A Note on Raising the Mandatory Retirement Age and. Its Effect on Long-run Income and Pay As You Go (PAYG) Pensions

Transcription

1 The Sociey for Economic Sudies The Universiy of Kiakyushu Working Paper Series No (acceped in March, 2018) A Noe on Raising he Mandaory Reiremen Age and Is Effec on Long-run Income and Pay As You Go (PAYG) Pensions Jumpei Tanaka * The Universiy of Kiakyusyu Absrac Fani (2014) showed ha raising he mandaory reiremen age always reduces capial accumulaion and may lower per young income and pension benefi, under he assumpion ha old and young labor is homogenous (namely, perfec subsiues). However, empirical sudies cas doub on his assumpion. Thus, in his paper, we reexamine his analysis by assuming ha he wo labors are heerogeneous (namely, imperfec subsiues), and prove ha his resuls no longer hold when he elasiciy of subsiuion is no sufficienly high. * j-anaka@kiakyu-u.ac.jp 1

2 1. Inroducion Faced wih rapid populaion aging and resuling fiscal pressure on he social securiy sysem, many developed counries seek o raise boh he eligibiliy age of pension benefi and he mandaory reiremen age. Generally, hese policies are considered unavoidable in order o miigae he slowdown of economic growh caused by he rapid drop in he producion-age populaion and o srenghen he susainabiliy of he social securiy sysem. However, is such a convenional view really righ? Fani (2014) heoreically invesigaed his quesion by using a simple overlapping generaions model, where he old households allocaed a par of heir endowed ime o he labor supply, and demonsraed ha raising he mandaory reiremen age always reduces capial accumulaion and lowers per young income and pension benefi when he capial share is sufficienly high. If his resul is valid, such a policy is harmful o boh economic growh and fiscal susainabiliy of he pension sysem, conrary o he convenional view. The purpose of his paper is o reexamine his resul. Fani (2014) derived he resul by assuming ha old and young labor is homogenous (namely, perfec subsiues). However, a considerable amoun of empirical research cass doub on his assumpion. Gruber e.al (2010) invesigaed he relaion beween he labor force paricipaion of he old and he young in welve OECD counries and showed ha he labor paricipaion of he young is no negaively bu raher posiively associaed wih ha of he old. The same conclusions are found in he subsequen researches by Zhang (2012), Munnel and Wu (2012), and Kondo (2016), who respecively sudied he cases of China, USA, and Japan 1. These resuls imply ha i is more 1 Cerainly, no all empirical sudies suppor such a resul. Marins e.al (2009) showed 2

3 realisic o assume ha old and young labor is heerogeneous (namely, imperfec subsiues). Taking hese empirical evidences ino accoun, we reexamine Fani s (2014) analysis by adoping he more general CES echnology for he wo inpus (old labor and young labor). Under such a specificaion, Fani s (2014) analysis corresponds o a special case where is elasiciy of subsiuion is infinie. I shows ha he resuls derived by Fani (2014) are valid only when he elasiciy of subsiuion is sufficienly high. In oher words, if i is no sufficienly high, which seems likely, raising he mandaory reiremen age is a proper policy because i simulaes capial accumulaion and raises per-young income and pension benefi. 2. Model and Resul The model examined here is almos he same as ha used by Fani (2014) excep ha he more general producion funcion is assumed, and hus, we describe he srucure of he model only briefly. As for he household secor, he individuals who are born a ime are homogenous and is populaion (N ) grows a a consan rae of n. The uiliy maximizaion problem of he individual is formulaed as follows: max y c,c+1 o s.. c y + s = (1 τ)w y o, c +1 U = ln c y o + γ ln c +1 = R +1 s + (1 τ)w o +1 λ + z +1 (1 λ) where c y o is he young-aged consumpion, c +1 is he old-aged consumpion, s is he ha in Porugal, firms employing older female workers significanly reduced he hiring of younger female workers by a gradual increase in he legal reiremen age of female workers. Vesad (2013) found almos one-o-one replacemen for reired elderly workers and newly hired young workers in Norway. 3

