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1 ,, A B C A (B Negishi(1960) (C) ( ) fujii@econ.kobe-u.ac.jp 082e527e@stu.kobe-u.ac.jp iritani@econ.kobe-u.ac.jp 1

2 Auerbach and Kotlikoff(1987) (1987) (1988) 4 (2004) 5 Diamond(1965) Auerbach and Kotlikoff(1987) 1 ( ) , , , % 4 (2001) 2000 (2007) (2001) 2

3 3 A B C (A) (B) (C) 6 (1988) (1999) 2004 (2005) 7 (1988) ( ) (2004) 8 (2001) (2001) 3

4 3 ( ) Negishi(1960) ( ) ( ) Negishi A B C (1988) (1996) Kato(1998) (1988) 9 9 4

5 (1996) ( ) Kato(1998) (1988) (2001) (2004) (2008) (1988) (2001) 10 ( ) (2004) (2001) ( ) 3 11 Shimono and Tachibanaki(1985) (1990) (2001) (2002a) (2006) (2008) Shimono and Tachibanaki(1985) (2004) 5

6 (1990) (2001) 12 (2002a) 13 (2006) (2008) 14 A B C (2002a) 14 (2006) (2008) 6

7 3.1 W τw T bw B r (1 τ)w T + bw + B 1 + r (1) (1 τ)w T (bw + B)/(1 + r) t j W j t t T t, B t t (τw j t + T t) = (bw j t 1 + B t) (2) j j 15 τ, b (2) τ, T t, b, B t h a 1, a 2,, a h, (a j < a j+1 ) d 1, d 2,, d h h d i = 1, d i 0 a j, d j t N t a j d j N t, j = 1, 2..., h t L t L t = (a 1 d 1 + a 2 d a h d h )N t = a j d j N t (3) 15 7

8 3.2 K L F (K, L) F 2 F 1 ( ) F K = F K (K, L) > 0, F L = F L (K, L) > 0, F KK = 2 F K 2 (K, L) < 0, F LL = 2 F L 2 (K, L) < 0, f(k) = F (k, 1), k = K/L f (k) > 0 f (k) < 0 F (K, L) lim f (k) = k 0 lim f (k) = 0. k 1 w t, r t t t F K = F K (Kd t, L d t ) = r t, F L = F L (Kd t, L d t ) = w t (4) (Kt d, L d t ) L d t = L t (4) 1 Kt d (r t ) 2 w t k = K/L Lf(k) = F (K, L) f(k) kf (k) = F L f(k(r t )) k(r t )f (k(r t )) = w t, k(r t )L t = K d t (r t ) 1 w t F K (Kt d (r t ), L t ) = r t dk d t dr t < 0, dw t dr t < 0 (5) 8

9 3.3 t t N t 1, N t S t 1 K t (= S t 1 ) L t r t w t t a i Wt i = a i w t (1 τ)a i w t T t a i ba i w t 1 + B t r t s t 1 t (1 τ)a i w t T t d i N t, i = 1,..., h ba i w t 1 + B t + r t s i t 1 d in t 1, i = 1,..., h u(c t, c t+1 ) u : R 2 + R 2 2 ( ) (1) R 2 + (quasi-concave) (2) x 1 0, x 2 0, u(x 1, 0) = u(0, x 2 ) = inf{u(x 1, x 2 ) (x 1, x 2 ) R 2 +} a i t max u(c yi t, coi t+1) sub to c yi t + s i t = (1 τ)a i w t T t, R2 ++ c oi t+1 = (1 + r t+1)s i t + ba i w t + B t+1 (6) c y t, co t+1 t a i c y t co t+1 si t s i t t + 1 9

10 max u(c yi t, coi t+1) sub to c yi t + coi t r t+1 = I i t (7) I i a i 16 I i t def 1 = (1 τ)a i w t T t + (ba i w t + B t+1 ) (8) 1 + r t+1 r t+1, w t, T t B t+1 It(r i t+1, w t, T t, B t+1 ) c o t = c oi t d i N t 1 = { (1 + rt )s i } t 1 + ba i w t 1 + B t di N t 1 (τa i w t + T t )d i N t = (ba i w t 1 + B t )d i N t 1 (9) 3.4 t (c yi t + s i t)d i N t + c oi t d i N t 1 = F (K t, L t ) + K t (10) ( (a i w t τa i w t T t )d i N t + bai w t 1 + B + (1 + r t )s i ) t 1 di N t 1 = a i w t d i N t +(1 + r t ) (τa i w t + T )d i N t + s i t 1d i N t 1 = w t (ba i w t 1 + B)d i N t 1 a i d i N t + (1 + r t )K t = w t L t + r t K t + K t = F (K t, L t ) + K t = 16 def = 10

