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1 ,, (i) (ii) (i) (ii) Negishi (1960) 2010 (2010) ( ) ( ) (2010) [email protected] [email protected] [email protected]

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

3 (A), (B) Auerbach and Kotlikoff(1987) (2004) (2007) Diamond(1965) (1988) ( ) (2004) 5 (2001) (2001) 2

4 3 3 (A) (B) (C) 4 ( ) Negishi(1960) ( ) Negishi(1960) ( ) Negishi(1960) 3

5 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, B t t (τ t W j t + T ) = (bw j t 1 + B t) (2) j j 6 T, b (2) τ t, T, b, B t h a 1, a 2,, a h, (a j < a j+1 ) 6 4

6 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) 2.2 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) 5

7 (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 (K d t (r t ), L t ) = r t dk d t dr t < 0, dw t dr t < 0 (5) 2.3 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 t )a i w t T a i ba i w t 1 + B t r t s t 1 t (1 τ t )a i w 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 6

8 2 ( ) (1) R 2 + (quasi-concave) R2 ++ (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, c yi coi t + s i t = (1 τ t )a i w t T, t+1) sub to (6) c oi t+1 = (1 + r t+1)s i t + ba i w t + B t+1 c y t, co t+1 t a i c y t co t+1 si t s i t t + 1 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 I i t def 1 = (1 τ t )a i w t T + (ba i w t + B t+1 ) (8) 1 + r t+1 r t+1, w t, τ t B t+1 I i t(r t+1, w t, τ 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 (τ t a i w t + T )d i N t = (ba i w t 1 + B t )d i N t 1 (9) 7

9 3 B τ t (c yi t + s i t)d i N t + = (a i w t τa i w t T t )d i N t + a i w t d i N t +(1 + r t ) c oi t d i N t 1 = F (K t, L t ) + K t (10) (τa i w t + T )d i N t + s i t 1d i N t 1 = w t ( bai w t 1 + B + (1 + r t )s i ) t 1 di N t 1 (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 = 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 (K t+1, L t+1 ) = r t+1 K d t+1 (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 8

10 t = 1 K 1, w 0, s i 0, N t(t = 0, 1, 2,... ), d i, a i T, 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 ) (τ 1 a i w 1 + T )d i N 1 = (ba i w 0 + B 1 )d i N 0 τ 1 1 t 1 t 2 K d 2 (r 2 ) = h si 1 (r 2)d i N 1 (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 (τ t a i w t + T )d i N t = (ba i w t 1 + B t )d i N t 1 Kt+1(r d t+1 ) = s i t(r t+1 )d i N t w t, τ t, r t+1 Kt+1 d (r t+1) F K (Kt+1 d, L t+1) = r t+1 T, b, B t d i, a i, N t (t = 0, 1, 2... ) w t, τ t (11) (12) 9

11 F K (K t+1, L t+1 ) = r t+1 K d t+1 (r t+1) k(r t+1 ) = K d t+1 (r t+1)/l t+1 K d t+1 (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 ) 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 = 1 t = 2 10

12 8 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, τ t = τ, 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, τ, r, B)d i N = 0 (13b) (13c) 9 (w, r, K d, τ) 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) 8 Diamond(1965) 9 s i 11

13 (13b) (13c) 13c 1 + r (τa i w(r) + T )d i (1 + r)k d (r) (1 + r) (ba i w(r) + B)d i = 0 s i (w(r), τ, r, B)d i N = 0 r, τ 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 = 14b (1 + r)k d (r) s i d i N (14a) (14b) { c oi ( r, I i) (ba i w(r) + B) } d i N = 0 (15) 14a 15 r, T h (τ b)a dw h id i J = dr a iw(r)d i ( K d + (1 + r) dkd dr S + (1 + r) S ) h c oi r I i a iwd i N K d = S J J = (τ b)a i d i dw dr c oj I j a jwd j N (1 + r) ( dk d a i w(r)d i dr S ) r J (14a) (14b) r, τ ψ 1 (r, τ), ψ 2 (r, τ) ṙ = ψ 2 (r, τ), τ = ψ 1 (r, τ) J > 0 5 J > 0 12

14 5 (B) (τ) (τ) T ) 5 6 c > Negishi(1960) 3. 3 r, I j, w, K, τ, c kj, k = y, o B Ĩ j (B) = I j (r(b), w(b), τ(b), B), w(b) = w(r(b)) c kj (B) = c kj (r(b), Ĩj (B)), k = y, o, K(B) = K(r(B)) 13

15 5.1 J = 0 r, τ B 14a 15 dr J db 1 dτ = ( ) h c oi 1 db I i 1 + r 1 d i N 5 c oi dr db = I i a iwd i N ( ) c oi 1 a i wd i I i 1 + r 1 d i N J 2 0 < c oi / I i < 1 dr db > 0 (16) (14a) (16) dτ db a i wd i = (b τ) a j dw dr 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 14