4 savings, 0 < γ < 1 is he subjecive discoun facor, w y is he young worker s real wage, o w +1 is he old worker s real wage, τ is he social securiy ax rae, and 0 λ 1 is a fracion of ime devoed o he labor supply a he old period. Noe ha in our model, an old worker s real wage is no equal o a young worker s wage because he wo workers are assumed o be heerogeneous (namely, imperfec subsiues). By solving he above problem, we have he following savings funcion: s = γ 1 + γ (1 τ)w y γ (1 τ)w o +1 λ + z +1 (1 λ). (1) R +1 As for he public secor, he governmen runs he Pay As You Go (PAYG) pension sysem, and he budge consrain a ime is given by z (1 λ)n 1 = τw y N + τw o λn 1, (2) where z is he pension benefi. The lef-hand side of (2) represens he social securiy expendiure and he righ-hand side represens he ax receips. As for he producion secor, firms are supposed o be idenical and ac compeiively. We assume he following producion funcion: Y = K a b L y ρ + (1 b)(l o ) ρ ρ ( < ρ 1), (3) where Y is he oupu, K is he capial, L y is he young labor, and L o is he old labor. The elasiciy of subsiuion beween L y and L o is σ = 1/(1 ρ). If ρ = (or σ = 1) holds, L y and L o are perfec subsiues, which correspond o he case examined by Fani (2014). If ρ = 0 (or σ = 1 ) holds, he producion funcion reduces o he Cobb Douglas ype funcion, i.e., Y = K a b() (1 b)() L 1, L2,. Under he assumpion ha capial fully depreciaes a he end of each period, he firs order condiions of profi maximizaion are as follows: 4

5 R = a K a 1 N b L y ρ + (1 b) Lo ρ ρ, (4.a) N N w y = b(1 a) K a N L y ρ 1 N w o = (1 b)(1 a) K a Lo ρ 1 N N b L y ρ + (1 b) Lo ρ ρ 1, N N b L y ρ + (1 b) Lo ρ ρ 1, N N (4.b) (4.c) where R is he gross ineres rae. Having formulaed he behaviors of individuals, he governmen, and firms, we can derive he equilibrium dynamics of he economy. The equilibrium condiions of he labor markes (young and old) are respecively given as follows: N = L y, λn 1 = L o. (5) Subsiuing (5) and N +1 = (1 + n)n ino (4.a), (4.b), and (4.c), we ge he following: R = ab(λ) ρ k a 1, w 1, = b(1 a)b(λ) ρ 1 k a, w 2, = (1 b)(1 a) λ ρ n where B(λ) = b + (1 b) λ ρ 1 + n B(λ) ρ 1 k a,, k = K N. (6.a) (6.b) (6.c) The equilibrium condiion of he capial marke is expressed by he equaion: (1 + n)k +1 = s. Subsiuing (1), (2), (6.a), (6.b), and (6.c) ino his condiion and arranging, we can derive he dynamics of capial accumulaion: where A 1 = k +1 = A ρ 1B(λ) k a A 2 B A, (7) 3 b(1 a)(1 τ)γ (1 + n)(1 + γ), A 2 = a a(1 + γ), A 3 = (1 a)b(1 τ). a(1 + γ) From (7) we can easily confirm ha he seady sae exiss uniquely and is globally 5

6 sable. The seady sae level of per young capial is k = A ρ 1B(λ) A 2 B(λ) A 3 1. (8) In he seady sae, wha effecs does a rise in λ (he mandaory reiremen age) have on per young capial (k ), income (y ), and pension benefi (z )? Concerning he effec on k, we have k λ 0 when Ω 1 = B (λ) 1 ρ A 2B A 3 A 2 B 1 0. (9) 1 a I can be easily confirmed ha Ω 1 > 0 holds in he case of ρ 0, which means ha in such a case k λ > 0 always holds. In he case of ρ > 0, on he oher hand, he condiion Ω 1 0 can be rewrien as follows: ρ ρ ρ = (1 a) A 2B(λ) A 3, 0 < ρ A 2 B(λ) < 1. (10) Thus, we have he following resul: Resul 1: When ρ (he elasiciy of subsiuion beween young labor and old labor) is smaller han he criical value ρ (0 < ρ < 1), raising he mandaory reiremen age has a posiive impac on per young capial. Figure 1 below illusraes his resul. From (10), we can see ha a rise in λ lowers k only when ρ (he elasiciy of subsiuion beween young labor and old labor) is higher han he criical value ρ (0 < ρ < 1).Therefore, if he wo labors are perfec subsiues (ρ = 1), which correspond o he case invesigaed by Fani (2014), a rise in λ has a negaive impac on k. However, he opposie resul holds if ρ is no so high. For example, in he case of ρ = 0 (Cobb Douglas producion funcion), such a change 6

