservaion wages, differences in earning capaciy migh also be imporan as a source of heerogeneiy ha affecs he reiremen decision. Earning capaciy differences can be modeled by endowing agens wih differen effecive unis of labor. Such an exension would generae some ineresing predicions. For example, if he raio of he effecive unis endowed when 8
young and when old is he same across agens, hose wih higher abiliy would are more likely o reire han hose wih less abiliy. On he oher hand, if hose wih lower earnings when young happen o be hose whose earning capaciy depreciaes faser as hey age, (such an assumpion may be a reasonable way of capuring he siuaion ha he job held by unskilled may be more physically demanding), hen his resul can be reversed, and he poor may reire early, a predicion consisen wih he evidence ha he beer educaed ends o reire laer (see, for example, Fuchs 983). Ye hese changes will no aler he basic feaure of he model, i.e., earning a higher wage income when young induces more workers o reire early. Second, he model can be exended o allow sochasic shocks. For example, suppose ha he oal facor produciviy, A, is subjec o i.i.d. shocks. If here is a negaive shock in period, he old generaion may find hemselves shor of he reiremen income, and be forced o work. The negaive shock would hus reduce he young generaion s wage income in period no only hrough he direc effec of lower produciviy bu also hrough he indirec effec of higher labor force paricipaion by he old generaion. The lower wage income when young in urn force his generaion o work in period +, which depresses he nex generaion s wage income. Thus, a emporary echnology shock propagaes across periods. Anoher possibiliy is ha populaion growh may be subjec o shocks. A baby boom generaion migh have o work longer, which affecs he nex generaion. Similarly, he impacs of a large inflow of migraion in one period may affec no only he curren generaion, bu also he fuure generaions. Likewise, wars, plagues and oher shocks ha decimae one generaion could boos he wage income for he nex generaion so much ha he economy may escape from he lower seady sae, and all he fuure generaions may enjoy higher sandard of living. (For example, some hisorians sugges ha Black Deah se up he sage for he fuure European Miracle, by boosing he wage income.) I should be poined ou ha he presen model has neiher exernaliies nor dynamic inefficiencies. The above argumen merely suggess ha he loss in one generaion is accompanied by he gains in he fuure generaions, and does no imply any Pareo improvemen. Third, he model can be applied o he analysis of social securiy. The previous conribuions on his subjec, such as Hu (979), made assumpions ha ensure he uniqueness of he seady sae, which would imply ha policy could only change he seady sae locally, hereby imposing subsanial resricions on poenial impacs of social securiy. 9
Fourh, he preferences used in he above analysis assume ha he income effec of a higher wage, which encourages early reiremen, dominaes he price effec of higher wage, which discourages early reiremen. This feaure may be useful for capuring some poliical economy aspecs of he social securiy sysem, because i suggess ha here are sronger poliical demands for social securiy in more developed counries. I is hoped ha he model presened in his paper will simulae furher research along hese lines. 20
References: Blanchard, O.J., and S. Fischer (989) Lecures on Macroeconomics, Cambridge, MIT Press. Cosa, D. L. (998) The Evoluion of Reiremen: An American Economic Hisory, 880-990, Chicago, Universiy of Chicago Press. Diamond, P. A. (965) Naional Deb in a Neoclassical Growh Model, American Economic Review, Vol. 55, pp. 026-050. Feldsein, M. S. (974) Social Securiy, Induced Reiremen, and Aggregae Capial Accumulaion, Journal of Poliical Economy, Vol. 82, pp. 905-926. Fuchs, V. R. (983) How We Live: An Economic Perspecive on Americans from Birh o Deah, Cambridge, Harvard Universiy Press. Gong, L., and N. Liu (2006), One Secor Neoclassical Growh Model wih Endogenous Reiremen: A Coninuous Case, unpublished ypescrip, Guanghua School of Managemen, Peking Universiy. Gruber, J., and D. Wise, eds. (999) Social Securiy and Reiremen Around he World, Chicago, Universiy of Chicago Press. Hu, S. C. (979) Social Securiy, he Supply of Labor, and Capial Accumulaion, American Economic Review, Vol. 69, pp. 274-283. Masuyama, K. (2005) Povery Traps, forhcoming in L. Blume and S. Durlauf eds., The New Palgrave Dicionary of Economics, 2 nd Ediion, Macmillan. Reichlin, P. (986) Equilibrium Cycles in an Overlapping Generaions Economy wih Producion, Vol. 40, pp. 89-02. Romer, P. M. (986) Increasing Reurns and Long Run Growh, Journal of Poliical Economy, Vol. 94, pp. 002-037. 2
Figure : Growh Dynamics wih Exogenous Reiremen K + = w 45 x = 0 x = O K = w 22
Figure 2: Growh Dynamics wih Endogenous Reiremen K + = w 45 x = 0 Figure 2a: (αµ < ) x = O w w + K = w K + = w 45 x = 0 Figure 2b: (αµ > ) x = O w w + K = w 23
Figure 3: Growh Dynamics wih Endogenous Reiremen A Case of Heerogeneous Agens w 45 x = 0 x = O w 24