16 5.2 Blanchard and Fisher(1989) De la Croix(2002) Heijdra(2009) 1 Negishi(1960) Negishi(1960) 1 2 u i (c yi, c oi ) = {c yi } α i {c oi } β i, i = 1, 2 0 < α i < 1, 0 < β i < 1, α i + β i < 1, i = 1, 2 CK δ L 1 δ, 0 < δ < 1 W Ben = 2 u id i W R = min(u 1, u 2 ) 1 2 τ B W Ben W R 1 G(u i )di, P G(u i ) = exp βu i i β u i 1 ρ a 1 ρ 1 15

17 1: B W Ben W R 1 2 (α 1, β 1 ) (0.2919, ) (0.3496, ) (α 2, β 2 ) (0.6620, ) ( , ) (a 1, a 2 ) (6.834, ) (8.714, ) (d 1, d 2 ) (0.1370, ) (0.2475, ) (C, δ) (10, ) (10, ) (τ, b, B, T ) (0.21,0.2,0.5,0.4) (0.2037,0.2,0.5,0.4) dw Ben db > > 0 dw R db > > 0 Negishi(1960) Negishi(1960) c yj (B)d j N + K(B) = c oj (B)d j N = F ( K(B), L) s j (B)d j N, B (17) (1 + r) d c yj db d j + d c oj db d j = r s j (B) def = (1 τ)a j w(b) T (B) c yj (B) ( dτ ) db a d w j w + (1 τ)a j d j N < 0 db 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) (19) α j u j ( c yj, c oj )d j N, α j = 1/λ j, j = 1, 2,..., h (20) 16

18 (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 Negishi (1960) (20) Negishi Ĩj µ, σ, CV B 11 Ĩ i def = (1 τ(b))a i w(b) T + ba i w(b) + B, i = 1, 2,..., h 1 + r(b) dĩj db = dτ db a jw + (( 1 τ + b ) dw a j 1 + r dr ba ) jw + B dr (1 + r) 2 db r (21) (2005) (2006) (2008) 17

19 τ b > 0 dr/db > 0 (17) dτ/db > 0 (21) B µ = h Ii d i (17) dµ db = = < 0 dĩj db d i (( 1 τ + b ) dw a j 1 + r dr ba ) jw + B dr (1 + r) 2 d j db r dτ a j wd j db 1 dσ 2 2 db < 0 (22) 12 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) 13 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 φ(b) a i w ρ(b)

20 φ(b) > 0, φ (B) < 0, ρ (B) < 0 ρ(b) Bφ /φ B Bρ /ρ B 4 5,6 Bφ /φ > Bρ /ρ, 4 2 τ) B 5.4 B Negishi(1960) B Negishi(1960) B Negishi(1960) B B Negishi(1960) B Negishi(1960) Negishi(1960) 19

21 7 u i (x i ), x i = (c yi, c oi ) m (0 < m < 1) 0 < m < 1 Negishi(1960) u i (x i ) u(x i ) (1, 1/(1 + r), I j ) I j j j i x j c ji (1, 1/(1 + r), I i ) = I i c ji (1, 1/(1 + r)), j = y, o c ji (1, 1/(1 + r)) = c ji (1, 1/(1 + r), 1), j, i v j v j = V (1, 1/(1 + r), I j ) = {I j } m V (1, 1/(1 + r), 1) a j λ j λ j = m{i j } m 1 V (1, 1/(1 + r), 1) Negishi(1960) α j 1 α j = {I j } 1 m {I 1 } 1 m d {I h } 1 m d h I 1 ( B Ik Ik φ ) (B) B I1 = φ(b)ρ(b)(a 1 a k ) φ(b) ρ (B) ρ(b) α 1 B = (1 m)φ(b)ρ(b) ({I 1 } 1 m d {I h } 1 m d h ) 2 B {I 1 } m {I k } m d k (a 1 a k ) α h B = k=2 (1 m)φ(b)ρ(b) ({I 1 } 1 m d {I h } 1 m d h ) 2 B {I h } m {I k } m d k (a h a k ) h 1 k=1 { } ρ (B) ρ(b) B + φ (B) φ(b) B { } ρ (B) ρ(b) B + φ (B) φ(b) B 20

22 j = 1, 2,..., h α j B = (1 m)φ(b)ρ(b) ({I 1 } 1 m d {I h } 1 m d h ) 2 B k j{i j } m {I k } m d k (a j a k ) { } ρ (B) ρ(b) B + φ (B) φ(b) B x j = k j {Ij } m {I k } m d k (a j a k ) x j < x j+1 a 1 < a 2 < < a h x 1 < 0, x h > 0 n, 1 < n < h x n 1 < 0 < x n (i) (ii) B n (1 < n < h) (1)α 1,..., α n 1 (2)α n,..., α h Negishi(1960) (i) 4 2) 4, B negishi 2: 1 2 ( φ B φ, B ρ ) ρ ( , 2.045) ( , 3.470) 6 (A) (B) (C) 14 j x j 0 B 21