7 posiively affecs k. When ρ is relaively small, why does a rise in λ have he posiive impac on k? The inuiive reason is as follows. From (6.b) and (6.c), we have he following: (w y ) (w o ) = b 1 ρ λ 1 b 1 + n. This means ha a rise in λ generally widens he wage gap beween he wo labors and is exen is larger when ρ is smaller. Namely, when ρ is relaively small, a rise in λ causes a relaively large increase in w y and a relaively large decrease in w o. Such a change in he wage profile induces each individual o save more, which resuls in a higher per young capial. (Figure 1 around here) Nex, we discuss he effec of a rise in he mandaory reiremen age (λ) on per young income (y ). Subsiuing (8) ino (3) and arranging, is seady sae level is y = B(λ) ρ (k ) a = Afer some calculaions, we obain he following: y a A 1 1 B(λ) ρ (A 2 B(λ) A 3 ) a λ 0 when Ω 2 = B (λ) 1 ρ A 2B A 3 A 2 B. (11) a 0. (12) 1 a I can be easily confirmed ha Ω 2 > 0 holds in he case of ρ 0, which means ha in such a case, y λ > 0 always holds, whereas in he case of ρ > 0, he condiion Ω 2 0 can be rewrien as follows: ρ ρ a, (13) where ρ is he criical value appeared in (10). Accordingly, he following resul holds. 7

8 Resul 2: When ρ (he elasiciy of subsiuion) is smaller han he criical value ρ /a (> ρ ), raising he mandaory reiremen age has a posiive impac on per young income. This resul means ha a rise in λ lowers y only when ρ (he elasiciy of subsiuion) is higher han he criical value ρ a, which is a larger value han ρ because is denominaor is he capial share (0 < a < 1). Therefore, if he capial share is sufficienly large and accordingly ρ a < 1 holds, a rise in λ lowers y in he case of ρ = 1, as shown by Fani (2014) 2. Figure 2 illusraes his case. However, when ρ is smaller han ρ a, he opposie resul holds. Furhermore, as depiced in Figure 2, he range of ρ under which y λ > 0 holds becomes broader han he range under which k λ > 0 holds. The inuiive reason is as follows. From (11), a rise in λ affecs y hrough wo channels: per young capial (k ) and elderly labor supply (B(λ)). As he laer effec is always posiive, y λ > 0 holds in he broader range of ρ han he range under which k λ > 0 holds. (Figure 2 around here) Finally, we discuss he effec of a rise in he mandaory reiremen age (λ) on per-young pension benefi (z ). From (2), (6.b), (6.c), and (8), is seady sae level can be derived as follows: 2 On he conrary, if he capial share is no sufficienly large and herefore ρ a > 1 holds, a rise in λ always has a posiive impac on y. 8