11 t t + 1 r t+1 t + 1 L t+1 = a i d i N t+1 K t+1 F (K t+1, L t+1 ) = r t+1 K t+1 Kt+1 d (r t+1) h si t(r t+1 )d i N t K d t+1(r t+1 ) = s i t(r t+1 )d i N t (11) r t+1 1 t = 1 K 1, w 0, s i 0, N t(t = 0, 1, 2,... ), d i, a i τ, b, B t (t = 0, 1, 2,... ), 1 w 0 h si 0 (r 1)d i N 0 F K (K 1, L 1 ) r 1 r 1 = F K (K 1, L 1 ) r 1 w 1 = F L (K 1, L 1 ) (τa i w 1 + T 1 )d i N 1 = (ba i w 0 + B 1 )d i N 0 T 1 1 t 1 t 2 K d 2 (r 2 ) = h si 1 (r 2)d i N 1 (12) 11

12 r2 2 K 2 = K2 d(r 2 ) 2 t 2 r t, L t, K t (4) (9) (11) F L (Kt d, L t ) = w t (τa i w t + T t )d i N t = (ba i w t 1 + B t )d i N t 1 K d t+1(r t+1 ) = s i t(r t+1 )d i N t w t, T t, r t+1 Kt+1 d (r t+1) F K (Kt+1 d, L t+1) = r t+1 τ, b, B t d i, a i, N t (t = 0, 1, 2... ) w t, T t (11) (12) F K (K t+1, L t+1 ) = r t+1 Kt+1 d (r t+1) k(r t+1 ) = Kt+1 d (r t+1)/l t+1 Kt+1 d (r t+1) = L t+1 k(r t+1 ) r t+1 = f (k(r t+1 )), 1 = f dk dr t+1 r t+1 0 k(r t+1 ) r t+1 k(r t+1 ) 0 t+1 h si t(r t+1 )d i N t = S t (r t+1 ) 12

13 3 ds t (r t+1 )/dr t+1 0 ˆr S t (ˆr) > 0 r t+1 S t (r t+1 ) > Kt+1 d (r t+1) r t+1 Kt+1 d (r t+1) > S t (r t+1 ) r t+1 Kt+1 d (r t+1) < S t (r t+1 ) r Kt+1 d (r ) = S t (r ) t s i t = s i, c y t = cy, c o t = c o, K t = K, N t = N, L t = L, w t = w, r t = r, T t = T, B t = B, I i t = I i t = 1, 2, L = h a id i N r F K (K d, L) = 0, w F L (K d, L) = 0 (13a) (τa i w + T )d i N K d (ba i w + B)d i N = 0 s i (w, T, r, B)d i N = 0 (13b) (13c) t = 1 t = 2 18 Diamond(1965) 13

14 19 (w, r, K d, T ) B c y (r, I j ), c o (r, I j ) (13a) (13c) 4 ( ) τ > b. 4 (13b) (τ b) h a iwd i = B T 4 B > T a 13c (13a) K d (r) 13a 2 w = F L (K d (r), L) = w(r) (13b) (13c) 13c 1 + r (τa i w(r) + T )d i (ba i w(r) + B)d i = 0 (1 + r)k d (r) (1 + r) s i (w(r), T, r, B)d i N = 0 (14a) (14b) r, T 2 (7) I i def = (1 τ)a i w(r) T + ba iw(r) + B, i = 1, 2,..., h 1 + r (1 + r)s i = c oi (r, I i ) (ba i w + B), S = s i d i N 19 s i 14