23 (A) (B) 1 1 (i) (ii) (i) (ii) Negishi(1960) (ii) (C) 3 1 (2002) (2004) 15 2 (2002b) r w 15 (2002) (2004) 22

24 3 (A) (B) (C) A 5 (22) (23) B Ĩi Ĩj ( 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 (( db + φ (B) = dφ dτ (B) = w db ρ (B) = dρ db (B) = B dr (1 + r) 2 db r 1 τ + b 1 + r ) dw dr bw ) dr (1 + r) 2 db dĩj db = φ (B)a j + ρ (B), dµ db = φ (B)ā + ρ (B) 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 ) 23

25 Ĩ i a id i ā = Ĩ 1 a id i ā µ = + φ ( Ĩ i d i = Ĩ1 + φ(b) a i d i ā a i = 1 ā ( ) ( ) 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 2 i d i = 1 ā (ā2 + σ 2 a ) > ā = a i d i 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 B 16 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 ) (25) d i d j (Ĩj ) 2 Ĩi B = = 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 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

26 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 ρσ a 2 (( Bρ ρ ) ( Bφ )) φ B u(x i ), x i = (c yi, c oi ) m 0 < m < 1 V j λ j V j (1, 1/(1 + r), I j ) = {I j } m V (1, 1/(1 + r), 1), λ j = m{i j } m 1 V (1, 1/(1 + r), 1) α j 1 α j = {I j } 1 m {I 1 } 1 m d {I h } 1 m d h α j B = 1 m ({I 1 } 1 m d {I h } 1 m d h ) 2 ( Ĩ j = (1 τ) w + b w 1 + r ) a j T + Ĩ j = Ĩ1 + φ(b)(a j a 1 ), j = 1, 2,..., h ( ) {I j } m {I k } m I j d k B Ik Ik B Ij k j B 1 + r = φ(b)a j + ρ(b), φ(b) > 0 I j B Ik Ik B Ij = (a j a k ) φ(b)ρ(b) B { } ρ (B) ρ(b) B + φ (B) φ(b) B 25

27 j = 1, 2,..., h α j B = (1 m)φ(b)ρ(b) ({I 1 } 1 m d {I h } 1 m d h ) 2 B k j{i j } m {I k } m d k (a j a k ) { } ρ (B) ρ(b) B + φ (B) φ(b) B x j = k j {Ij } m {I k } m d k (a j a k ) a 1 < a k < a h, k = 2,..., h 1 j 1 x j x j+1 = {I k } m d k ((a j a k ){I j } m (a j+1 a k ){I j+1 } m ) k=1 {I j+1 } m {I j } m d j+1 (a j+1 a j ) + {I j } m {I j+1 } m d j (a j a j+1 ) + k=j+2 k j, j + 1 {I k } m d k ((a j a k ){I j } m (a j+1 a k ){I j+1 } m ) (a j a k ){I j } m (a j+1 a k ){I j+1 } m = {Ij } 1 m {I j+1 } 1 m + I k ( {I j+1 } m {I j } m) φ(b) < 0 x j < x j+1 x 1 = {I 1 } m I k m (a 1 a k ) < 0 k=2 h 1 x h = {I h } m I k m (a h a k ) > 0 k=1 n, 1 < n < h x n 1 < 0 < x n 17 α j B > 0 (B) ρ ρ(b) B < (B) φ φ(b) B, j = 1,..., n 1 α k B < 0, k = n,..., h 17 j x j 0 26

28 [1] (2002),No.37,pp [2] (2001),42,pp [3] (2002b) - -,28,pp [4] (2004),167,pp.1-17 [5] (2004) Discussion Paper,No.408,pp.1-19 [6] (2005) [7] (2006) -,, 3 [8] (2008),, 6 [9] (2005),pp [10] (2007) 7,pp [11] (2010) Discussion Paper,No

29 [12] (1987) 6,pp [13] (1988),pp.1-15 [14] (2001),37,pp [15] (1999) [16] Auerbach, A. and Kotlikoff, L.J.(1987), Dynamic Fiscal Policy, Cambridge University Press. [17] Blanchard,O. and Fischer,S.(1989), Lectures on Macroeconomics, MIT Press. [18] Diamond, P.A.(1965), National Debt in a Neoclassical Growth Model, American Economic Review, Vol.55, No.5, pp [19] Heijdra,B.(2009), Foundations of Modern Macroeconomis Second Edition, Oxford University Press. [20] Negishi, T.(1960), Welfare Economics and Existence of an Equilibrium for a Competitive Economy, Metroeconomica, Vol.12, No.2-3, pp [21] 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

Auerbach and Kotlikoff(1987) (1987) (1988) 4 (2004) 5 Diamond(1965) Auerbach and Kotlikoff(1987) 1 ( ) ,

,, 2010 8 24 2010 9 14 A B C A (B Negishi(1960) (C) ( 22 3 27 ) E-mail:[email protected] E-mail:[email protected] E-mail:[email protected] 1 1 1 2 3 Auerbach and Kotlikoff(1987) (1987)