9 z = τ(1 + n)(wy ) + τλ(w o ) 1 λ = (1 + n)(1 a) y 1 λ (14) = 1 A 4 B(λ) ρ (1 λ)(a 2 B(λ) A 3 ) a A 4 = τ(1 + n)(1 a)a 1 a. Afer some calculaions, we have z λ 0 when Ω 3 = (1 λ)b (λ) 1 ρ A 2B A 3 A 2 B a 1 a + B(λ) A 2B A 3 0 (15) A 2 B I can be easily confirmed ha Ω 3 > 0 holds in he case of ρ 0, which means ha in such a case, z λ > 0 always holds. We can also show ha Ω 3 > 0 always holds, even in he case of ρ > 0 if he following condiion is saisfied. ρ a (1 λ)b (λ) B (16) Here, ρ a is he criical value appeared in (13). On he oher hand, if (16) is no saisfied, he condiion Ω 3 0 can be rewrien as follows: ρ ρ ρ = ρ a B(λ) 1 (1 λ)b (λ) ρ a > ρ a. (17) Summarizing hese poins, we have he following resul. Resul 3: If (16) is saisfied, raising he mandaory reiremen age always has a posiive impac on per young pension benefi. Even if (16) is no saisfied, when ρ (he elasiciy of subsiuion) is smaller han he criical value ρ (> ρ /a > ρ ), he same resul holds. From Resul 3, we can see ha a rise in λ lowers z only when (16) is no saisfied and ρ is higher han ρ. Therefore, if ρ < 1 holds, raising he mandaory reiremen age has a negaive impac on z (namely, i worsens he susainabiliy of pension 9

10 sysem) in he case of ρ = 1, as poined ou by Fani (2014) 3. Figure 3 illusraes his case. However, when ρ is smaller han ρ, he opposie resul holds. As depiced in Figure 3, he range of ρ under which z λ > 0 holds becomes broader han he range under which y λ > 0 holds. The inuiive reason is as clear. As z = (1 + n)(1 a) y 1 λ holds from (14), a rise in λ affecs z hrough wo channels: he numeraor (y ) and he denominaor (1 λ). As he laer effec is always posiive, z λ > 0 holds in he broader range of ρ han he range under which y λ > 0 holds. (Figure 3 around here) 3. Conclusion Fani (2014) demonsraed ha raising he mandaory reiremen age always reduces capial accumulaion and may lower per young income and pension benefi under he assumpion ha young labor and old labor are perfec subsiues. However, we proved ha he opposie resul holds in he more realisic assumpion ha he wo labors are imperfec subsiues. Our resul indicaes ha he convenional view ha raising boh he eligibiliy age of pension benefi and he mandaory reiremen age is necessary is proper. References Fani, L. (2014) Raising he mandaory reiremen age and is effec on long-run 3 On he oher hand, if ρ > 1 holds, a rise in λ always has a posiive impac on z. 10

11 income and pay-as-you-go (PAYG) pensions, Meroeconomica, Vol.65 (4), pp Gruber, J. and D. A. Wise (Eds.) (2010) Social Securiy Programs and Reiremen around he World: The Relaionship o Youh Employmen, Universiy of Chicago Press. Kondo, A. (2016) Effecs of increased elderly employmen on oher worker s employmen and elderly s earnings in Japan, IZA Journal of Labor Policy, Vol.5 (2) Marin, P. S., Novo, A. A. and Porugal, P. (2009) Increasing he legal reiremen age: he impac on wages, worker flows and firm performance, IZA Discussion Paper, No.4187 Munnel, A. H. and Wu, A. Y. (2012) Will delayed reiremen by he baby boomers lead o higher unemploymen among younger workers?, Cener for Reiremen Research a Boson College Working Paper Vesad, O. L. (2013) Early reiremen and youh employmen in Norway, Presened a European Associaion of Labour Economics Zhang, C. (2012) The relaionship beween elderly employmen and youh employmen: evidence from China, Universiy Library of Munich. MPRA Paper

12 ρ = 0 ρ = ρ ρ = 1 (Cobb Douglas) (perfec subsiues) k λ > 0 k λ = 0 k λ < 0 (Figure 1: The effec of a rise in he mandaory reiremen age (λ) on per young capial (k )) ρ = 0 ρ = ρ ρ = ρ /a ρ = 1 (Cobb Douglas) (perfec subsiues) y λ > 0 y λ = 0 y λ < 0 (Figure 2: The effec of a rise in he mandaory reiremen age (λ) on per young income (y )) ρ = 0 ρ = ρ ρ = ρ /a ρ = ρ ρ = 1 12

13 (Cobb Douglas) (perfec subsiues) z λ > 0 z λ = 0 z λ < 0 (Figure 3: The effec of a rise in he mandaory reiremen age (λ) on per young pension benefi (z )) 13