15 14b (1 + r)k d (r) { c oi ( r, I i) (ba i w(r) + B) } d i N = 0 (15) 14a 15 r, T h (τ b)a dw id i 1 J = dr ( K d + (1 + r) dkd dr S + (1 + r) S ) h c oi I i r I i T d in K d = S J dw J = (τ b)a i d i dr { =K d (τ b) c r (ε + η) r c oj ( dk d I j d jn (1 + r) dr } S ) r c, ε, η c = c oj I j d j, ε = r dk d K d dr η = r S S r 0 < ε, 0 < η J (14a) (14b) r, T ψ 1 (r, T ), ψ 2 (r, T ) ṙ = ψ 2 (r, T ), T = ψ 1 (r, T ) J > 0 5 J >

16 (B) (T ) 5 6 c > Negishi(1960) 3. 3 r, I j, w, K, T, c k j, k = y, o B Ĩ j (B) = I j (r(b), w(b), T (B), B), w(b) = w(r(b)) c kj (B) = c kj (r(b), Ĩj (B)), k = y, o, K(B) = K(r(B)) 16

17 J = 0 r, T B 14a 15 dr J db 1 dt = ( ) h c oi 1 db I i 1 + r 1 d i N 5 ( 1 1 ) c oi dr db = 1 + r I i d in + N > 0 (16) J (14a) (16) dt db = dw dr (τ b)a j dr db d j + 1 > 1 (17) d K(B) db = dk dr dr db = 1 dr F KK db < 0 (18) 2 5, 6 B B c yj (B)d j N + K(B) = c oj (B)d j N = F ( K(B), L) s j (B)d j N, s j (B) def = (1 τ)a j w(b) T (B) c yj (B) B (17) (1 + r) d c yj db d j + d c oj db d j = r d w (1 τ)a j db d j r dt db < 0 (19) 17

18 j B V j (B) = u j ( c yj (B), c oj (B)) j dv j db = λ d c yj j db + λ j r d c oj db λ j j Negishi(1960) α j u j ( c yj, c oj )d j N, α j = 1/λ j, j = 1, 2,..., h (20) (19) d db α j u j ( c yj (B), c oj (B))d j N = r ) ((1 + r) d cyj db + d coj d j N < 0 db 3 5, 6 Negishi(1960) Negishi (1960) (20) 20 Ĩ j µ, σ, CV B 21 Ĩ i def = (1 τ)a i w(b) T (B) + ba i w(b) + B, i = 1, 2,..., h 1 + r(b) (2005) (2006) (2008) 18

19 τ b > 0 dr/db > 0 dt db = dw dr (τ b)a j dr db d j + 1 > 1 (( dĩj db = 1 τ + b 1 + r ) dw a j dr ba jw + B (1 + r) 2 ) dr db r dt db < 0 (21) B dµ/db < 0 1 dσ 2 2 db < 0 (22) 22 CV = h ) 2 (Ĩj d j 1 µ B CV 2 dcv 2 db = 2 (( ) ( )) Bµ 3 φ(b)2 2 ρ(b)σ a Bρ (B) Bφ (B) ρ(b) φ(b) (23) 23 φ(b) = (1 τ)w + bw/(1 + r), φ d((1 τ)w + bw/(1 + r)) (B) =, db ρ(b) = T + B/(1 + r), ρ d( T + B/(1 + r)) (B) = db Ĩ i = φ(b)a i + ρ(b), i = 1, 2,..., h φ(b) a i w ρ(b) φ(b) >

20 0, φ (B) < 0, ρ (B) < 0 ρ(b) Bφ /φ B Bρ /ρ B 4 5, 6 ρ < 0 ρ > 0 Bφ /φ > Bρ /ρ, 6 A B C 3 (2004) (2002) (2002) 20

21 Negishi(1960) (2002) (2004) 24 2 (2002b) r w (2002) (2004) 21

22 5 (22) (23) B Ĩ i Ĩ j ( Ĩ j = (1 τ) w + b w ) a j T + B 1 + r 1 + r = φ(b)a j + ρ(b) φ(b) > 0 dĩi /db, dµ/db (( 1 τ + b φ (B) = dφ ) dw db (B) = 1 + r dr ρ (B) = dρ db (B) = B dr (1 + r) 2 db r dt db dĩj db = φ (B)a j + ρ (B), dµ db = φ (B)ā + ρ (B) bw (1 + r) 2 ) dr db ā = h a id i φ (B) < 0, ρ (B) < 0 σ 1 dσ 2 2 db = I i dii db d i I i dµ db d i ( = φ I i a i d i + ρ µ φ āµ ρ µ = φ ā I i a id i ā µ ( Ĩ j = Ĩ1 + (1 τ) w + b w ) (a j a 1 ) = 1 + r Ĩ1 + φ(b)(a j a 1 ), j = 1, 2,..., h Ĩ i a id i ā = Ĩ 1 a id i ā µ = + φ ( Ĩ i d i = Ĩ1 + φ(b) ( ) ( ) ai d i ā a a i d i i a 1 = Ĩ1 + φ(b) ā a i a 1 ) a i d i a 1 ) a i d i ā a i = 1 ā a 2 i d i = 1 ā (ā2 + σ 2 a ) > ā = a i d i 22

23 σ a a j, j = 1, 2,..., h Ĩ i a id i ā > µ (24) φ (B) < 0 1 dσ 2 2 db < 0 25 B dcv 2 db = 2 µ 3 d i d j Ĩ j Ĩ i Ĩi B d i d j (Ĩj ) 2 Ĩi (25) B (25) d i d j Ĩ j Ĩ i Ĩi B = µ d i Ĩ i Ĩi B = µ d i Ĩ i (φ a i + ρ ) = µφ d i a i Ĩ i + µ 2 ρ = µφ ( āĩ1 + φ(b) ( ā 2 + σ a 2 a 1 ā )) + µ 2 ρ (25) d i d j (Ĩj ) 2 Ĩi B = d j (Ĩj ) 2 d i Ĩi B = d j (Ĩj ) 2 d i (φ a i + ρ ) = (µ 2 + σ 2 )(φ ā + ρ ) = φ µ 2 ā + ρ µ 2 + φ σ 2 ā + ρ σ 2 dcv 2 db = 2 ( ( µ 3 µφ āĩ1 + φ(b) ( ā 2 + σ 2 a a 1 ā )) ) (φ µ 2 ā + φ σ 2 ā + ρ σ 2 ) µ = φā + ρ = Ĩ1 + φ(b)(ā a 1 ), σ 2 = φ 2 σ a 2. dcv 2 db = 2 ) ( µ 3 B φ2 2 ρσ a (( Bρ Bφ )) ρ φ

24 [1] (1996),,pp [2] (2002),No.37,pp [3] (2001),42,pp [4] (2002a),10,pp [5] (2002b) -,28,pp [6] (2004),167,pp.1-17 [7] (2008) -,43,pp [11] (2004) Discussion Paper,No.408,pp.1-19 [11] (2005) [11] (2006) -,, 3 [11] (2008),, 6 24

25 [12] (2005),pp [13] (2007) ,43,pp [14] (1990), 118,pp [15] (1987) 6,pp [16] (1988),pp.1-15 [17] (2001),37,pp [18] (1999) [19] Auerbach, A. and Kotlikoff, L.J.(1987), Dynamic Fiscal Policy, Cambridge University Press. [20] Diamond, P.A.(1965), National Debt in a Neoclassical Growth Model, American Economic Review, Vol.55, No.5, pp [21] Kato, R.(1998), Transition to an Aging Japan Public Pension, Savings, and Capital Taxation, Journal of the Japanese and International Economies, Vol.12, pp [22] Negishi, T.(1960), Welfare Economics and Existence of an Equilibrium for a Competitive Economy, Metroeconomica, Vol.12, No.2-3, pp [23] Shimono, K. and Tachibanaki, T.(1985), Lifetime Income and Public Pension An Analysis of the Effect on Redistribution Using a Two-period Analysis, Journal of Public Economics, Vol.26, pp

(2004 ) 2 (A) (B) (C) 3 (1987) (1988) Shimono and Tachibanaki(1985) (2008) , % 2 (1999) (2005) 3 (2005) (2006) (2008)

(2004 ) 2 (A) (B) (C) 3 (1987) (1988) Shimono and Tachibanaki(1985) (2008) , % 2 (1999) (2005) 3 (2005) (2006) (2008) ,, 23 4 30 (i) (ii) (i) (ii) Negishi (1960) 2010 (2010) ( ) ( ) (2010) E-mail:fujii@econ.kobe-u.ac.jp E-mail:082e527e@stu.kobe-u.ac.jp E-mail:iritani@econ.kobe-u.ac.jp 1 1 16 (2004 ) 2 (A) (B) (C) 3 (1987)

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